BAER STRUCTURES, UNITALS

AND

ASSOCIATED FINITE GEOMETRIES

by

Catherine Thérèse Quinn, B.Sc.(Ma. & Comp. Sc.) (Hons.) (Adelaide)

Thesis submitted for the degree of

Doctor of Philosophy at the University of Adelaide

Department of Pure Mathematics July, 1997

Contents

Abstract 5

Signed Statement 7

Acknowledgements 8

1 Preliminary Results 9 1.1 Incidence Structures and Designs . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Projective, Affine and Translation Planes . . . . . . . . . . . . . . . . . . 10 1.3 Projective Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 Quadrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5 Arcs, Curves and Normal Rational Curves in PG(n, q) . . . . . . . . . . 18 1.6 Varieties and Plane Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.7 The ruled cubic surface V_{2}^{3} . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.8 Segre varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.9 Spreads and Reguli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.10 The Bruck and Bose representation . . . . . . . . . . . . . . . . . . . . . 33

1.10.1 The construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.10.2 Some Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.10.3 A regular spread for Σ_{∞} = PG(2h − 1, q) . . . . . . . . . . . . . 35 1.10.4 Coordinates for the projective plane Π = PG(2, q^{h}) . . . . . . . . 39

1.11 Plane {k; n}−arcs and sets of type (m, n) . . . . . . . . . . . . . . . . . 42 1.12 Caps, Ovoids and Spreads of PG(3, q) . . . . . . . . . . . . . . . . . . . 45 1.13 Unitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

1.13.1 The Classical Unitals . . . . . . . . . . . . . . . . . . . . . . . . . 46 1.13.2 Unitals embedded in Finite Projective planes . . . . . . . . . . . . 48 1.13.3 Buekenhout-Metz Unitals . . . . . . . . . . . . . . . . . . . . . . 50

1.14 Inversive Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 1.15 Maximal Arcs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 1.16 Generalized Quadrangles . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2 The Bruck and Bose representation in PG(4, q) 63 2.1 Bruck-Bose in PG(4, q) . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.2 Known Bruck-Bose representations of Baer substructures . . . . . . . . . 65 2.3 The Bruck-Bose representation of Conics in Baer subplanes of PG(2, q^{2}) 69 2.4 A characterisation of Baer ruled cubic surfaces . . . . . . . . . . . . . . . 81

2.4.1 The extended ruled cubic surface . . . . . . . . . . . . . . . . . . 82 2.5 Additional properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 2.6 An Alternative Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 2.7 A look at PG(2, q^{4}) in Bruck-Bose . . . . . . . . . . . . . . . . . . . . . 102

3 The Bruck and Bose representation in PG(n, q), n > 4 108 3.1 Some properties concerning (h − 1)−spreads of PG(2h − 1, q) . . . . . . 108 3.2 The Bose representation of PG(2, q^{2}) in PG(5, q) . . . . . . . . . . . . 117 3.3 The Bose representation of Conics in PG(2, q^{2}) . . . . . . . . . . . . . . 128 3.4 The Bruck-Bose representation of PG(2, q^{4}) in PG(8, q) revisited …. 133

4 Baer subplanes and Buekenhout-Metz Unitals 136

4.1 The intersection of a Baer subplane and a Buekenhout-Metz unital in

PG(2, q^{2}) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

4.2 Examples and quartic curves . . . . . . . . . . . . . . . . . . . . . . . . . 148

4.3 Concerning Classical Unitals in PG(2, q^{2}) . . . . . . . . . . . . . . . . . 152

5 A characterisation of Buekenhout-Metz unitals 154 5.1 Proof of theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6 Maximal Arcs, Inversive planes and T_{3}(O) 158 6.1 Maximal arcs and Thas maximal arcs . . . . . . . . . . . . . . . . . . . . 158 6.2 Excursion into T_{3}(O) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.3 Thas maximal arcs and Inversive planes . . . . . . . . . . . . . . . . . . 161 6.4 A characterisation of Thas maximal arcs . . . . . . . . . . . . . . . . . . 164

6.4.1 A sequence of lemmata . . . . . . . . . . . . . . . . . . . . . . . . 164

Bibliography 172

Abstract

In this thesis we study the representation of finite translation planes in projective spaces introduced by André [1]. This theory was also developed by Bruck and Bose [21, 22] in a distinct but equivalent form. Throughout this thesis we refer to this representation as the Bruck and Bose representation or simply Bruck-Bose. Of particular importance is the representation of Baer subplanes of translation planes π_{q^{2}} of order q^{2}; the importance is due to the crucial role Baer subplanes have in the characterisation of various substructures, including unitals and maximal arcs, of projective planes, as will be evident in the text. In Chapter 1 we present the necessary preliminary material required for the later chapters. In particular we present in detail the Bruck and Bose representation [21, 22] of the Desarguesian plane P G(2, q^{h}) and the associated coordinatisation. In Chapter 2 we begin by reviewing the known results concerning the representation of Baer subplanes of P G(2, q^{2}) in the Bruck and Bose representation in P G(4, q). We provide a new proof of the result of Vincenti [90] and Bose, Freeman and Glynn [19], that the non-affine Baer subplanes of P G(2, q^{2}) are represented in Bruck-Bose by certain ruled cubic surfaces in P G(4, q) which we term Baer ruled cubic surfaces. We characterise Baer ruled cubic surfaces in P G(4, q) for a general fixed Bruck and Bose representation of P G(2, q^{2}) in P G(4, q). We determine that non-degenerate conics in Baer subplanes of P G(2, q^{2}) are represented in Bruck-Bose by normal rational curves; a normal rational curve which arises in this way is of order 2, 3 or 4 and is therefore properly contained in a plane, hyperplane or no hyperplane of P G(4, q) respectively. We apply these results to prove the existence of certain (q^{2} + 1)−caps in P G(4, q) which are not contained in any hyperplane of P G(4, q) and which contain many normal rational curves of order 4. Further properties of these caps are determined in Chapter 3. We also include a discussion of the ruled cubic surface obtained as the projection from a point P of the Veronese Surface in P G(5, q) onto a hyperplane not containing P; in this setting we determine some alternative proofs for our results and prove some extensions. In Chapter 3 we investigate the Bruck and Bose representation in P G(n, q) with n > 4. We prove various results concerning the regular (h − 1)−spreads of P G(2h − 1, q) which determine the Bruck and Bose representation of P G(2, q^{h}) in P G(2h, q), treating the case h = 4 in greater detail. In particular, we prove the existence of induced spreads and show how the induced spreads are closely related to the Bruck and Bose representation of the Baer substructures of P G(2, q^{h}). To obtain further properties of the higher dimensional Bruck-Bose representation of the non-affine Baer substructures of the Desarguesian plane, we make use of the Bose representation [18] of P G(2, q^{2}). In this chapter, we also prove results concerning the Bruck and Bose representation of nondegenerate conics in P G(2, q^{2}) and we discuss the relationship between these results and the Bruck-Bose representation of non-affine Baer sublines of P G(2, q^{4}) in P G(8, q). In Chapter 4 we investigate Baer subplanes and Buekenhout-Metz unitals in P G(2, q^{2}). In particular we improve the known results by showing that in P G(2, q^{2}), with q > 13, a Baer subplane and a Buekenhout-Metz unital with elliptic quadric as base have at least 1 point and at most 2q +1 distinct points in their intersection. Our method of proof makes use of the Bruck and Bose representation of P G(2, q^{2}) in P G(4, q) and the properties of a certain irreducible sextic curve in P G(4, q). We also prove that the non-classical Buekenhout-Metz unitals, with an elliptic quadric base, in P G(2, q^{2}) are inherited from the classical unitals in P G(2, q^{2}) by a certain procedure of swapping regular 1−spreads of P G(3, q) in the Bruck and Bose representation of P G(2, q^{2}). In Chapter 5 we prove that a unital in P G(2, q^{2}) is a Buekenhout-Metz unital if and only if there exists a point T of the unital such that each secant line of the unital through T intersects the unital in a Baer subline. This is an improvement of the characterisation of Lefèvre-Percsy [56] and an improvement of the characterisation of Casse, O’Keefe and Penttila [26] for the cases q > 3. In the final chapter we investigate the relationships between Thas maximal arcs, the generalized quadrangle T_{3}(O) and egglike inversive planes. This work was motivated by the approach of Barwick and O’Keefe [13] in investigating the relationship between Buekenhout-Metz unitals and inversive planes (see also [6, Section 5.] and [92]). We attempt to characterise the Thas maximal arcs in those translation planes where they exist using two configurational properties; we do not succeed in this, but prove a characterisation of Thas maximal arcs in P G(2, q^{2}) for certain values of q.

Statement

This work contains no material which has been accepted for the award of any other degree or diploma in any university or other tertiary institution and, to the best of my knowledge and belief, contains no material previously published or written by another person, except where due reference has been made in the text. I give consent to this copy of my thesis, when deposited in the University Library, being available for loan and photocopying.

................................................ Catherine Thérèse Quinn

Acknowledgements

I wish to thank my supervisor Assoc Prof Rey Casse for his guidance, encouragement and for many motivating discussions. I also wish to thank Dr S.G. Barwick for many helpful discussions. I am grateful for the financial support provided by an Australian Postgraduate Research Award (APRA) in the first year of my candidature. Thanks to Lino for his constant support and also to my family.

Chapter 1

Preliminary Results

In this chapter we collect together the main definitions, known results and constructions we require for our original work presented in later chapters.

1.1 Incidence Structures and Designs

In this section we follow Hughes and Piper [53]. An incidence structure S = (V, B, I) is two sets V and B called varieties (or points) and blocks (or lines) respectively, with an incidence relation I ⊆ V × B; a point P is incident with a block ℓ if and only if (P, ℓ) ∈ I. An incidence structure S is finite if the sets V and B are both finite. From now on our incidence structures are finite incidence structures. Given any block in an incidence structure S, there is a set of points incident with it and it will be convenient to identify the block with this pointset. An incidence structure has repeated blocks if there exist two blocks identified with the same pointset. If a point P is incident with a block ℓ then we shall write P Iℓ or P ∈ ℓ and we shall use the expressions “P is on ℓ”, “ℓ contains P ”, “ℓ passes through P ” and similar convenient expressions. A t − (v, k, λ) design is an incidence structure with exactly v points, no repeated blocks, each block is incident with exactly k distinct points and each subset of t distinct points is incident with exactly λ common blocks. A t − (v, k, λ) design has parameters v, k, b, r, t, λ where, b equals the number of blocks, r equals the number of blocks incident with a point and v, k, t, λ are defined as above. If S is a t − (v, k, λ) design, then for any integer s satisfying 0 ≤ s < t, there are exactly λ_{s} blocks of S which are incident with any given subset of s distinct points of S, where

λ_{s} = λ\begin{matrix}( & {v−s \atop t−s} & ) \\ ( & {k−s \atop t−s} & ) \end{matrix}.

Moreover we have the following identities for the parameters of S:

1. b = λ\begin{matrix}( & {v \atop t} & ) \\ ( & {k \atop t} & ) \end{matrix};

2. if t > 0 then bk = vr;

3. if t > 1 then r(k − 1) = λ_{2}(v − 1).

For an incidence structure S(V, B, I) and P a point of S, we define the internal structure, S_{P}, of S at P to be the set of all blocks of S which contain P and the set of all points of S, except P, which lie on at least one of those blocks and the incidence in S_{P} is inherited from the incidence in S. In particular if S = (V, B, I) is a t−(v, k, λ) design, then for any point P of S the internal structure S_{P} of S at P is a (t − 1) − (v − 1, k − 1, λ) design with parameters

\begin{aligned}v′ & = & v − 1 \\ k′ & = & k − 1 \\ t′ & = & t − 1 \\ λ′ & = & λ \\ b′ & = & r \\ r′ & = & λ_{2} \end{aligned}

where v, k, t, λ, b, r are the parameters of S.

1.2 Projective, Affine and Translation Planes

In this section we briefly present some familiar results from [52]; for further detail concerning the material in this section consult [52]. A projective plane is a set of points and lines together with an incidence relation between the points and lines such that,

(i) Any two distinct points are incident with a unique line.

(ii) Any two distinct lines are incident with a unique point.

(iii) There exist four points such that no three are incident with one line.

If one line of a projective plane contains only a finite number of points, then every line in the plane contains a finite number of points. A projective plane with this property is called finite. We shall consider only finite projective planes in this thesis. If π is a projective plane, then a set π^{d} of points and lines together with an incidence relation such that the points (lines) of π^{d} are the lines (points) of π and two elements of π^{d} are incident in π^{d} if and only if they are incident in π, is a projective plane. π^{d} is called the dual plane of π. Projective planes satisfy the Principle of Duality: Let A be any theorem about projective planes. If A^{∗} is the statement obtained by interchanging the words points and lines, then A^{∗} is a theorem about dual planes. Hence A^{∗} is a theorem about projective planes (see [52, Theorem 3.2]). Let π be a finite projective plane, then there exists a positive integer n ≥ 2 such that each line of π is incident with exactly n + 1 points and each point of π is incident with exactly n + 1 lines. π contains exactly n^{2} + n + 1 points and n^{2} + n + 1 lines. The integer n is then called the order of π. All known finite projective planes have prime power order (see [52, Section III.2]). For a fixed line ℓ_{∞} of π, denote by aff(π) the set of points and lines of π obtained by deleting ℓ_{∞} and all its points; the incidence in aff(π) is inherited from π. We write aff(π) = π\ℓ_{∞} or aff(π) = π^{ℓ_{∞}}. Then aff(π) is an affine plane of order n and we call ℓ_{∞} the line at infinity of aff(π) and call the points on ℓ_{∞} the points at infinity. Two lines in aff(π) are parallel if they do not intersect in aff(π); parallelism in aff(π) is an equivalence relation and in this way each point at infinity in π corresponds to a unique parallel class of lines in aff(π). A collineation of a finite projective plane π is a bijection from points to points and lines to lines; a collineation preserves collinearity. An elation with axis ℓ_{∞} and centre X is a collineation of π which fixes all points of a line ℓ_{∞} of π and fixes all lines through a point X ∈ ℓ_{∞}. If the group of elations with axis ℓ_{∞} in π is transitive on the points of π not incident with ℓ_{∞}, then the finite projective plane π is a translation plane with translation line ℓ_{∞}. In this case the affine plane aff(π) = π\ℓ_{∞} is also referred to as a translation plane, however the context in which these terms are used in the text should make the meaning clear. (See [52, Chapter IV, section 5.] or [33, Section 3.1.22] for further details). Throughout this thesis we call the translation line ℓ_{∞} of a translation plane π the line at infinity. A finite projective plane π is called Desarguesian if it satisfies a certain configurational property for any choice of point V and line ℓ in π. A theorem of Baer, [52, Theorem 4.29], relates this configurational property of the plane to a collineation group property of the plane. Moreover, Baer’s result states that a projective plane π is Desarguesian if and only if π is a translation plane with respect to a line ℓ for every choice of line ℓ in π. Consequently, for a Desarguesian projective plane π and for any line ℓ_{∞} in π, the affine plane aff(π) = π\ℓ_{∞} is a translation plane. For q a prime power, the Galois field plane P G(2, q) is the unique Desarguesian finite projective plane of order q (see [33, Section 1.4]). A subplane π_{m} of a finite projective plane π_{n} of order n is a subset of the elements of π_{n} which form a projective plane having the same incidence as π_{n}. A subplane π_{m} of π_{n} is called proper if π_{m} ≠ π_{n}. Bruck’s theorem ([52, Theorem 3.7]) states that if a finite projective plane π_{n} of order n contains a proper subplane π_{m} of order m, then either n = m^{2} or n ≥ m^{2} +m. If n = m^{2} then the subplane π_{m} of order m is called a Baer subplane of the finite projective plane π_{m^{2}} of order n = m^{2}. In this case each point in π_{m^{2}}\π_{m} is incident with a unique line of π_{m} and each line ℓ of π_{m^{2}} is either a line of π_{m} (and so intersects π_{m} in m + 1 < n + 1 points) or ℓ intersects π_{m} in a unique point. If a line ℓ intersects a Baer subplane B of π_{m^{2}} in m + 1 points, then the intersection ℓ ∩ B is called a Baer subline (of ℓ) in B. Much of this thesis is devoted to examining Baer subplanes and utilising the properties of Baer subplanes; particularly in the case of the Desarguesian projective plane P G(2, q^{2}) of square order q^{2}. We include the following results for later reference.

Theorem 1.2.1 [29] [33, Result 3.2.17] In P G(2, q^{2}), a quadrangle (four distinct points no three collinear) is contained in a unique Baer subplane of P G(2, q^{2}). □

It follows from Theorem 1.2.1 that for a line ℓ in P G(2, q^{2}), any three distinct points of ℓ are contained in a unique Baer subline of ℓ. Moreover, by Theorem 1.2.1, the Desarguesian plane P G(2, q^{2}) contains exactly q^{3}(q^{3} +1)(q^{2} +1) distinct Baer subplanes. In the following characterisation of Baer subplanes of P G(2, q^{2}), a blocking k−set in P G(2, q^{2}) is a subset of k points of P G(2, q^{2}) which meets every line but contains no line completely (see [48, Section 13.1]).

Theorem 1.2.2 [48, Theorem 13.2.2] In P G(2, q^{2}), if B is a blocking (q^{2} + q + 1)−set, then B is a Baer subplane of P G(2, q^{2}). □

1.3 Projective Spaces

We briefly present some results of [48, Chapter 2] to establish terminology and notation. Let V = GF (q^{n+1}) be the (n + 1)−dimensional vector space over GF (q) with origin 0. With respect to some basis of V, the elements of V are of the form X = (x_{0}, x_{1}, …, x_{n}), where x_{i} ∈ GF (q). For X = (x_{0}, x_{1}, …, x_{n}), Y = (y_{0}, y_{1}, …, y_{n}) in V, the relation

X ≡ Y if and only if x_{i} = λy_{i}, for all i = 0, 1, …, n
for some λ ∈ GF (q)\{0}

is an equivalence relation on the vectors in V \{0} with equivalence classes the onedimensional subspaces of V with the origin deleted. The set of equivalence classes is an n−dimensional projective space over GF(q) and is denoted by PG(n, q). For each X ∈ V \{0}, the equivalence class containing X is a point in P G(n, q). Consequently, the number θ(n) of points in P G(n, q) equals

|P G(n, q)| = θ(n) = \frac{q^{n+1} − 1}{q − 1} = q^{n} + q^{n−1} + … + q + 1.

It will be convenient to take the standard basis for V over GF (q) and to use X = (x_{0}, x_{1}, …, x_{n}) to denote the point of P G(n, q) which contains X ∈ V; we write X is a point of P G(n, q) with homogeneous coordinates X = (x_{0}, x_{1}, …, x_{n}) to mean that the coordinates λ(x_{0}, x_{1}, …, x_{n}), λ ∈ GF (q)\{0}, represent the same point X in P G(n, q). A subspace of dimension m, or m−space, of P G(n, q) is a set Π_{m} of points all of whose coordinates form (together with the origin) a subspace of dimension m + 1 of V. For 0 < m ≤ n, the number ϕ(m; n, q) of m−spaces of P G(n, q) is given by,

ϕ(m; n, q) = \frac{(q^{n+1} − 1)(q^{n} − 1) … (q^{n−m+1} − 1)}{(q^{m+1} − 1)(q^{m} − 1) … (q − 1)}.

An (n − 1)−space of P G(n, q) is called a hyperplane (or prime); the set of points X = (x_{0}, x_{1}, …, x_{n}) in P G(n, q) in a hyperplane Σ_{n−1} of P G(n, q) satisfy an equation

a_{0}x_{0} + a_{1}x_{1} + … + a_{n}x_{n} = 0 where the coefficients a_{i} ∈ GF (q) are not all zero. We shall call [a_{0}, a_{1}, …, a_{n}] the (hyperlane) coordinates of the hyperplane Σ_{n−1}. If Π_{r} and Π_{s} are subspaces of P G(n, q) of dimension r and s respectively, then:

1. the intersection of Π_{r} and Π_{s} is written Π_{r} ∩ Π_{s};

2. the join or span of Π_{r} and Π_{s} is written Π_{r}Π_{s} or ⟨Π_{r}, Π_{s}⟩ and is the smallest subspace of P G(n, q) which contains Π_{r} and Π_{s};

3. Dimension Theorem (Grassman’s Identity) Let dimΠ denote the dimension of a subspace Π of P G(n, q), then

dimΠ_{r} + dimΠ_{s} = dim(Π_{r} ∩ Π_{s}) + dim⟨Π_{r}, Π_{s}⟩.

Note that distinct subspaces Π_{r} and Π_{s} are disjoint in P G(n, q) if and only if as subspaces of V they intersect in the origin; by the definition of dimension of subspaces of P G(n, q) we have Π_{r} ∩ Π_{s} = ∅ implies dim(Π_{r} ∩ Π_{s}) = −1. If S and S′ are two subspaces in P G(n, q) then a collineation σ : S → S′ is a bijection which preserves incidence; that is, if Π_{r} ⊂ Π_{s} then Π_{r}^{σ} ⊂ Π_{s}^{σ}. A projectivity σ : S → S′ is a bijection given by an (n + 1) × (n + 1) matrix H ∈ P GL(n, q): for X, Y ∈ P G(n, q), if Y = X^{σ} then the corresponding (column) homogeneous coordinates satisfy λY = HX, for some λ ∈ GF (q)\{0}. The matrix H is non-singular. With respect to a fixed basis of V over GF (q), an automorphism ϕ of GF (q) induces an automorphism ϕ of P G(n, q); this collineation is given by X^{ϕ} = (x_{0}^{ϕ}, x_{1}^{ϕ}, …, x_{n}^{ϕ}) for each point X in P G(n, q). In particular, in P G(n, q^{2}) the automorphism,

\begin{matrix}P G(n, q^{2}) & ⟶ & P G(n, q^{2}) \\ X = (x_{0}, x_{1}, …, x_{n}) & ↦ & \overline{X} = X^{q} = (x_{0}^{q}, x_{1}^{q}, …, x_{n}^{q}) \end{matrix}

is called the Fröbenius automorphism. For each subspace Π_{r} of P G(n, q^{2}) we call \overline{Π}_{r}, the image of Π_{r} under the Fröbenius automorphism, the conjugate space of Π_{r} with respect to the extension GF (q^{2}) of GF (q). If Σ_{n−1} is any hyperplane in P G(n, q), then AG(n, q) = P G(n, q)\Σ_{n−1} is an n−dimensional affine space over GF(q). The subspaces of AG(n, q) are the subspaces of P G(n, q) with the points of Σ_{n−1} deleted. If the affine space AG(n, q) is obtained from P G(n, q) in this way, for a fixed hyperplane Σ_{n−1} of P G(n, q), we call Σ_{n−1} the hyperplane at infinity. For any projective space P G(n, q) there is a dual space P G(n, q)^{d} whose points and hyperplanes are respectively the hyperplanes and points of P G(n, q); P G(n, q)^{d} is an n−dimensional projective space over GF (q), that is, P G(n, q)^{d} is isomorphic to P G(n, q). The projective space P G(n, q) satisfies the Principle of Duality: For any theorem true in P G(n, q), there is an equivalent theorem true in P G(n, q)^{d}; in particular, if T is a theorem in P G(n, q) stated in terms of points, hyperplanes and incidence the same theorem is true in P G(n, q)^{d} and gives a dual theorem T^{d} in P G(n, q) by interchanging point and hyperplane whenever they occur. Thus join and meet are dual. Hence the dual of an m−space in P G(n, q) is an (n − m − 1)−space (see [48, Section 2.1]). We now present results concerning subgeometries of P G(n, q) which generalise the properties of subplanes of finite projective planes. Since GF (q) is a subfield of GF (q^{k}) for k > 1 a positive integer, the projective space P G(n, q) is naturally embedded in P G(n, q^{k}) once the coordinate system is fixed. Any P G(n, q) embedded in P G(n, q^{k}) is a subgeometry of P G(n, q^{k}). We are particularly interested in the case k = 2 and any P G(n, q) embedded in P G(n, q^{2}) is called a Baer subgeometry of P G(n, q^{2}). Once the coordinate system is fixed, P G(n, q) is called the Baer n−space or real Baer n−space of P G(n, q^{2}). As mentioned above, the Fröbenius automorphism in P G(n, q^{2}) fixes P G(n, q) pointwise. A Baer subgeometry P G(n, q) of P G(n, q^{2}) has properties analagous to those of a Baer subplane of a finite projective plane, as follows.

Theorem 1.3.1 [73, Theorems 3.1, 3.2] Let B = P G(n, q) be embedded as a Baer subgeometry of P G(n, q^{2}).

(i) Each point P in P G(n, q^{2})\B is incident with a unique line of B;

(ii) Each hyperplane Π_{n−1} of P G(n, q^{2}) intersects B in either a (n − 1)−space or an (n − 2)−space of B. □

1.4 Quadrics

In this section we follow [50, Sections 22.1, 22.2]. A quadric Q_{n} in P G(n, q) is any set of points (x_{0}, x_{1}, …, x_{n}) ∈ P G(n, q) such that F (x_{0}, x_{1}, …, x_{n}) = 0 for some quadratic form F ∈ GF (q)[x_{0}, x_{1}, …, x_{n}]. We write Q_{n} = V (F) and F has form

F (x_{0}, x_{1}, …, x_{n}) = ∑_{i=0}^{n} a_{i}x_{i}^{2} + ∑_{i<j} a_{ij}x_{i}x_{j}

where the a_{i}, a_{ij} ∈ GF (q) are not all zero. If there is no change in coordinate system which reduces the form F to one in fewer variables, then F is non-degenerate and Q_{n} is non-singular; otherwise F is degenerate and Q_{n} is singular. The projective linear group P GL(n+1, q) acting on all non-singular quadrics in P G(n, q) has one or two orbits according as n is even or odd. For n even, the non-singular quadrics in P G(n, q) are projectively equivalent and are called parabolic. For n odd, a nonsingular quadric Q_{n} in P G(n, q) is either hyperbolic or elliptic. See [50, Section 22.1] for the canonical forms of these quadrics. Let W_{n} = V (F) be a quadric in P G(n, q) with

F (x_{0}, x_{1}, …, x_{n}) = ∑_{i=0}^{n} a_{i}x_{i}^{2} + ∑_{i<j} a_{ij}x_{i}x_{j}.

Define A = [a_{ij}], where a_{ii} = 2a_{i}, a_{ji} = a_{ij} for i < j. Let B = [b_{ij}], where b_{ii} = 0, b_{ji} = −b_{ij} = −a_{ij} for i < j. If q = 2^{h}, then define trace(t)= t + t^{2} + t^{2^{2}} + … + t^{2^{h−1}}, t ∈ GF (q). Let C_{0} = {t ∈ GF (q) | trace(t) = 0} and let C_{1} = {t ∈ GF (q) | trace(t) = 1}. For q odd, C_{0} will denote the non-zero squares in GF (q) and C_{1} will denote the non-squares in GF (q). In the following theorem, for q even, Δ and α are evaluated as rational functions over the set of integers where a_{i}, a_{ij} are treated as indeterminates z_{i}, z_{ij}; then z_{i}, z_{ij} are specialised to a_{i}, a_{ij} to give the result.

Theorem 1.4.1 [50, Theorem 22.2.1]

(i) W_{n} is singular or not according as Δ is zero or not, where

Δ = \{ \begin{matrix}\frac{1}{2}|A|, & n even \\ |A|, & n odd \end{matrix}

(ii) For n odd, the non-singular quadric W_{n} is hyperbolic or elliptic according as α ∈ C_{0} or C_{1}, where

α = \{ \begin{matrix}(−1)^{(n+1)/2}|A|, & q odd \\ \{|B| − (−1)^{(n+1)/2}|A|\}/\{4|B|\}, & q even. \end{matrix}

□

For the quadric W_{n} = V (F) in P G(n, q) a line ℓ is a tangent to W_{n} if ℓ contains a unique point of W_{n}. Let X = (x_{0}, x_{1}, …, x_{n}), Y = (y_{0}, y_{1}, …, y_{n}) ∈ P G(n, q) with X ≠ Y and define

G(X, Y) = F (X + Y) − F (X) − F (Y).

If X is a point of the quadric and Y is not a point of the quadric, then XY is a tangent to W_{n} if and only if G(X, Y) = 0. If X, Y are both points of the quadric, then G(X, Y) = 0 if and only if the line XY lies on the quadric. Moreover for q even, if one of X and Y is not on the quadric, then XY is a tangent if and only if G(X, Y) = 0. An alternative expression for G(Y, X) is given by,

G(Y, X) = ∑ \frac{∂F}{∂x_{i}} (Y)x_{i}.

For q even and n even, if Q_{n} is a non-singular (parabolic) quadric, then Q_{n} has a nucleus, that is, there exists a unique point Y ∉ Q_{n} such that G(Y, X) = 0 for all points X; that is, a point Y for which \frac{∂F}{∂x_{i}} (Y) = 0, for all i. Let Q_{n} be a non-singular quadric in P G(n, q) and let P be a point of the quadric. The set of points X for which G(P, X) = 0 is the tangent hyperplane to Q_{n} at P. The tangent hyperplane at P contains any m−space on P which is contained in Q_{n}. If W_{n} is singular in P G(n, q), then W_{n} is a cone Π_{k}Q_{s}, the join of a vertex k−space Π_{k} to a non-singular quadric base Q_{s} contained in an s−space Π_{s} with Π_{k} ∩ Π_{s} = Π_{−1} and k + s = n − 1. Let P be a point of W_{n}. The set of points X for which G(P, X) = 0 is the tangent space to W_{n} at P. The tangent space at P contains the vertex Π_{k} and if P ∈ Π_{k}, then the tangent space at P is the whole space P G(n, q).

1.5 Arcs, Curves and Normal Rational Curves in

PG(n, q)

In this section, unless stated otherwise, the material can be found in [50, Sections 27.1, 27.5]. A k_{n}−arc in P G(n, q) is a set of k points not contained in a hyperplane with at most n points in any hyperplane of P G(n, q). A rational curve C^{d} of order d in P G(n, q) is the set of points

{P (t_{0}, t_{1}) = P (g_{0}(t_{0}, t_{1}), …, g_{n}(t_{0}, t_{1}))| t_{0}, t_{1} ∈ GF (q) }

where each g_{i} is a binary form of degree d and the highest common factor of the g_{i} is 1. The curve C^{d} may also be written

{P (t) = P (f_{0}(t), …, f_{n}(t))| t ∈ GF (q) ∪ {∞} }

where f_{i}(t) = g_{i}(1, t). As the g_{i} have no non-trivial common factor, at least one of the f_{i} has degree d. Also C^{d} is normal if it is not the projection of a rational curve C′^{d}, of order d, in P G(n + 1, q), where C′^{d} is not contained in a hyperplane. Let C^{d} be a normal rational curve in P G(n, q) not contained in a hyperplane. Then

(i) q ≥ n ;
(ii) d = n ;
(iii) C^{n} is projectively equivalent to {P (t) = P (t^{n}, t^{n−1}, …, t, 1) | t ∈ GF (q) ∪ {∞}}.
(iv) C^{n} consists of q + 1 points no n + 1 in a hyperplane.

Note that if C^{n} is a normal rational curve in P G(n, q), then C^{n} has form

{P (t) = P (f_{0}(t), f_{1}(t), …, f_{n}(t)) | t ∈ GF (q) ∪ {∞} }

where at least one of the polynomials f_{0}, f_{1}, …, f_{n} has degree n. Also, since (f_{0}(t), …, f_{n}(t)) is the image of (t^{n}, t^{n−1}, …, t, 1) under some projectivity H ∈ P GL(n + 1, q), the polynomials f_{0}(t), …, f_{n}(t) are linearly independent. A normal rational curve C^{2} in P G(2, q) is a non-degenerate conic and a normal rational curve C^{3} in P G(3, q) is called a twisted cubic curve.

By property (iv) above, a normal rational curve C^{n} of order n in P G(n, q) is a (q + 1)_{n}−arc. In certain cases the converse result is true; for example, for (q + 1)_{4}−arcs in P G(4, q) we have the following results.

Theorem 1.5.1 [25] In P G(4, q), q = 2^{h}, every (q + 1)_{4}-arc is the pointset of a normal rational curve. □

Theorem 1.5.2 [50, Theorem 27.6.5] In P G(4, q), q odd, q ≤ 7, every (q + 1)_{4}-arc is the pointset of a normal rational curve. □

Theorem 1.5.3 [76] [81] In P G(n, q), q odd, with q > (4n − 23/4)^{2}, every (q + 1)_{n}-arc is the pointset of a normal rational curve. In particular, in P G(4, q), q odd, q > (10.25)^{2}, every (q + 1)_{4}-arc is the pointset of a normal rational curve. □

Note the following result due to Glynn [43].

Theorem 1.5.4 [43] In P G(4, 9), there exists a 10_{4}−arc which is not the pointset of a normal rational curve. □

1.6 Varieties and Plane Curves

In this section we follow Semple and Roth, Introduction to Algebraic Geometry [71]. The setting is P G(n, q).

Definition 1.6.1 A primal or hypersurface is the locus of points V whose coordinates (x_{0}, …, x_{n}) satisfy an equation of the form:

F (x_{0}, …, x_{n}) = ∑_{i_{0}+...+i_{n}=r} ρ_{j}x_{0}^{i_{0}}x_{1}^{i_{1}} … x_{n}^{i_{n}} = 0, j = 1, …, (_{n}^{n+r})

where the ρ_{j} ∈ GF (q) are not all zero and F is a homogeneous polynomial. If F is of degree r, the primal is said to be of order r and is denoted by V = V_{n−1}^{r} . If F is of degree 2, the primal is called a quadric.

There are (_{n}^{n+r}) coefficients in such a polynomial F of degree r, therefore (_{n}^{n+r})−1 points of P G(n, q) (in general position) determine a unique primal V_{n−1}^{r}. Example: In P G(n, q), (_{n}^{n+2}) − 1 = (_{2}^{n+2}) − 1 = \frac{n(n+3)}{2} generic points determine a unique quadric. In other words, a quadric can be made to pass through \frac{n(n+3)}{2} points; the quadric is unique if and only if these points result in \frac{n(n+3)}{2} linearly independent conditions on the coefficients of the defining polynomial F.

Theorem 1.6.2 An arbitrary line meets a V_{n−1}^{r} in r points (some may coincide, some may belong to an extension of the field GF (q)). □

In P G(n, q), n generic primals have, in general, points in common whose coordinates are found by solving their equations simultaneously for x_{0} : x_{1} : …: x_{n}. When the primals are generally situated with respect to each other, we obtain the following result:

Bézout’s Theorem 1.6.3 In P G(n, q), n generic irreducible primals V_{n−1}^{r_{i}} (i = 1, 2, …, n), of orders r_{1}, …, r_{n} respectively, have r_{1}r_{2} … r_{n} common points. □

Here irreducible means the polynomial defining a given primal is irreducible in the field and in any extension of the field. Example: In P G(2, q), two generic conics have 4 points in common; of these four points some may coincide, some may belong to an extension of the field. Similarly, in P G(3, q), three generic quadrics have 8 points in common. Note that when the primals are not in general position to one another the intersection need not be the number of points prescribed by Bézout’s Theorem 1.6.3, for example: Consider the three quadrics x_{0}x_{3}−x_{1}x_{2} = 0, x_{1}^{2}−x_{2}x_{0} = 0 and x_{1}x_{3}−x_{2}^{2} = 0 in P G(3, q); the complete intersection is the set ζ = {(1, θ, θ^{2}, θ^{3}); θ ∈ GF (q) ∪ {∞}} which contains more that 8 points if q > 7. So here the quadrics are not generic, are not in general position with respect to one another. Dimension of a variety

Definition 1.6.4 A point-locus in P G(n, q) is said to be an irreducible algebraic manifold V_{k} of dimension k if its points can be shown to be in algebraic 1-1 correspondence with the points of an irreducible primal M_{k} of a space P G(k + 1, q).

Algebraically this means that if x̃ = (x_{0}, x_{1}, …, x_{n}) is a general point in P G(n, q) and if ỹ = (y_{0}, …, y_{k+1}) is a general point in P G(k + 1, q), then there exists a set of n + 1 polynomials F_{0}, …, F_{n}, homogeneous and of the same degree in y_{0}, …, y_{k+1}, and a further irreducible homogeneous polynomial M(y_{0}, …, y_{k+1}), such that as ỹ describes the primal M_{k} defined by M(y_{0}, …, y_{k+1}) = 0, the point x̃ = (x_{0}, …, x_{n}) given by ρx_{i} = F_{i}(y_{0}, …, y_{k+1}), i = 0, …, n, describes V_{k}; and the correspondence is such that the generic point of V_{k} arises from only one point of M_{k}.

The equations

ρx_{i} = F_{i}(y_{0}, …, y_{k+1}) (i = 0, …, n)  (1.1)
and M(y_{0}, …, y_{k+1}) = 0  (1.2)

are called the parametric equations of V_{k}; the parameters are the k + 1 ratios y_{i} : y_{0} (i = 1, …, k + 1). A variety V_{1} of dimension 1 is called a curve and a variety V_{2} of dimension 2 is called a surface. Example: In P G(3, q) consider the plane M_{2}^{1} defined by M(y_{0}, y_{1}, y_{2}, y_{3}) = y_{3} = 0. Consider:

ρx_{0} = F_{0}(y_{0}, y_{1}, y_{2}, y_{3}) = y_{2}^{2},
ρx_{1} = F_{1}(y_{0}, y_{1}, y_{2}, y_{3}) = y_{0}y_{2},
ρx_{2} = F_{2}(y_{0}, y_{1}, y_{2}, y_{3}) = y_{1}y_{2},
ρx_{3} = F_{3}(y_{0}, y_{1}, y_{2}, y_{3}) = y_{0}y_{1}.

So (y_{0}, y_{1}, y_{2}, y_{3}) ⟶ (y_{2}^{2}, y_{0}y_{2}, y_{1}y_{2}, y_{0}y_{1}) is an algebraic 1-1 correspondence and therefore the quadric with equation x_{1}x_{2}−x_{0}x_{3} = 0 in P G(3, q) has dimension 2 and so is a (primal) variety V_{2}^{2}. Order of a variety An (n − k)−space in P G(n, q) can be represented as the complete intersection of k hyperplanes in general position, that is, the solution set to a system of k linear equations of the form

∑_{i=0}^{n} a_{ij}x_{i} = 0 (j = 1, …, k).  (1.3)

These k equations, combined with equations (1.1), represent k polynomials in y_{0}, y_{1}, …, y_{k+1}. Together with equation (1.2) we have k + 1 polynomials in y_{0}, y_{1}, …, y_{k+1} which represent k + 1 primals in P G(k + 1, q) and by Bézout’s Theorem these k + 1 primals intersect in a given number, r say, of points. We have

Theorem 1.6.5 A generic (n − k)−space in P G(n, q) meets a variety V_{k} of dimension k in a fixed number r of points. We call r the order of the variety. □

We shall write V_{k}^{r} for a variety of dimension k and order r in P G(n, q). Note: From above, V_{n−1}^{r} denotes a primal in P G(n, q). Now n − (n − 1) = 1 and a subspace of dimension 1 is a line and by Theorem 1.6.2, a line intersects V_{n−1}^{r} in r points which is consistent with our definition of order. Example: The points of a h−space in P G(n, q) can be put in 1 − 1 correspondence with those of a h−space in a P G(h + 1, q), with the latter being a primal defined by an equation order 1. Hence any projective subspace of dimension h is a variety of dimension h. By Grassman’s identity (Dimension Theorem), a generic (n − h)−space in P G(n, q) intersects a h−space in a unique point. Thus a h−space of P G(n, q) is a variety of dimension h and order 1 and is denoted by S_{h}^{1}. Intersections of varieties

Theorem 1.6.6 In P G(n, q), the intersection of two varieties V_{k} and V_{h}, of dimension k and h respectively, in general forms a manifold V_{k+h−n} of dimension k + h − n (where k + h ≥ n). □

If two varieties V_{k} and V_{h} intersect properly in a variety V_{k+h−n} of dimension k + h − n the intersection is called normal.

Theorem 1.6.7 (Generalized theorem of Bézout) If two varieties V^{r} and V^{s}, of orders r and s respectively, intersect normally, then the intersection is a variety V^{rs} of order rs. □

Examples:

(1) The intersection of a primal V_{n−1}^{r} and a h−space S_{h}^{1} in P G(n, q) is in general a primal, V_{h−1}^{r} say, of order r in the h−space.

(2) The intersection of a (n−k)−space S_{n−k}^{1} and a variety V_{k}^{r} in P G(n, q) is in general S_{n−k}^{1} ∩ V_{k}^{r} = V_{0}^{r}. In other words a generic (n − k)−space intersects a variety of dimension k in r points, where r is the order of the variety (as promised by our definition of order).

Consider a normal rational curve C_{1}^{n} in P G(n, q) as discussed in Section 1.5. Then by Section 1.5, the pointset of C_{1}^{n} in canonical form is

{P (t) = P (t^{n}, t^{n−1}, …, t, 1) | t ∈ GF (q) ∪ {∞} }.

The points of C_{1}^{n} are in algebraic 1-1 correspondence with the points of the (primal) line in P G(2, q) with points {(1, t, 1) | t ∈ GF (q) ∪ {∞} } and equation M(y_{0}, y_{1}, y_{2}) = y_{0} − y_{2} = 0, for example. Hence the normal rational curve indeed has dimension 1 as a variety. Consider a generic hyperplane, that is an S_{n−1}^{1} in P G(n, q), with equation a_{0}x_{0} + a_{1}x_{1} + … + a_{n}x_{n} = 0, where a_{i} ∈ GF (q) are not all zero. Then S_{h−1}^{1} intersects the curve in precisely n points, namely the points of C_{1}^{n} with parameter t such that t satisfies a_{0}t^{n} + a_{1}t^{n−1} + … + a_{n} = 0; note that some of these points may coincide or belong to an extension of GF (q). Hence, by the definition of order of a variety, the normal rational curve is a variety C_{1}^{n} of dimension 1 and order n. We include some additional results for later reference. For a discussion of genus, the reader is referred to [71], [5] and [48]; note that the genus g of a curve is a nonnegative integer. Here we state a few isolated results which we require in the proof of Theorem 4.1.4 in Chapter 4.

Theorem 1.6.8 [5, Chapter VIII, Part VI]

(i) An irreducible algebraic curve of order 6 in P G(4, q), which is not contained in a hyperplane, has maximum possible genus 2, that is, g ≤ 2.

(ii) An irreducible algebraic curve of order 4 in P G(4, q) which is not contained in a hyperplane has genus g = 0 (in fact this is a normal rational curve in P G(4, q)).

Result 1.6.9 [5, Page 239, Example 8.] Let l be a line in P G(4, q) skew to the plane of a conic C in P G(4, q). Let θ be a projectivity between the points P of the conic C and the points P^{θ} of the line l. The set of points on the lines P P^{θ} is a rational ruled cubic surface. The curve which is the intersection of this surface with a general quadric is a sextic curve, of genus 2.

Theorem 1.6.10 Hasse-Weil Theorem [48, Theorem 10.2.1] [81] [2, Corollary 2.4] If C^{r} is an absolutely irreducible curve of order r and genus g in P G(n, q), n ≥ 2 and if R is the number of points of C^{r} then,

|R − (q + 1)| ≤ 2g\sqrt{q}
□

Finally we discuss plane curves specifically, that is, curves in P G(2, q). From above, in P G(2, q), a plane curve C^{n} of order n is represented by an equation

f(x, y, z) = 0

such that f is a polynomial of degree n, homogeneous in x, y, z. The equation of a general C^{n} may be written in the form

f(x, y, z) ≡ u_{0}z^{n} + u_{1}z^{n−1} + … + u_{n} = 0,

where u_{i} ≡ u_{i}(x, y) is homogeneous of degree i in x and y. A multiple point of order k (or k-fold point, k > 1) of C^{n} is a point P of the curve such that a generic line through P meets the curve in only n−k further points. A k−fold point is a singular point of the curve. If P′(0, 0, 1) is a k-fold point of C^{n}, the equation of C^{n} may be written in the form

z^{n−k}u_{k}(x, y) + z^{n−k−1}u_{k+1}(x, y) + … + u_{n}(x, y) = 0  (1.4)

where u_{k}(x, y) = 0 is the equation of the k tangents of C^{n} at P′(0, 0, 1); note that these tangents are not necessarily distinct or belonging to GF (q).

1.7 The ruled cubic surface V_{2}^{3}

The ruled cubic surface which we denote by V_{2}^{3} is a variety which plays an important role in most of the work in this thesis; this was briefly introduced in Result 1.6.9. In this section we define this surface and briefly summarise the important properties required for later chapters. The following material can be found in Bernasconi and Vincenti [15]. Consider the space P G(4, q) and let C be a non-degenerate conic in a plane S_{2} of P G(4, q); let ℓ be a line of P G(4, q) such that ℓ ∩ S_{2} = ∅. Let ϕ be a projectivity between ℓ and C, that is if we denote the non-homogeneous coordinates of points of C (respectively ℓ) by θ (respectively λ) then for a fixed projectivity ϕ ∈ P GL(2, q) consider the one-to-one correspondence between points P of ℓ with points P^{ϕ} of C given by the relationship θ = ϕ(λ), λ ∈ GF (q)∪{∞}. Note that since ϕ ∈ P GL(2, q), the projectivity is determined by the images of three distinct points of ℓ (see [48, page 119]). Consider the set G of q + 1 lines P P^{ϕ} where the point P varies over ℓ.

Theorem 1.7.1 [15] The set of points incident with the lines of G is a rational ruled variety, of order 3 and dimension 2, of P G(4, q).

We call such a variety a ruled cubic surface and denote it by V_{2}^{3}. The line ℓ shall be referred to as the line directrix of the ruled cubic surface, the conic C as the base conic, the projectivity ϕ as the associated projectivity and the q + 1 lines in G shall be called generators of V_{2}^{3}.

Theorem 1.7.2 [15] Let V_{2}^{3} be a ruled cubic surface in P G(4, q) with line directrix ℓ base conic C and associated projectivity ϕ. The following properties are satisfied by V_{2}^{3}:

1. Any three distinct generators of V_{2}^{3} are not contained in a hyperplane of P G(4, q),

2. [15, Proposition 1.2] For any hyperplane S_{3}^{1} of P G(4, q) the intersection S_{3}^{1} ∩ V_{2}^{3} is a cubic curve C_{1}^{3}; note that the cubic curve C_{1}^{3} may be reducible or may have some component(s) in P G(4, F), where F is a field extension of GF (q),

3. ( The proof of Theorem 2.1 in [15])

(a) In P G(4, q) there exist precisely q^{2} conics on V_{2}^{3} and each such conic is disjoint from the line directrix ℓ,

(b) Each conic on V_{2}^{3} contains a unique point on each generator of V_{2}^{3},
(c) Two distinct conics on V_{2}^{3} intersect in a unique point (of V_{2}^{3}).

Note that since any three distinct generators of V_{2}^{3} are not contained in a hyperplane of P G(4, q) the ruled cubic surface V_{2}^{3} is not contained in a hyperplane of P G(4, q). Finally we note that the ruled cubic surfaces in P G(4, q) are projectively equivalent,

Theorem 1.7.3 [15, Theorem 1.2, Corollary] If V and V′ are two ruled cubic surfaces in P G(4, q), then there exists at least one projectivity Φ of P G(4, q) such that V^{Φ} = V′.

1.8 Segre varieties

In this section we follow Section 25.5 of [50]; we include some proofs to clarify the geometric properties of Segre varieties. In [50, Section 25.5] a Segre variety is defined in terms of k projective spaces P G(n_{1}, q), P G(n_{2}, q), …, P G(n_{k}, q) where n_{i} ≥ 1; here we consider only the special case of a Segre variety defined in terms of two projective spaces P G(n_{1}, q) and P G(n_{2}, q) which according to [50] is the Segre variety most studied. Consider two projective spaces P G(n_{1}, q) and P G(n_{2}, q), with n_{1}, n_{2} ≥ 1. Denote the points of P G(n_{i}, q) by P_{i} = (x_{0}^{(i)}, x_{1}^{(i)}, …, x_{n_{i}}^{(i)}), for i = 1, 2. Let N_{r} = {0, 1, 2, …, r} for any integer r ≥ 1 and let η be a bijection of N_{n_{1}} × N_{n_{2}} onto N_{m}, where m + 1 = (n_{1} + 1)(n_{2} + 1). Then the Segre variety of the two given projective spaces is the variety ρ_{n_{1};n_{2}} with pointset

{P (x_{0}, …, x_{m})|x_{j} = x_{η(i_{1},i_{2})} = x_{i_{1}}^{(1)}x_{i_{2}}^{(2)} with P_{i} = (x_{0}^{(i)}, …, x_{n_{i}}^{(i)}) ∈ P G(n_{i}, q), i = 1, 2}

of P G(m, q) = P G(n_{1}n_{2} + n_{1} + n_{2}, q). Thus, a typical point P (x_{0}, x_{1}, …, x_{m}) of ρ_{n_{1};n_{2}} is determined by a point P_{1}(x_{0}^{(1)}, x_{1}^{(1)}, …, x_{n_{1}}^{(1)}) in P G(n_{1}, q) and a point P_{2}(x_{0}^{(2)}, x_{1}^{(2)}, …, x_{n_{2}}^{(2)}) in P G(n_{2}, q). The m + 1 = (n_{1} + 1)(n_{2} + 1) components x_{0}, x_{1}, …, x_{m} of the coordinates of P are given by x_{j} = x_{η(i_{1},i_{2})} = x_{i_{1}}^{(1)}x_{i_{2}}^{(2)} and therefore the components x_{j} are in one-to-one correspondence with all possible products of the form x_{i_{1}}^{(1)}x_{i_{2}}^{(2)}, where i_{1} ∈ {0, 1, …, n_{1}} and i_{2} ∈ {0, 1, …, n_{2}}. Since P_{i}(x_{0}^{(i)}, x_{1}^{(i)}, …, x_{n_{i}}^{(i)}) ≠ (0, 0, …, 0), i = 1, 2, each P_{i} contains at least one non-zero component; the product of these non-zero components (one from P_{1} and one from P_{2}) therefore occurs as a component of P (x_{0}, x_{1}, …, x_{m}). Hence for any point P in the Segre variety ρ_{n_{1};n_{2}} we have P (x_{0}, x_{1}, …, x_{m}) ≠ (0, 0, …, 0). The integers n_{1}, n_{2} are called the indices of the variety. It can be shown that this Segre variety is absolutely irreducible and non-singular, with order equal to

\frac{(n_{1} + n_{2})!}{n_{1}!n_{2}!}.

Any point P (x_{0}, x_{1}, …, x_{m}) of the Segre variety satisfies the following equations

x_{η(i_{1},i_{2})}x_{η(j_{1},j_{2})} − x_{η(i_{1},j_{2})}x_{η(j_{1},i_{2})} = 0.  (1.5)

Theorem 1.8.1 [50, Theorem 25.5.1] The Segre variety ρ_{n_{1};n_{2}} is the intersection of all quadrics defined by the equations (1.5), and conversely any point of P G(n_{1}n_{2}+n_{1}+n_{2}, q) satisfying the equations (1.5) corresponds to a unique element of P G(n_{1}, q) × P G(n_{2}, q). □

Let

δ : P G(n_{1}, q) × P G(n_{2}, q) ⟶ ρ_{n_{1};n_{2}}

be defined by

(P_{1}(x_{0}^{(1)}, x_{1}^{(1)}, …, x_{n_{1}}^{(1)}), P_{2}(x_{0}^{(2)}, x_{1}^{(2)}, …, x_{n_{2}}^{(2)})) ⟼ P (x_{0}, x_{1}, …, x_{m})
with x_{j} = x_{η(i_{1},i_{2})} = x_{i_{1}}^{(1)}x_{i_{2}}^{(2)}.

By Theorem 1.8.1 the mapping δ is a bijection.

Theorem 1.8.2 [50, Theorem 25.5.2] For a given fixed point P_{1} of P G(n_{1}, q), the set of all points δ(P_{1}, P_{2}) with P_{2} ∈ P G(n_{2}, q), is an n_{2}−dimensional projective space contained in ρ_{n_{1};n_{2}}. Similarly, for a given fixed point P_{2} of P G(n_{2}, q), the set of all points δ(P_{1}, P_{2}) with P_{1} ∈ P G(n_{1}, q), is an n_{1}−dimensional projective space contained in ρ_{n_{1};n_{2}}.

Proof We prove the first statement; the second is proved analogously. P_{1}(x_{0}^{(1)}, x_{1}^{(1)}, …, x_{n_{1}}^{(1)}) ≠ (0, 0, …, 0) is a fixed point of P G(n_{1}, q). For any point P_{2}(x_{0}^{(2)}, x_{1}^{(2)}, …, x_{n_{2}}^{(2)}) in P G(n_{2}, q), the components x_{0}, x_{1}, …, x_{m} of the coordinates of P (x_{0}, x_{1}, …, x_{m}) = δ(P_{1}, P_{2}) are, up to order,

x_{0}^{(1)}x_{0}^{(2)}, x_{0}^{(1)}, x_{1}^{(2)}, …, x_{0}^{(1)}x_{n_{2}}^{(2)},
x_{1}^{(1)}x_{0}^{(2)}, x_{1}^{(1)}, x_{1}^{(2)}, …, x_{1}^{(1)}x_{n_{2}}^{(2)},
⋮
x_{n_{1}}^{(1)}x_{0}^{(2)}, x_{n_{1}}^{(1)}, x_{1}^{(2)}, …, x_{n_{1}}^{(1)}x_{n_{2}}^{(2)},

The set {P (x_{0}, x_{1}, …, x_{m}) = δ(P_{1}, P_{2})} is therefore a n_{2}−dimensional space. □ By Theorem 1.8.2, the Segre variety ρ_{n_{1};n_{2}} has a system Σ_{1} of n_{1}−dimensional subspaces and a system Σ_{2} of n_{2}−dimensional subspaces.

Theorem 1.8.3 [50, Theorem 25.5.3, Theorem 25.5.5] Any two distinct elements of Σ_{i} are skew, for i = 1, 2. Each point of ρ_{n_{1};n_{2}} is contained in exactly one point of Σ_{i}, for i = 1, 2. Each element of Σ_{i} intersects each element of Σ_{j}, i ≠ j, in exactly one point.

Proof Let Π_{n_{1}} ∈ Σ_{1} correspond to the point P_{2} ∈ P G(n_{2}, q), that is Π_{n_{1}} = {δ(P_{1}, P_{2}) | P_{1} ∈ P G(n_{1}, q)}. Similarly, let Πn_{1}^{′} ∈ Σ_{1} correspond to the point

P2^{′} ∈ P G(n_{2}, q) and suppose P_{2} ≠ P2^{′}.

For any points P_{1}, P1^{′} ∈ P G(n_{1}, q), (P_{1}, P_{2}) ≠ (P1^{′}, P2^{′}) and therefore δ(P_{1}, P_{2}) ≠ δ(P1^{′}, P2^{′})

since δ is a bijection. It then follows from Theorem 1.8.2 that Π_{n_{1}} ∩ Πn_{1}^{′} = ∅. Similarly for two distinct elements Π_{n_{2}}, Πn_{2}^{′} ∈ Σ_{2}, Π_{n_{2}} ∩ Πn_{2}^{′} = ∅. Consider the space Π_{n_{1}} ∈ Σ_{1} which corresponds to the point P_{2} ∈ P G(n_{2}, q) and the space Π_{n_{2}} ∈ Σ_{2} which corresponds to the point P_{1} ∈ P G(n_{1}, q), then δ(P_{1}, P_{2}) is the unique point contained in the intersection Π_{n_{1}} ∩ Π_{n_{2}}. □

Corollary 1.8.4 [50, Theorem 25.5.4]

(i) The number of points of ρ_{n_{1};n_{2}} is |ρ_{n_{1};n_{2}}| = θ(n_{1})θ(n_{2}) = |P G(n_{1}, q)| |P G(n_{2}, q)|.

(ii) The number of n_{1}−dimensional subspaces in the system Σ_{1} is |Σ_{1}| = θ(n_{2}) = |P G(n_{2}, q)|. The number of n_{2}−dimensional subspaces in the system Σ_{2} is |Σ_{2}| = θ(n_{1}) = |P G(n_{1}, q)|. □

The main example of a Segre variety which we shall use in this thesis is the Segre variety ρ_{1;n} in P G(2n + 1, q) with n_{1} = 1 and n_{2} = n ≥ 1. The variety ρ_{1;n} has order n + 1 and has |ρ_{1;n}| = (q + 1)θ(n) points. The variety has a system Σ_{1} of |Σ_{1}| = θ(n) lines and a system Σ_{2} of |Σ_{2}| = θ(1) = q + 1 n−spaces. If n = 1, then the variety ρ_{1;1} in P G(3, q) is a hyperbolic quadric. If n = 2, then the variety ρ_{1;2} in P G(5, q) has a system Σ_{1} of q^{2} + q + 1 pairwise disjoint lines and a system Σ_{2} of q + 1 pairwise disjoint planes; in this case each line in Σ_{1} intersects each plane in Σ_{2} in exactly a point. We include a few additional properties of Segre varieties, in particular, some results concerning the existence of Segre subvarieties of a Segre variety.

Theorem 1.8.5 [50, Theorem 25.5.6] No hyperplane of P G(m, q) contains the Segre variety ρ_{n_{1};n_{2}}. □

It is convenient to use the following notation and to choose η as the following bijection in the definition of a Segre variety. The element x_{j} = x_{η(i_{1},i_{2})} will be denoted by x_{i_{1}i_{2}}. Let η(i_{1}, i_{2}) = i_{1}(n_{1} + 1) + i_{2}. The equations (1.5) become

x_{i_{1}i_{2}}x_{j_{1}j_{2}} − x_{j_{1}i_{2}}x_{i_{1}j_{2}} = 0.  (1.6)

Theorem 1.8.6 [50, Theorem 25.5.7] The Segre variety ρ_{n_{1};n_{2}} consists of all points P (x_{00}, x_{01}, …, x_{0n_{2}}, x_{10}, x_{11}, . . . . . . , x_{1n_{2}}, …, x_{n_{1}0}, x_{n_{1}1}, …, x_{n_{1}n_{2}}) of P G(m, q) for which rank[x_{ij}] = 1. □

By Theorem 1.8.6, for example, the Segre variety ρ_{1;2} in P G(5, q) consists of all points (x_{00}, x_{01}, x_{10}, x_{11}, x_{20}, x_{21}) for which

rank [ \begin{matrix}x_{00} & x_{01} \\ x_{10} & x_{11} \\ x_{20} & x_{21} \end{matrix} ] = 1

that is, for which

\begin{aligned}x_{00}x_{11} & − & x_{10}x_{01} & = & 0 \\ x_{00}x_{21} & − & x_{20}x_{01} & = & 0 \\ x_{10}x_{21} & − & x_{20}x_{11} & = & 0. \end{aligned}

A Segre variety ρ_{n_{1};n_{2}} in P G(m, q) is the intersection of quadrics with equations (1.6). Therefore any line ℓ of P G(m, q) intersects ρ_{n_{1};n_{2}} in 0, 1, 2 or q + 1 points. By the following result, any line contained in the Segre variety ρ_{n_{1};n_{2}} must be contained in either an element of Σ_{1} or an element of Σ_{2}.

Theorem 1.8.7 [50, Lemma 25.5.10] Any line ℓ of ρ_{n_{1};n_{2}} is contained in an element of Σ_{1} or Σ_{2}. □

An s−space Π_{s} which is contained in ρ_{n_{1};n_{2}} and such that Π_{s} is contained in no (s + 1)−space Π_{s+1} of ρ_{n_{1};n_{2}}, is called a maximal space or maximal subspace of ρ_{n_{1};n_{2}}. Using Theorem 1.8.7 it can be shown that,

Theorem 1.8.8 [50, Theorem 25.5.11, Corollary 1]

(i) The maximal spaces of the Segre variety ρ_{n_{1};n_{2}} are the elements of Σ_{1} and Σ_{2};

(ii) Each s−space of ρ_{n_{1};n_{2}}, with s > 0, is contained in either a unique element of Σ_{1} or a unique element of Σ_{2}. □

By Theorems 1.8.4 and 1.8.8, it is possible to count the number of subspaces contained in ρ_{n_{1};n_{2}} as follows.

Corollary 1.8.9 [50, Theorem 25.5.11, Corollary 2] Let n_{1} ≤ n_{2}. The number of s−spaces contained in ρ_{n_{1};n_{2}} is

(i) θ(n_{1})ϕ(s; n_{2}, q) + θ(n_{2})ϕ(s; n_{1}, q), for 0 < s ≤ n_{1};

(ii) θ(n_{1})ϕ(s; n_{2}, q), for n_{1} < s ≤ n_{2}. □

Finally, we present the results from [50, Section 25.5] on Segre subvarieties.

Theorem 1.8.10 [50, Theorem 25.5.12] Let P_{i} ∈ P G(n_{i}, q) and let P G(d_{i}, q) be a d_{i}−space of P G(n_{i}, q), i = 1, 2. Then

(i) δ({P_{1}} × P G(d_{2}, q)) is a d_{2}−subspace and δ(P G(d_{1}, q) × {P_{2}}) is a d_{1}−subspace of ρ_{n_{1};n_{2}};

(ii) all subspaces of ρ_{n_{1};n_{2}} are obtained as in (i);

(iii) when d_{i} > 0, i = 1, 2, δ(P G(d_{1}, q) × P G(d_{2}, q)) is a Segre variety ρ_{d_{1};d_{2}} contained in ρ_{n_{1};n_{2}};

(iv) ρ_{d_{1};d_{2}} = ρ_{n_{1};n_{2}}∩P G(m′, q), where m′ = d_{1}d_{2}+d_{1}+d_{2} and P G(m′, q) is the m′−space generated by ρ_{d_{1};d_{2}}; (v) all Segre subvarieties of ρ_{n_{1};n_{2}} are obtained as in (iii). □

Note that by considering the number of subspaces of P G(n_{1}, q) and P G(n_{2}, q), by Theorem 1.8.10 the number of Segre subvarieties of ρ_{n_{1};n_{2}} can be calculated (see [50, Theorem 25.5.12, Corollary 1]).

Theorem 1.8.11 [50, Theorem 25.5.12, Corollary 2] Let Π_{s} be an s−space of ρ_{n_{1};n_{2}}, s ≥ 1, contained in an element Π_{n_{1}} of Σ_{1}. Then the elements of Σ_{2} meeting Π_{s} in a point are the elements of a system of maximal subspaces of a Segre subvariety ρ_{s;n_{2}} of ρ_{n_{1};n_{2}}. □

1.9 Spreads and Reguli

The following definitions and results are found in [48, Chapter 4] and [50, Section 25.6]; for further detail the reader should refer to these texts. A spread S_{r} of r−spaces of P G(n, q) is a set of r−spaces which partitions P G(n, q); that is, every point of P G(n, q) lies in some r−space of S_{r} and every two r−spaces of S_{r} are disjoint. The r−spaces in S_{r} are the elements of S_{r}. A spread of r−spaces in P G(n, q) will also be called an r−spread of P G(n, q). In the case r = 1 and n = 3, a 1−spread of P G(3, q) will sometimes be called a line spread of P G(3, q) or simply a spread of P G(3, q).

Theorem 1.9.1 [48, Theorem 4.1.1] The following are equivalent:

(i) there exists a spread S_{r} of r−spaces of P G(n, q);

(ii) |P G(r, q)| divides |P G(n, q)|;

(iii) (r + 1) divides (n + 1). □

Consider a Segre variety ρ_{1;n} in P G(2n + 1, q). The system of maximal n−spaces of ρ_{1;n} will be called an n−regulus. In the case n = 1, the Segre variety ρ_{1;1} is a hyperbolic quadric in P G(3, q); a 1−regulus is also called a regulus.

Theorem 1.9.2 [50, Theorem 25.6.1, Corollary] If Π_{n}, Πn^{′}, Πn^{′′} are mutually skew n−spaces in P G(2n+1, q), n ≥ 1, then the set of all lines having non-empty intersection with Π_{n}, Πn^{′} and Πn^{′′} is a system of maximal spaces of a Segre variety ρ_{1;n}. Moreover, the n−regulus in ρ_{1;n} which contains Π_{n}, Πn^{′} and Πn^{′′} is the unique n−regulus in P G(2n+1, q) which contains Π_{n}, Πn^{′} and Πn^{′′}. □

For mutually skew n−spaces Π_{n}, Πn^{′}, Πn^{′′} in P G(2n + 1, q) the (unique) n−regulus containing Π_{n}, Πn^{′}, and Πn^{′′} is denoted by R(Π_{n}, Πn^{′}, Πn^{′′}). In this thesis we shall use the following definition of a regular n−spread.

Definition 1.9.3 For q > 2 an n−spread S_{n} of P G(2n+1, q) is called regular if for any three distinct elements Π_{n}, Πn^{′}, Πn^{′′} of S_{n} the whole n−regulus R(Π_{n}, Πn^{′}, Πn^{′′}) is contained in S_{n}.

Theorem 1.9.4 [50, Theorems 25.6.4, 25.6.5] For q > 2 an n−spread S_{n} of P G(2n + 1, q) is regular if and only if the n−spaces of S_{n} meeting any line not in an element of S_{n} form an n−regulus. □

In the case n = 1, the above definition and theorem concerning regular 1−spreads in P G(3, q) are also valid in the case q = 2; by [49, Chapter 17] for q = 2, every 1−spread in P G(3, q) is regular. For n > 1 and q = 2 every n−spread in P G(2n + 1, 2) satisfies the property that for any three distinct elements Π_{n}, Πn^{′}, Πn^{′′} of the spread the whole n−regulus R(Π_{n}, Πn^{′}, Πn^{′′}) is contained in the spread. (See [50, Section 25.6] for more detail on the case n > 1 and q = 2.) The regular n−spreads in P G(2n + 1, q) are projectively equivalent by the following theorem.

Theorem 1.9.5 [50, Theorem 25.6.7] The group P GL(2n+2, q) acts transitively on the set of all regular n−spreads of P G(2n + 1, q). □

Finally we include the well known characterisation by Bruck of regular 1−spreads of P G(3, q).

Theorem 1.9.6 [20, Theorem 5.3] Let P G(3, q) be embedded as a subgeometry of P G(3, q^{2}). Let denote the Fröbenius automorphism of P G(3, q^{2}) which fixes every point in P G(3, q). Let g be any line of P G(3, q^{2}) which contains no point of P G(3, q). For each such line, g, let S_{g} denote the set of all lines of P G(3, q) which meet g. Then, S_{g} = S\overline{g} is a regular spread of P G(3, q). Every regular spread of P G(3, q) can be represented in this manner for a unique pair of lines g, \overline{g}. □

1.10 The Bruck and Bose representation

In this section we present the results of Bruck and Bose ([21] and [22]) which provide us with a representation of translation planes of order q^{h} in the projective space P G(2h, q). In particular, we obtain a representation of the Desarguesian plane P G(2, q^{h}). We also obtain a convenient and natural coordinate system for P G(2, q^{h}) in this Bruck and Bose representation.

1.10.1 The construction

In this section we follow [21, section 4.]. Let S be a (h − 1)−spread of Σ_{∞} = P G(2h − 1, q) and embed Σ_{∞} as a hyperplane in P G(2h, q). Define an incidence structure aff(Π) = (P, B, I) as follows: The points of aff(Π) are the points of P G(2h, q)\Σ_{∞}. The lines of aff(Π) are the h−spaces of P G(2h, q) which intersect Σ_{∞} in a unique element of S. (Note that this implies that each such h−space is not contained in Σ_{∞}.) The incidence relation of aff(Π) is that induced by the incidence relation of P G(2h, q).

Theorem 1.10.1.1 [21, Theorem 4.1 and its Corollary] aff(Π) is an affine plane of order q^{h}. □

The affine plane aff(Π) may be embedded in a projective plane Π by adjoining the spread S to aff(Π) as a line at infinity which we denote by ℓ_{∞}. Each element of S corresponds to a class of parallel lines of aff(Π), thus each element of S is adjoined to Π as a point at infinity.

Hence the corresponding projective plane Π has a perfectly concrete representation in terms of the above construction.

Theorem 1.10.1.2 [21, Theorem 7.1, Corollary] aff(Π) is a translation plane with translation line the line at infinity. Moreover, every finite translation plane is isomorphic to at least one plane aff(Π).

Theorem 1.10.1.3 [22, Theorem 12.1, Corollary] The finite projective plane Π is Desarguesian if and only if the (h − 1)−spread S of Σ_{∞} is a regular spread. □

Finally we note:

Theorem 1.10.1.4 [50, Theorem 25.6.7] The group P GL(2h, q) acts transitively on the set of all regular (h − 1)−spreads of P G(2h − 1, q). □

1.10.2 Some Galois theory

Before we present a coordinatisation for the projective plane Π, we review some well known Galois theory. The following information can be found for example in chapter 7 of A first course in Abstract Algebra, [39], by John B. Fraleigh. Let GF (q) denote the (finite) Galois field of order q, where q = p^{r}, p is prime and r ≥ 1 is an integer. The integral domain of all polynomials in an indeterminate x with coefficients in the field GF (q) is denoted by GF (q)[x]. Let h be a positive integer, h > 1; then there exists a monic polynomial of degree h in GF (q)[x] which is irreducible over GF (q). Denote this polynomial by,

p_{α}(x) = x^{h} − c_{h−1}x^{h−1} − … − c_{1}x − c_{0}

where the c_{i} are in GF (q). There exists an extension field E of GF (q) and an element α ∈ E such that p_{α}(α) = 0. Hence α ∈ E is algebraic over GF (q) of degree h, and the polynomial p_{α}(x) is called the minimal polynomial for α over GF (q). Each element b in GF (q)(α), a simple extension field of GF (q), can be uniquely expressed in the form,

b = b_{0} + b_{1}α + … + b_{h−1}α^{h−1}

where the b_{i} are in GF (q). Thus, GF (q)(α) is a finite field extension of degree h over GF (q) and therefore has q^{h} elements. It follows that GF (q)(α) is isomorphic to the unique finite field with q^{h} elements, GF (q)(α) ≅ GF (q^{h}). We shall identify GF (q)(α) with GF (q^{h}). Consider the field extension GF (q^{h}) = GF (q)(α) of GF (q), which from above is an algebraic extension of degree h; it is also a vector space of dimension h over GF (q) with basis {1, α, α^{2}, …, α^{h−1}} where addition of vectors is the usual addition in GF (q^{h}) and scalar multiplication λb is the usual field multiplication in GF (q^{h}) with λ ∈ GF (q) and b ∈ GF (q^{h}). We shall often identify GF (q^{h}) (as a vector space of dimension h over GF (q)) with the vector space GF (q)^{h}, since we have the following isomorphism of vector spaces,

\begin{aligned}ϕ : & GF (q^{h}) = GF (q)(α) & ⟶ & GF (q)^{h} \\ & b = b_{0} + b_{1}α + … + b_{h−1}α^{h−1} & ⟼ & (b_{0}, b_{1}, …, b_{h−1}) \end{aligned}

where the b_{i} are in GF (q) and {1, α, …, α^{h−1}} is the basis, mentioned above, for GF (q^{h}) as a vector space over GF (q). By the above theory, there exists an element β ∈ GF (q^{2h}), β ∉ GF (q^{h}), such that β is algebraic over GF (q^{h}) and hence GF (q^{2h}) is a vector space of dimension 2 over GF (q^{h}) with basis {1, β}. The field GF (q^{2h}) is also a finite field extension of GF (q) of dimension 2h. Moreover, since {1, α, …, α^{h−1}} is a basis for GF (q^{h}) as a vector space over GF (q), and {1, β} is a basis for GF (q^{2h}) as a vector space over GF (q^{h}), the 2h elements,

{1, α, …, α^{h−1}, β, βα, …, βα^{h−1}}

form a basis for GF (q^{2h}) as a vector space over GF (q).

1.10.3 A regular spread for Σ_{∞} = PG(2h − 1, q)

Our aim is to obtain a convenient coordinate representation of P G(2, q^{h}) in the Bruck-Bose setting with construction Π as given in Section 1.10.1. By Theorems 1.10.1.1 and 1.10.1.3 we require a regular (h − 1)−spread S of Σ_{∞} = P G(2h − 1, q). The following determination of a regular spread S is a special case of the work of Bruck and Bose given in [21, section 5.]. Throughout this section we shall use the results of Section 1.10.2 and the notation introduced there. Represent Σ_{∞} = P G(2h − 1, q) as the (2h)−dimensional vector space GF (q^{2h}) over GF (q); the points of P G(2h − 1, q) corresponding to the 1-dimensional vector subspaces of GF (q^{2h}). By Section 1.10.2 and the notation introduced there, GF (q^{2h}) has basis,

{1, α, …, α^{h−1}, β, βα, …, βα^{h−1}}

as a vector space over GF (q). Let J(∞), J(0), J(1) be three distinct (h − 1)−subspaces of P G(2h − 1, q), chosen so that as vector subspaces of GF (q^{2h}), J(∞) has basis {1, α, …, α^{h−1}}, J(0) has basis {β, βα, …, βα^{h−1}}, and J(1) has basis {1 + β, α + βα, …, α^{h−1} + βα^{h−1}}. Denote by ′ the following linear transformation of J(∞) onto J(0),

′ : a ⟼ a′ = βa

and, consequently, the following linear transformation maps J(∞) onto J(1),

a ⟼ a + a′.

Note that the vector space GF (q^{2h}) is the direct sum of J(∞) and J(0). The three vector subspaces J(∞), J(0), J(1) intersect pairwise in the zero vector and hence, when considered as (h − 1)−dimensional subspaces of P G(2h − 1, q), the three subspaces are pairwise disjoint. Since J(∞) is the h−dimensional vector space GF (q^{h}) over GF (q) with basis {1, α, …, α^{h−1}}, each element a ∈ J(∞) can be uniquely expressed in the form,

a = a_{0} + a_{1}α + … + a_{h−1}α^{h−1}

where the a_{i} are in GF (q). Note that α^{h} is an element of GF (q^{h}) and, by Section 1.10.2,

α^{h} = c_{0} + c_{1}α + … + c_{h−1}α^{h−1}.  (1.7)

since α ∈ GF (q^{h}) has minimal polynomial p_{α}(x) = x^{h} − c_{h−1}x^{h−1} − … − c_{0}, where the c_{i} are in GF (q). Similarly, for each power α^{h+i}, i = 1, …, h − 2, the element α^{h+i} is of course also an element of GF (q^{h}) and therefore can be uniquely expressed as a linear combination of the basis elements {1, α, …, α^{h−1}}. Hence, let

α^{h+i} = g_{i,0} + g_{i,1}α + … + g_{i,h−1}α^{h−1}  (1.8)

where the g_{i,j} are in GF (q). Consider the product ba of two elements b, a ∈ J(∞). We have,

\begin{aligned}b & = & b_{0} + b_{1}α + … + b_{h−1}α^{h−1} \\ a & = & a_{0} + a_{1}α + … + a_{h−1}α^{h−1} \end{aligned}

where b_{i} and a_{i} are elements of GF (q). Therefore ba is given by,

(b_{0} + b_{1}α + … + b_{h−1}α^{h−1})(a_{0} + a_{1}α + … + a_{h−1}α^{h−1})  (1.9)

and by substituting the expressions (1.7) and (1.8) into the product (1.9), we can simplify (1.9) and determine ba as a (unique) linear combination of {1, α, …, α^{h−1}}. Denote this linear combination by,

ba = (b_{0} + b_{1}α + … + b_{h−1}α^{h−1})(a_{0} + a_{1}α + … + a_{h−1}α^{h−1})
= (d_{0} + d_{1}α + … + d_{h−1}α^{h−1})
= d

where the d_{i} are in GF (q) and d ∈ J(∞) = GF (q^{h}). For convenience, we represent each element a ∈ J(∞) as a h−dimensional vector (a_{0}, a_{1}, …, a_{h−1}), where a = a_{0} + a_{1}α + … + a_{h−1}α^{h−1} with the a_{i} ∈ GF (q) as usual. Then for each element b ∈ J(∞), b = b_{0} + b_{1}α + … + b_{h−1}α^{h−1} = (b_{0}, b_{1}, …, b_{h−1}), the product (1.9) is equivalent to a linear transformation of J(∞) defined by a h × h matrix, which we shall denote by B_{b}, with entries in GF (q), as follows,

\begin{matrix}J(∞) & ⟶ & J(∞) \\ a = (a_{0}, a_{1}, …, a_{h−1}) & ⟼ & (a_{0}, a_{1}, …, a_{h−1})B_{b} = (d_{0}, d_{1}, …, d_{h−1}). \end{matrix}

and we use the convention that for a and B_{b} as above, the product aB_{b} is the element d = d_{0} + d_{1}α + … + d_{h−1}α^{h−1} of J(∞) = GF (q^{h}).

For each of these h × h matrices B_{b} over GF (q) defined above, let

J(b) = {aB_{b} + a′ | a ∈ J(∞)}  (1.10)

so that J(b) is a h−dimensional vector subspace of GF (q^{2h}) and so represents a (h − 1)−space in Σ_{∞} = P G(2h − 1, q). Let C denote the collection of the q^{h} matrices B_{b} over GF (q), so that,

C = {B_{b} | b ∈ GF (q^{h})}.

Let S be the collection

{J(∞)} ∪ {J(b) | b ∈ GF (q^{h})}

of q^{h} + 1 (h − 1)−spaces in P G(2h − 1, q). Note that for b = 0 and b = 1 the definition of spaces J(0) and J(1) is consistent with our earlier definition of these spaces. We can also note by (1.9) and the following remarks, that J(0) is defined by the zero matrix B_{0} = 0 in C and J(1) is defined by the identity matrix B_{1} = I in C. We now show that S is a regular (h − 1)−spread of Σ_{∞} = P G(2h − 1, q). First we note that since J(∞) has basis {1, α, …, α^{h−1}} as a vector subspace of GF (q^{2h}) and given the Definition (1.10) of J(b), the subspaces J(∞) and J(b) have only the zero vector in common and hence as (h − 1)−spaces in P G(2h − 1, q) they are disjoint. Consider a matrix B_{b} in C. For any element a ∈ J(∞) the product aB_{b} corresponds to the element ba in J(∞) = GF (q^{h}). Hence aB_{b} = 0, for a ∈ J(∞) and a ≠ 0, if and only if b = 0. It follows that for every non-zero matrix B_{b} in C, B_{b} is non-singular. Moreover we note that for distinct matrices B_{b_{1}}, B_{b_{2}} in C,

B_{b_{1}} − B_{b_{2}} = B_{b_{1}−b_{2}}

is an element of C since b_{1} − b_{2} ∈ GF (q^{h}). Similarly, C is closed under matrix multiplication. In fact (C, +, ·) is isomorphic to the field GF (q^{h}) under the isomorphism B_{b} ↦ b from C to GF (q^{h}). For distinct matrices B_{b_{1}}, B_{b_{2}}, since B_{b_{1}} −B_{b_{2}} is an element of C, by the above discussion B_{b_{1}} −B_{b_{2}} is non-singular. Next suppose that the two vector subspaces J(b_{1}) and J(b_{2}) of GF (q^{2h}), corresponding to the distinct matrices B_{b_{1}}, B_{b_{2}} ∈ C respectively, have a non-zero vector x in common. By Definition (1.10), for some elements a_{1}, a_{2} ∈ J(∞) = GF (q^{h}),

x = a_{1}B_{b_{1}} + a1^{′} = a_{2}B_{b_{2}} + a2^{′}

and by equating coefficients of the basis elements of GF (q^{2h}), we obtain a1^{′} = a2^{′} and therefore a_{1} = a_{2}. Hence we have the equality a_{1}B_{b_{1}} = a_{1}B_{b_{2}} which implies a_{1}(B_{b_{1}} − B_{b_{2}}) = 0. Since B_{b_{1}} − B_{b_{2}} is non-singular we have a_{1} = 0 and so x = 0, a contradiction. Hence S is a collection of q^{h} + 1 pairwise disjoint (h − 1)−spaces in Σ_{∞} = P G(2h − 1, q), that is , S is a (h − 1)−spread of Σ_{∞}. Finally, by [22, Theorem 11.3] and since (C, +, ·) is a field, the spread S is a regular spread of Σ_{∞}. By Theorem 1.10.1.3 and since S is a regular spread, the Bruck-Bose construction Π, of Section 1.10.1 with spread S, is a Desarguesian projective plane of order q^{h}.

1.10.4 Coordinates for the projective plane Π = PG(2, q^{h})

Let Π be a finite projective plane with the construction of Section 1.10.1 with the notation introduced there. Let S be the regular (h − 1)−spread of Σ_{∞} = P G(2h − 1, q) determined in the previous section and with the notation introduced there. By Theorems 1.10.1.1 and 1.10.1.3, Π is the Desarguesian projective plane P G(2, q^{h}) since S is a regular (h − 1)−spread. In this section we use the results of [21, section 6.] to obtain a coordinate system for this Desarguesian projective plane Π determined by S. We shall utilise this coordinatisation in later chapters in examination of varieties, specified by their equations in P G(2, q^{h}), in the Bruck-Bose setting. First we recall a familiar coordinatisation of P G(2, q^{h}). The points of P G(2, q^{h}) have homogeneous coordinates (x, y, z), where x, y, z ∈ GF (q^{h}) and x, y, z are not all equal to zero. Let ℓ_{∞}, the line at infinity, be the line with equation z = 0, or in line coordinates, ℓ_{∞} is the line [0, 0, 1]. Let AG(2, q^{h}) = P G(2, q^{h})\ℓ_{∞} be the affine plane obtained from P G(2, q^{h}) by removing ℓ_{∞} and all of its points. The points of AG(2, q^{h}) have coordinates of the form (x, y, 1) or occasionally for convenience we shall write these affine coordinates in the form (x, y). The lines of AG(2, q^{h}) may be divided into two types:

(i) Lines with equation y = γ or, equivalently, with line coordinates [0, 1, −γ], where γ ∈ GF (q^{h}). These lines constitute a parallel class of lines in AG(2, q^{h}) with point at infinity (1, 0, 0) in P G(2, q^{h}).

(ii) Lines with equation x = by + s or, equivalently, with line coordinates [1, −b, −s], where b, s ∈ GF (q^{h}). For each b ∈ GF (q^{h}) these lines constitute a parallel class of lines in AG(2, q^{h}) with point at infinity (b, 1, 0) in P G(2, q^{h}).

We work in the Bruck-Bose setting to obtain a natural coordinatisation of the incidence structure Π, natural in the sense that the coordinatisation will correspond to the above coordinatisation of the plane P G(2, q^{h}) in a convenient way. We have Σ_{∞} = P G(2h − 1, q) embedded as a hyperplane in the projective space P G(2h, q). We represent P G(2h − 1, q) as a 2h−dimensional vector space GF (q^{2h}) over the field GF (q) with basis,

{1, α, α^{2}, …, α^{h−1}, β, βα, …, βα^{h−1}}.

Embed GF (q^{2h}) as a hyperplane in the (2h + 1)−dimensional vector space GF (q^{2h+1}), and we only need to add a single element e^{∗} say of GF (q^{2h+1}) which is not in GF (q^{2h}) in order to obtain a basis

{1, α, α^{2}, …, α^{h−1}, β, βα, …, βα^{h−1}, e^{∗}}

for GF (q^{2h+1}). The regular (h − 1)-spread S of P G(2h − 1, q) is the collection of q^{h} + 1 h−dimensional vector subspaces of GF (q^{2h}) defined in the previous section, with the notation introduced there,

S = {J(∞)} ∪ {J(b) | b ∈ GF (q^{h})}.

Consider the construction in Section 1.10.1 of the finite Desarguesian projective plane Π. Each affine point of Π is a 1-dimensional vector subspace of GF (q^{2h+1}) not contained in the hyperplane GF (q^{2h}) and so has a unique basis element of the form

x + y′ + e^{∗} or, equivalently, (x_{0}, x_{1}, …, x_{h−1}, y_{0}, y_{1}, …, y_{h−1}, 1)

where y′ ∈ J(0) so that x, y ∈ J(∞) = GF (q^{h}) and have unique representation in the form x = ∑_{i=0}^{h−1} x_{i}α^{i}, y = ∑_{i=0}^{h−1} y_{i}α^{i}, where the x_{i}, y_{i} are in GF (q). (Note that we have used the fact that GF (q^{2h}) is the direct sum of J(∞) and J(0).) Thus we define the coordinates of the affine point of Π with this basis element to be (x, y, 1) for every ordered pair of elements x, y ∈ J(∞) = GF (q^{h}). We have defined,

\begin{aligned}(x, y, 1) & = & \{x + y′ + e^{∗}\} \\ & = & \{(x_{0}, x_{1}, …, x_{h−1}, y_{0}, y_{1}, …, y_{h−1}, 1)\}. \end{aligned}

A line of Π, distinct from the line at infinity, is a (h + 1)−dimensional vector subspace of GF (q^{2h+1}) over GF (q) which intersects GF (q^{2h}) in a unique element J of S and so has the form,

\begin{aligned}⟨J, (x, y, 1)⟩ & = & ⟨J, x + y′ + e^{∗}⟩ \\ & = & ⟨J, (x_{0}, x_{1}, …, x_{h−1}, y_{0}, y_{1}, …, y_{h−1}, 1)⟩ \end{aligned}

provided (x, y, 1) is one of its points. We divide these lines into two types:

(i) Lines with equation y = γ. If γ is in J(∞) = GF (q^{h}), the point (x, y, 1) of Π

lies on the line

⟨J(∞), (0, γ, 1)⟩

if and only if y = γ.

These lines constitute a parallel class of lines in aff(Π) with point at infinity J(∞) in Π.

(ii) Lines with equation x = by + s. If s is in J(∞) = GF (q^{h}) and J(b) is in S, the point (x, y, 1) lies on the line

⟨J(b), (s, 0, 1)⟩ = ⟨{aB_{b} + a′ | a ∈ J(∞)}, s + 0′ + e^{∗}⟩

if and only if (x − s) + y′ is in J(b), that is, if and only if

(x_{0} − s_{0}, x_{1} − s_{1}, …, x_{h−1} − s_{h−1}) = (y_{0}, y_{1}, …, y_{h−1})B_{b}

where s = s_{0} + s_{1}α + … + s_{h−1}α^{h−1}.

For each b ∈ GF (q^{h}) these lines constitute a parallel class of lines in aff(Π) with point at infinity J(b) in Π.

Now if we wish we can consider the line at infinity ℓ_{∞} of Π as being the line with equation z = 0, or in line coordinates the line [0, 0, 1]. Each element of the regular spread S = {J(∞)} ∪ {J(b) | b ∈ GF (q^{h})} is a point on the line at infinity and it is convenient to associate J(b) with the coordinates (b, 1, 0) for all b ∈ GF (q) ∪ {∞}, so that in particular J(∞) is associated with (1, 0, 0).

1.11 Plane {k; n}−arcs and sets of type (m, n)

A {k; n}−arc K in a finite projective plane π_{q}, of order q, is a set of k = |K| points in the plane such that no n + 1 are collinear but some n are collinear. Barlotti introduced this definition of a {k; n}−arc in 1956. If n = 2 we call K a k−arc and in Desarguesian projective planes of odd order the (q + 1)−arcs are characterised in Segre’s Theorem as follows.

Segre’s Theorem 1.11.1 [69] In P G(2, q), q odd, every (q + 1)−arc is a conic. □

For later reference we include:

Theorem 1.11.2 [48, Theorem 12.2.5, Corollary 2] Any {k; 3}−arc in P G(2, q), q > 3, satisfies k ≤ 2q + 1. □

Let K be a {k; n}−arc in the finite projective plane π_{q}, of order q. If a line ℓ contains exactly s points of K we call ℓ an s−secant of K (0-secants are also called external lines and 1-secants are often called tangents of K). Denote by t_{s} (s = 0, …, n) the number of s−secants of K in π_{q}. The following identities are proved in [75].

∑_{s=0}^{n} t_{s} = q^{2} + q + 1  (1.11)
∑_{s=1}^{n} st_{s} = k(q + 1).  (1.12)
∑_{s=2}^{n} s(s − 1)t_{s} = k(k − 1)  (1.13)

K is said to be of type (m_{1}, m_{2}, …, n), with m_{1} < m_{2} < … < n, if t_{m_{1}}, t_{m_{2}}, …, t_{n} are non-zero, that is if every line of π_{q} intersects K in exactly m_{1}, m_{2}, … or n points. Note that the points of a conic in P G(2, q) is a set of type (0, 1, 2) since each line of the plane is external, tangent or secant to the conic.

We now give some results concerning sets of type (m, n) in π_{q} which can be found in [75], [74]. Let K be a set of type (m, n), 0 ≤ m < n ≤ q +1 in π_{q}; denote by k the number of points of K. Using the relationships (1.11), (1.12) and (1.13) we obtain t_{m} + t_{n} = q^{2} + q + 1, mt_{m} + nt_{n} = k(q + 1) and m(m − 1)t_{m} + n(n − 1)t_{n} = k(k − 1). Note that some terms will vanish when m = 0 or m = 1. These equations are easily solved and the parameters t_{m} and t_{n} are given by,

t_{m} = \frac{1}{(n−m)}[n(q^{2} + q + 1) − k(q + 1)]  t_{n} = \frac{1}{(n−m)}[k(q + 1) − m(q^{2} + q + 1)].

Moreover the integer k is found to satisfy

k^{2} − k[(m + n)(q + 1) − q] + mn(q^{2} + q + 1) = 0.

By counting points of K on lines through a point Q ∉ K, respectively a point P ∈ K, we obtain a bound for the cardinality k of a set of type (m, n),

\begin{matrix}for 1 ≤ m, & mq + n ≤ k ≤ (n − 1)q + m; \\ for m = 0, & k = (n − 1)q + n. \end{matrix}

These bounds are best possible in the sense that there exist examples of sets in π_{q}, for some values of q, m, n, where the cardinality k takes the extremal values. Examples will be discussed in the following sections. Let P be a point of K and denote by v_{m}, v_{n} the number m−secants, respectively n−secants, through P. Let Q be a point of π_{q} not in K and denote by u_{m}, u_{n} the number m−secants, respectively n−secants, through Q. Using the relationships,

\begin{aligned}v_{m} + v_{n} & = & q + 1 & u_{m} + u_{n} & = & q + 1 \\ (m − 1)v_{m} + (n − 1)v_{n} & = & k − 1 & mu_{m} + nu_{n} & = & k \end{aligned}

We can determine the parameters as,

\begin{aligned}v_{m} & = & u_{m} − q/(n − m) & u_{m} & = & (n(q + 1) − k)/(n − m) \\ v_{n} & = & u_{n} + q/(n − m) & u_{n} & = & (k − m(q + 1))/(n − m). \end{aligned}

Since these parameters are all integer valued, it follows that a necessary condition for the existence of a set K of type (m, n), 0 ≤ m < n ≤ q + 1, in the plane π_{q} of order q, is that (n − m) divides q.

Given a set K of type (m, n) of k points in a finite projective plane π_{q}, of order q, the following sets are related to K. The complement K̄ of K is the set of points of π_{q} not in K. K̄ has q^{2} + q + 1 − k points and is a set of type (q + 1 − n, q + 1 − m) in π_{q}. In the dual plane π_{q}^{d} of π_{q} the set K^{d_{1}} of m−secants of K constitute a set of type (v_{m}, u_{m}) of t_{m} points; similarly the n−secants of K constitute a set K^{d_{2}} of type (u_{n}, v_{n}) in the dual plane with t_{n} points. The trivial cases of sets of type (m, n) in π_{q} occur when n − m = q, where K is a line or the complement of a line, and when n − m = 1, where K is a point or the complement of a point. The non-trivial cases occur when n − m is a proper divisor of q. The sets of type (0, n) in π_{q} (where n divides q is a necessary condition for existence) are called maximal arcs; such sets necessarily have cardinality (n−1)q +n from above. We will leave the discussion of maximal arcs to a later section. It remains to consider sets of type (m, n), 1 ≤ m < n ≤ q in a finite projective plane π_{q} of order q. We have already determined (n − m) divides q is a necessary condition for existence. Using the parameters derived above of a set of type (m, n) as well as the associated sets in the dual plane and their parameters, Tallini-Scafati proved the following result in [75]:

Theorem 1.11.3 [75] Suppose K is a set of type (1, n), n ≤ q, in a finite projective plane π_{q}, of prime power order, then q is a square and K is either a set of type (1, \sqrt{q}+1) of q\sqrt{q} + 1 points OR a set of type (1, \sqrt{q} + 1) of q + \sqrt{q} + 1 points. □

The two sets in π_{q} identified in this characterisation are called a unital of order \sqrt{q} and a Baer subplane of order \sqrt{q} respectively (unitals will be defined and discussed in more detail in a later section). Tallini improved Tallini-Scafati’s result by removing the condition that the order of the plane must be a prime power.

Theorem 1.11.4 [74] Suppose K is a set of type (1, n), n ≤ q, in a finite projective plane π_{q}, of order q, and \frac{q}{(n − 1)} = p^{h} p prime and h > 0 integer. Then q = p^{2h}, n = \sqrt{q} + 1 and K is either a Baer subplane or a unital of order \sqrt{q}. □

1.12 Caps, Ovoids and Spreads of PG(3, q)

For further detail regarding this section consult [33, 1.4.47 to 1.4.62], [49] and [61]; we restrict our attention to P G(3, q). A k−cap in P G(3, q) is a set of k points no three of which are collinear. An ovaloid in P G(3, q) is a k−cap of maximum size. For a k−cap K in P G(3, q), each line ℓ in P G(3, q) is called an external, tangent or secant line of K according as the intersection ℓ ∩ K contains 0, 1 or 2 points of K. An ovoid is a k−cap in P G(3, q) such the tangent lines at each point form a plane. Moreover an ovoid in P G(3, q) has exactly q^{2} + 1 points. It is known that for q > 2 an ovoid in P G(3, q) is an ovaloid and conversely. Let O be an ovoid in P G(3, q). For each point P ∈ O there exist q + 1 tangent lines to O at P; these tangent lines lie in a plane about P called the tangent plane to O at P. Each plane of P G(3, q) intersects O in either 1 point or in a (q + 1)−arc and is called a tangent plane or secant plane of O respectively. For all values of q, the elliptic quadrics in P G(3, q) form an infinite class of ovoids known as the classical ovoids of P G(3, q). Each secant plane of an elliptic quadric in P G(3, q) intersects the elliptic quadric in q + 1 points of a non-degenerate conic. If q is odd, then every ovoid in P G(3, q) is an elliptic quadric [8]. The only other known class of ovoids in P G(3, q) are the Tits Ovoids which exist in P G(3, 2^{2r+1}), r ≥ 1 an integer. Their construction is given as follows. In P G(3, 2^{2r+1}), r ≥ 1 an integer, consider the automorphism defined by

σ : x ⟶ x^{2^{r+1}}

so that σ^{2} : x ⟶ x^{2^{2r+2}} = x^{2}. Let O_{T} be the set of points

O_{T} = {(1, z, y, x)| z = xy + x^{σ+2} + y^{σ}} ∪ {(0, 1, 0, 0)}

(see [49, Theorem 16.4.5]). For r = 1, O_{T} is the non-classical ovoid in P G(3, 8) discovered by Segre [69] in 1959. For all other r, the ovoid O_{T} was discovered by Tits [88] in 1960. The Tits ovoids are the only known non-classical ovoids of P G(3, q). Moreover for q ≤ 32 the ovoids of P G(3, q) are either classical or of Tits type (see [62], [63], [64]).

For σ the automorphism defined above, consider the following set of lines in P G(3, 2^{2r+1}):

g_{∞} = {(0, s, 0, t)| s, t ∈ GF (2^{2r+1})}
g_{a,b} = {(s, [ab + a^{σ+2} + b^{σ}]s + [a^{σ+1} + b]t, t + as, bs + a^{σ}t)| s, t ∈ GF (2^{2r+1})}

where a, b are any two elements in GF (2^{2r+1}). This set of q^{2} + 1 lines forms a spread of P G(3, 2^{2r+1}) called the Lüneburg Spread [58, Section 23]. It can be proved for q odd that no set of q^{2} + 1 tangents of an elliptic quadric in P G(3, q) can form a spread of P G(3, q); this is a recent result of Blokhuis and Wilbrink, a special case of which is a consequence of the main result of [7].

1.13 Unitals

A unital (or unitary block design) of order n is a 2 − (n^{3} + 1, n + 1, 1) design, for some integer n (see [33, section 2.4.21]). A unital is therefore an incidence structure with v = n^{3} + 1 points, k = n + 1 points on each block, such that any two distinct points are incident with a unique common block. A unital of order n has b = n^{2}(n^{2} − n + 1) blocks and each point is incident with exactly n^{2} blocks. The problem of determining for which values n a unital exists is only partially solved. The known examples of unitals are of order n where either n is a prime power or n = 6 (see [59] and [4]). We now discuss some known examples of unitals.

1.13.1 The Classical Unitals

A polarity in a projective plane π is a one-to-one and onto map α from the points (respectively lines) of π to the lines (respectively points) of π of order 2 and which preserves incidence, that is,

if P I ℓ then ℓ^{α} I P^{α}

for all points P and lines ℓ of π. A point P (respectively a line ℓ) of π is called absolute, with respect to a polarity α, if P is incident with its image P^{α} under α (respectively ℓ^{α} I ℓ).

Let a(α) denote the number of absolute points of a polarity α in P G(2, q). Since α has order 2, for a point P and a line ℓ,

P I ℓ if and only if ℓ^{α} I P^{α}

and if ℓ = P^{α} then

P I P^{α} if and only if ℓ^{α} I ℓ

and therefore the absolute points are in one-to-one correspondence with the absolute lines. It also follows that each absolute line ℓ contains a unique absolute point, namely the point ℓ^{α}, and conversely, each absolute point lies on a unique absolute line. Thus a(α) is also equal to the number of absolute lines of the polarity α in P G(2, q). The polarities of P G(2, q) are classified as follows:

Theorem 1.13.1.1 [48, Section 2.1(v)] A polarity α of P G(2, q), q = p^{h} p prime, is of one of the following types:

Name (also known as)

{GF (q) \atop = GF (p^{h})}

Locus of Absolute points

orthogonal (a) ordinary

p ≠ 2

q+1 points X of a non-degenerate conic with equation X^{t}AX = 0, where A is a symmetric matrix in GL(3, q)

(b) pseudo

p = 2

q + 1 points of a line.

unitary hermitian p arbitrary; h must be even so that q is a square q\sqrt{q} + 1 points of a (hermitian) curve with equation

X^{\sqrt{q}}HX = 0,

where H is a hermitian matrix (that is a matrix satisfying H^{t} = H^{\sqrt{q}} and H non-singular) in GL(3, q)

In P G(2, q^{2}), the Desarguesian projective plane of square order q^{2}, the set \overline{U} of absolute points of a unitary (or hermitian) polarity is a set of q^{3} + 1 points such that each line of the plane intersects \overline{U} in 1 or q + 1 points. The lines are called tangent (absolute) or secant (non-absolute) lines respectively. The structure \overline{U} is a 2 − (q^{3} + 1, q + 1, 1) design, that is, a unital of order q, where the blocks are the sets of q + 1 absolute points on the secant (non-absolute) lines of the polarity and incidence is the natural point-line incidence of P G(2, q^{2}). A unital in P G(2, q^{2}) which arises in this way from a unitary polarity is called a classical unital (or a Hermitian unital). The classical unitals are projectively equivalent under P GL(3, q) (see [48, Theorem 7.3.1]) and therefore up to isomorphism a classical unital has the equation,

x^{q+1} + y^{q+1} + z^{q+1} = 0,

which is the canonical form of a non-singular Hermitian curve where the matrix H is taken as the identity matrix. The classical unitals in P G(2, q^{2}) are also called Hermitian curves. For this reason in the literature a unital in P G(2, q^{2}), which is not necessarily classical, is sometimes called a Hermitian arc. The classical unitals have been characterised in a number of ways, for example:

Theorem 1.13.1.2 [57] [37] In P G(2, q^{2}), q > 2, a unital \overline{U} is classical if and only if each Baer subline in P G(2, q^{2}) intersects \overline{U} in 0,1,2 or q + 1 points . □

1.13.2 Unitals embedded in Finite Projective planes

A unital \overline{U} of order n is said to be embedded in a finite projective plane π_{q}, of order q, if the points of \overline{U} are a subset of the points of π_{q}, each block of \overline{U} is a set of points collinear in π_{q} (with distinct blocks on distinct lines) and incidence in \overline{U} is induced by the point-line incidence in π_{q}. If \overline{U} is embedded in π_{q} we sometimes say \overline{U} is a unital in π_{q}. The classical unitals are examples of unitals of order q (embedded) in the Desarguesian projective plane P G(2, q^{2}). Let \overline{U} be a unital of order s embedded in a finite projective plane π_{q} of order q. The points of \overline{U} are necessarily a set of type (0, 1, s + 1) or of type (1, s + 1) in π_{q}. Suppose \overline{U} is a set of type (1, s + 1) in π_{q} and if either q is a prime power or if q/s is a prime power then by Theorems 1.11.3 and 1.11.4 we have that q is a square and \overline{U} is a unital of order \sqrt{q} in π_{q}.

Consider a classical unital \overline{U} of order q in P G(2, q^{2}). Embed P G(2, q^{2}) as a Baer subplane in P G(2, q^{4}); then \overline{U} as a design has been embedded in P G(2, q^{4}) and the unital is a set of type (0, 1, q + 1) in P G(2, q^{4}). So we have examples of unitals (as designs) which arise naturally as structures in projective planes, but which may have external lines. However if the embedded unital has no external lines then by the Tallini-Scafati and Tallini characterisations in Theorem 1.11.3 and Theorem 1.11.4, with the appropriate condition on the order of the plane, we may restrict our attention to finite projective planes of square order q^{2} and (embedded) unitals of order q. Thus a unital \overline{U} embedded in a finite projective plane π_{q^{2}} of order q^{2} is a set of q^{3} + 1 points of the plane such that each line intersects \overline{U} in exactly 1 or q + 1 points; each line is called a tangent or secant line of \overline{U} respectively. Moreover, each point P ∈ \overline{U} is incident with a unique tangent line and q^{2} secant lines of \overline{U}. By the results of Section 1.11, since a unital \overline{U} in π_{q^{2}} is a (q^{3} + 1)−set of type (1, q + 1), the set of q^{3} + 1 tangent lines of \overline{U} are the points of a unital \overline{U}^{d} in the dual plane π_{q^{2}}^{d} of π_{q^{2}}; the unital \overline{U}^{d} is called the dual unital of \overline{U} in π_{q^{2}}^{d}.

Unitals from unitary polarities

Above we defined the classical unitals in P G(2, q^{2}) as those unitals (embedded in P G(2, q^{2})) which arise as the set of absolute points of a unitary polarity in P G(2, q^{2}). Let π_{q} be a finite projective plane (not necessarily Desarguesian) of order q. Due to the work of Baer [3] and Seib [70] we have the following results (statement taken from Hughes and Piper [52, Theorems 12.7, 12.11, 12.12]) concerning polarities in π_{q},

Theorem 1.13.2.1 Let σ be a polarity of a finite projective plane of order q. If q is not a square, then σ has a(σ) = q + 1 absolute points and

(a) if q is even, the absolute points are collinear
(b) if q is odd, the absolute points form a (q + 1)-arc

If q = s^{2} is a square, then σ has a(σ) ≤ s^{3} + 1 absolute points and if a(σ) = s^{3} + 1 then the set of absolute points and non-absolute lines forms a unital of order s = \sqrt{q}. □

In π_{q} a polarity σ is called orthogonal if a(σ) = q + 1 and unitary if a(σ) = q^{3/2} + 1. By the classification of polarities in the Desarguesian plane P G(2, q), given in Theorem 1.13.1.1, any polarity of P G(2, q) is either orthogonal or unitary. Note that there exist examples of non-Desarguesian planes π_{q} of order q and polarities σ in π_{q} whose number of absolute points satisfy q + 1 < a(σ) < q^{3/2} + 1 (see for example [52, Exercise 12.16]). Theorem 1.13.2.1 indicates one approach at finding new unitals, by finding unitary polarities in non-Desarguesian finite projective planes. See [32], [41], [42], [54], for example, for results concerning unitals constructed in this manner.

1.13.3 Buekenhout-Metz Unitals

The class of unitals known as Buekenhout-Metz unitals are defined in translation planes π_{q^{2}} of order q^{2} with kernel of order q. In this thesis we shall not define the term kernel, but note by [33, 5.1.11], a translation plane π_{q^{2}} of order q^{2} with kernel of order q is a translation plane which has a Bruck and Bose representation in P G(4, q) defined by a 1−spread S in a hyperplane Σ_{∞} = P G(3, q) of P G(4, q). In the Bruck and Bose representation, the line at infinity ℓ_{∞} (with points the elements of the spread S) is the translation line of the translation plane π_{q^{2}}. The construction is as follows: Let O be an ovoid in a hyperplane of P G(4, q)\Σ_{∞} intersecting Σ_{∞} in a unique point X, where the tangent plane to O at X does not contain the unique line t of S incident with X. Let V be a point of t distinct from X. Let \overline{U}^{∗} be the structure containing the spread line t and all points of P G(4, q)\Σ_{∞} on the ovoidal cone with vertex V and base O. The ovoidal cone \overline{U}^{∗} corresponds to a set \overline{U} of q^{3} + 1 points in the translation plane π_{q^{2}} which is defined by the spread S of Σ_{∞}. The set \overline{U} is a unital in π_{q^{2}} tangent to ℓ_{∞} at the point T which is represented by t in Bruck-Bose (see [24, Section 4. (4)]). We shall call a unital \overline{U} in π_{q^{2}} with the above construction a Buekenhout-Metz Unital, and we shall sometimes say \overline{U} is Buekenhout-Metz re (T, ℓ_{∞}). If a Buekenhout-Metz unital \overline{U} is constructed in a translation plane π_{q^{2}} as above, with the ovoid O an elliptic quadric, then we say \overline{U} is Buekenhout-Metz with elliptic quadric as base. (Note that we shall sometimes abbreviate Buekenhout-Metz to B-M.) Buekenhout proved in [24] that each classical unital \overline{U} in the Desarguesian plane P G(2, q^{2}) is Buekenhout-Metz re (T, ℓ_{T}) for any point T ∈ \overline{U} and ℓ_{T} the tangent line to \overline{U} at T. Moreover, Buekenhout showed that every classical unital in P G(2, q^{2}) is Buekenhout-Metz with elliptic quadric as base, that is, corresponds to an elliptic quadric cone in the Bruck-Bose representation of P G(2, q^{2}). Buekenhout constructed the first non-classical unitals in P G(2, 2^{4r+2}), r ≥ 1, by taking O to be a Tits ovoid in Σ_{∞} = P G(3, q), q = 2^{2r+1}, in the above construction. Metz [60] extended this class of non-classical unitals in P G(2, q^{2}) to all values of q > 2 by constructing Buekenhout-Metz unitals with base ovoid an elliptic quadric and such that the unitals did not arise from unitary polarities in P G(2, q^{2}). All known unitals in P G(2, q^{2}) are Buekenhout-Metz unitals (see for example [26]). Finally we state two characterisations; see Chapter 5 for a new characterisation of Buekenhout-Metz unitals in P G(2, q^{2}), for q > 3.

Theorem 1.13.3.1 [56, Section 2., Theorem] In P G(2, q^{2}), q > 2, a unital \overline{U} is Buekenhout-Metz re (T, ℓ_{∞}) if and only if every Baer subline with a point on ℓ_{∞} intersects \overline{U} in 0,1,2 or q + 1 points. □

Theorem 1.13.3.2 [57, Proposition 1] If \overline{U} is a Buekenhout-Metz unital re (T, ℓ_{∞}) in P G(2, q^{2}), with base ovoid an elliptic quadric and if there exists a secant line l of \overline{U}, not on T, such that l ∩ \overline{U} is a Baer subline, then \overline{U} is a classical unital. □

Unitals have been constructed in non-Desarguesian planes by using the construction of Buekenhout-Metz unitals given above, see for example [11, 12], [31].

1.14 Inversive Planes

A comprehensive introduction to inversive planes is given in Dembowski’s Finite Geometries [33, Chapter 6]. Recent results concerning this topic can be found in [61], [82], [83], [84], for example.

Definition 1.14.1 (Statement from [53]) An inversive plane I is a set of points with distinguished subsets of the points, called circles such that:

(I1) any three distinct points of I are in exactly one common circle;

(I2) if P, Q are points of I and ℓ is a circle with P ∈ ℓ but Q ∉ ℓ then there is a unique circle of I which contains both P and Q and meets ℓ only in the point P;

(I3) I contains four points which are not on a common circle.

Let I be an inversive plane and let P be a point of I. The set of points of I different from P together with the circles containing P (minus P) and with incidence given by inclusion, is called the internal structure I_{P} of I at P. For every point P of I the internal structure I_{P} is an affine plane called the internal plane of I at P. By (I1) for two distinct circles ℓ, m of I we have the number of points common to ℓ and m is 0,1 or 2 and in each case we say the circles ℓ and m are disjoint, tangent or intersecting respectively. Some subsets of circles in an inversive plane I are of particular importance and for reference later we have the following terminology: A bundle of circles is the set of all circles through two distinct points P, Q of I. The points P and Q are called the carriers of the bundle. A pencil is any maximal set of mutually tangent circles through a common point P, called the carrier of the pencil. (Note the pencils with given carrier P correspond to the parallel classes of lines in the affine plane I_{P}.) A flock is a set of mutually disjoint circles in I such that, with the exception of precisely two points P, Q every point of I is on a (necessarily unique) circle of the flock. These points P, Q are called the carriers of the flock. In the finite case an inversive plane can be defined in the following way.

Definition 1.14.2 A finite inversive plane I is a 3-(q^{2} + 1, q + 1, 1) design. We call q the order of I.

For every point P of a finite inversive plane I of order q the internal plane I_{P} is an (finite) affine plane of order q (see [33, Section 6.1(4)]). Up to isomorphism there is a unique inversive plane of order q, with q ∈ {2, 3, 4, 5, 7}. For q = 7 this was originally proved by R. F. Denniston with the aid of a computer; in [84], as a corollary of a theorem we shall mention below, Thas gives a computer-free proof of the uniqueness of the inversive plane of order 7. Let O be an ovoid in a 3-dimensional projective geometry. The points of O together with the intersections π ∩ O, with π a secant plane of O, is an inversive plane I(O). We call I(O) the inversive plane associated with the ovoid O. We call an inversive plane egglike if it is isomorphic to an I(O) for some 3-dimensional ovoid O (see [33, Section 1.2] for isomorphism of incidence structures). If O is an ovoid of P G(3, q), then the associated inversive plane I(O) is a finite (egglike) inversive plane of order q. Since the only known examples of ovoids in P G(3, q), with q > 2, fall into two infinite classes there are consequently two known infinite families of finite egglike inversive planes. If the ovoid is an elliptic quadric of P G(3, q), then the associated inversive plane is called classical or Miquelian since it satisfies the configurational condition known as the Theorem of Miquel (see [33, Chapter 6] for more detail.) The family of finite Miquelian inversive planes is denoted M(q). If the ovoid is a Tits ovoid in P G(3, 2^{2r+1}), with r ≥ 1 an integer, then the associated inversive plane belongs to the second known family of finite egglike inversive planes which is denoted by S(q). The only known finite inversive planes are the egglike inversive planes in the families M(q) and S(q). The problem of classification of ovoids of P G(3, q), with q > 2, is equivalent to the classification of finite egglike inversive planes. As stated in an earlier section, the ovoids of P G(3, q), with q > 2, have been classified for q ≤ 32. We now list some old and some recent important results concerning finite inversive planes.

1. [33, 6.1.3] For any point P of a finite egglike inversive plane I(O) the affine plane I_{P} is the Desarguesian plane AG(2, q).

2. [33, 6.2.14] Every (finite) inversive plane of even order q is egglike. Consequently q is a power of 2.

3. [33, 1.4.50] Every (finite) egglike inversive plane of odd order is Miquelian.

4. [82] [84] Let I be an inversive plane of odd order q, q ∉ {11, 23, 59}. If for at least one point P of I the internal plane I_{P} is Desarguesian, then I is Miquelian.

5. [77] [65] If F is a flock of a finite egglike inversive plane I = I(O), then F is linear (that is, the ovals of O in P G(3, q), which correspond to the circles of the flock F,

lie in planes of P G(3, q) about a common line.)

6. [61] Let I be a finite egglike inversive plane of order 2 < q ≤ 32. If q ∈ {8, 32} then I is Miquelian or of S(q) type. If q ∉ {8, 32} then I is Miquelian.

Finally, we shall mention the plane model of egglike inversive planes which is given in [84] for example. Let O be an ovoid of P G(3, q) and let I denote the corresponding inversive plane. The circles of I are in one-to-one correspondence with the secant plane sections of the ovoid in P G(3, q). As a consequence we shall interchange the setting between the incidence structure of the inversive plane and the geometry of the ovoid in P G(3, q). We shall even abuse the terminology and refer to “circles” of O in P G(3, q) when we mean secant plane sections of O which correspond to circles of the associated inversive plane. The context in which we do this should make our meaning clear. Let P be a point of O and let π be a plane of P G(3, q), not containing P. The intersection of π and the tangent plane π_{P} of O at P is denoted by ℓ_{∞}. By projection ζ of O − {P } from P onto π, the points of O − {P } are mapped onto the q^{2} points of π\ℓ_{∞}, the circles of O through P (minus P) are mapped onto the q^{2}+q affine lines of π. The Desarguesian affine plane π\ℓ_{∞} is isomorphic to the internal plane I_{P} of I at P. Moreover the circles of O not through P are mapped by ζ onto q^{3} − q^{2} ovals of π; each such oval is disjoint from ℓ_{∞}. If the ovoid O is an elliptic quadric, then the circles of O not through P are mapped by ζ onto the q^{3} − q^{2} non-degenerate conics of π containing two points X, \overline{X} ∈ ℓ_{∞}, which are the two points of O on ℓ_{∞} belonging to the quadratic extension GF (q^{2}) of GF (q). Example: Consider the projective line P G(1, q^{2}). The points of P G(1, q^{2}) together with the Baer sublines of P G(1, q^{2}), with incidence given by inclusion, forms a Miquelian inversive plane I of order q (see [33, page 273]). Fix a point P of P G(1, q^{2}) and consider the internal plane I_{P} ≅ AG(2, q) of I at P. By the above theory and using the same notation, denote by ℓ_{∞} the line at infinity of I_{P} and let π denote the projective completion of I_{P} so that I_{P} = π\ℓ_{∞}. In the correspondence between the inversive plane I defined on P G(1, q^{2}) and the internal plane I_{P} of I at P we have: The points of P G(1, q^{2})\{P } are the q^{2} points of π\ℓ_{∞}. The Baer sublines of P G(1, q^{2}) containing P (minus P) are the q^{2} + q lines of π\ℓ_{∞} and the Baer sublines of P G(1, q^{2}) not containing P are the q^{3} − q^{2} non-degenerate conics in π containing two fixed points X, \overline{X} ∈ ℓ_{∞}, with X, \overline{X} conjugate with respect to the quadratic extension GF (q^{2}) of GF (q). We represent the situation as follows, with C_{1}, C_{2} and C_{3} three Baer sublines of P G(1, q^{2}) disjoint from P. In π, the Baer sublines are represented as non-degenerate conics containing two points X, \overline{X} on ℓ_{∞} in the quadratic extension.

[figure: X; \overline{X}; ℓ_{∞}; π; P G(1, q^{2}); P; C_{3}; C_{2}; C_{1}; C_{1}; C_{2}; C_{3}]

1.15 Maximal Arcs

As discussed in Section 1.11, Barlotti [9] introduced the term {k; n}-arc for a set K of k points in a finite projective plane π_{q} of order q, where n, n ≠ 0, is the greatest number of collinear points in the set. {k; 2}-arcs are simply called k-arcs. A {k; n}-arc is complete if it is not contained in a {k + 1; n}-arc. Let K be a {k; n}-arc in π_{q}. By considering the points of K on the q + 1 lines through a point P of K, it is easy to see that the number k of points of K satisfies:

k ≤ (q + 1)(n − 1) + 1
= nq − q + n
= (n − 1)q + n.

A {nq − q + n; n}-arc in π_{q} is called a maximal arc. Equivalently, a maximal arc may be defined as a non-empty set K of points in π_{q} such that every line of π_{q} meets K in either exactly n points or in none at all; the lines of π_{q} are called secant or external lines of K.

Examples of maximal arcs: For the given value of n a maximal {nq − q + n; n}-arc K in π_{q} is:

n = 1,  A single point.
n = 2,  A (q + 2)-arc in π_{q}, q even, in other words K is a hyperoval in π_{q}, q even.
n = q,  The set of points π_{q}\ℓ for ℓ a line of π_{q}.
n = q + 1, The set of points of the plane π_{q}.

Note that the study of hyperovals (maximal arcs with n = 2) is an active field of study in its own right with an extensive literature. Since the maximal arcs with n = 1 or q + 1 are determined we consider maximal arcs with 1 < n < q + 1. Let K be a (maximal) {(n − 1)q + n; n}-arc in π_{q}, n ≤ q. Let Q be a point of π_{q} not in K. By definition every line through Q intersects K in 0 or n points therefore we have:

n divides (n − 1)q + n
hence n divides q.

Hence Barlotti obtained a necessary condition for the existence of a (maximal) {nq − q + n; n}-arc in π_{q}, n ≤ q, is that n divides q. Also in [9] it was shown that if a {nq−q+n; n}-arc K exists in π_{q} then the set of external lines of K is a (maximal) {q(q − n + 1)/n; q/n}-arc in the dual plane of π_{q}. It follows that a maximal {nq − q + n; n}-arc exists in P G(2, q), n ≤ q, if and only if a maximal {q(q − n + 1)/n; q/n}-arc exists in P G(2, q). Denniston [34] proved that Barlotti’s necessary condition for the existence of maximal arcs is sufficient in P G(2, q), q even, by constructing infinite families of maximal arcs in Desarguesian planes of even order. Cossu [30] showed the above necessary condition for existence of maximal arcs in π_{q} is not sufficient; he proved P G(2, 9) contains no {21; 3}-arc. Thas [79] generalised Cossu’s result by proving the following result.

Theorem 1.15.1 [79] In P G(2, q), q = 3^{h} and h > 1, there are no {2q + 3; 3}-arcs and (hence) no {q(q − 2)/3; q/3}-arcs. □

In this 1987 paper Thas made the following conjecture:

Conjecture 1.15.1 [79] In P G(2, q), q odd, the only maximal arcs are P G(2, q), AG(2, q) = P G(2, q)\ℓ_{∞} and the dual of AG(2, q).

This conjecture was recently proved by Ball, Blokhuis and Mazzocca [7].

Theorem 1.15.2 [7] For q an odd prime power, and 1 < n < q, the Desarguesian plane P G(2, q) does not contain a {nq − q + n; n}−arc. □

We now list some constructions and classes of maximal arcs. In [78] Thas constructed an infinite family of maximal arcs in certain translation planes of even order. In the literature this family of maximal arcs has been referred to as the Thas maximal arcs and we shall do so here. The Thas maximal arcs are defined in certain finite translation planes of order q^{2} with kernel of order q; each such translation plane corresponds to a 1−spread in a hyperplane Σ_{∞} of P G(4, q) by the 4−dimensional Bruck and Bose representation of the translation plane (Section 1.10). The construction is as follows. The construction of a Thas maximal arc: Let Σ_{∞} = P G(3, q) and consider an ovoid O and a spread S in Σ_{∞} such that each line of S is incident with a unique point of O. An ovoid O and a spread S in Σ_{∞} with this property will be called a Thas ovoid-spread pair (O, S). Let Σ_{∞} be embedded as a hyperplane in P G(4, q) and let X^{∗} be a point of P G(4, q)\Σ_{∞}. Denote by K^{∗} the set containing X^{∗} and all points of P G(4, q)\Σ_{∞} collinear with X^{∗} and a point of O. The set of points K^{∗} in P G(4, q) represents a maximal {q^{3} − q^{2} + q; q}-arc K in the translation plane π_{q^{2}} of order q^{2} with translation line ℓ_{∞} corresponding to the spread S. We call K a Thas maximal arc in π_{q^{2}} with base point X and axis line ℓ_{∞}. Note that the axis line ℓ_{∞} is an external line of K in π_{q^{2}}. Existence of Thas maximal arcs: By the above construction, a Thas maximal arc exists in a translation plane π_{q^{2}} of order q^{2} with translation line ℓ_{∞} if and only if a Thas ovoid-spread pair (O, S) exists in P G(3, q), for the spread S corresponding to π_{q^{2}} in the Bruck and Bose representation. The known examples are [78]:

Translation plane q  (O, S)
P G(2, q^{2}) even (elliptic quadric, Regular spread)
q = 2^{2r+1}, r ≥ 1 (Tits ovoid, Regular spread)
Lüneburg plane q = 2^{2r+1}, r ≥ 1 (Tits ovoid, Lüneburg spread)
q = 2^{2r+1}, r ≥ 1 (elliptic quadric, Lüneburg spread)

Note that for q odd, an ovoid in P G(3, q) is an elliptic quadric; by a recent result of Blokhuis and Wilbrink (see also Theorem 1.15.2 for certain cases) there exists no Thas ovoid-spread pair in P G(3, q), q odd. In [80] Thas generalised the above construction of a Thas maximal arc and constructed (maximal) {q^{2d−1} − q^{d} + q^{d−1}; q^{d−1}}-arcs in certain translation planes π_{q^{d}} of even order q^{d}. For further constructions of maximal arcs in planes other than the Desarguesian plane see for example [45, 46].

1.16 Generalized Quadrangles

We present here some preliminary results concerning generalized quadrangles, for later reference. Unless stated otherwise the definitions and results of this section are from Payne and Thas [67]. A (finite) generalized quadrangle (GQ) is an incidence structure S = (P, B, I) in which P and B are disjoint (non-empty) sets of objects called points and lines (respectively), and for which I is a symmetric point-line incidence relation satisfying the following three axioms: GQ axiom (i) Each point is incident with 1 + t lines (t ≥ 1) and two distinct

points are incident with at most one line.

GQ axiom (ii) Each line is incident with 1+s points (s ≥ 1) and two distinct lines

are incident with at most one point.

GQ axiom (iii) If X is a point and ℓ is a line not incident with X, then there is a

unique pair (Y, m) ∈ P × B for which X I m I Y I ℓ.

The integers s and t are the parameters of the GQ and S is said to have order (s, t); if s = t, then S is said to have order s.

Let S be a GQ of order (s, t). Let X, Y be two (not necessarily distinct) points of S. We write X ∼ Y and say that X and Y are collinear if there exists a line ℓ of S such that X I ℓ I Y. And X ≁ Y means X and Y are not collinear. Note that X ∼ X for any X in P. For X ∈ P the set {Y ∈ P; X ∼ Y } of points in the GQ collinear with X is denoted X^{⊥}. The trace of a pair (X, Y) of distinct points is the set X^{⊥} ∩ Y^{⊥} and is denoted {X, Y }^{⊥}. More generally, if A is a subset of points in P, then A “perp” is defined by A^{⊥} = ∩{X^{⊥} | X ∈ A}.

Result 1.16.1 For distinct points X and Y in a GQ S of order (s, t), the cardinality of {X, Y }^{⊥} is:

\begin{matrix}|\{X, Y \}^{⊥}| = s + 1 & if X ∼ Y, \\ |\{X, Y \}^{⊥}| = t + 1 & if X ≁ Y. \end{matrix}

□

For distinct points X and Y in S, the span of the pair (X, Y) is

{X, Y }^{⊥⊥} = {V ∈ P; V ∈ Z^{⊥} for all Z ∈ {X, Y }^{⊥}}.

If X ≁ Y, then {X, Y }^{⊥⊥} is also called the hyperbolic line defined by X and Y. A triad (of points) is a triple (X, Y, Z) of pairwise non-collinear points in P. Given a triad T = (X, Y, Z), a center of T is a point of T^{⊥}.

Result 1.16.2 [67, 1.2.4] Let S be a GQ of order (s, t). If s > 1 and t > 1, then s^{2} = t if and only if each triad (of points) has a constant number of centers, in which case this constant number of centers is s + 1. □

Let s^{2} = t > 1, so that S is a GQ of order (s, s^{2}) and by Result 1.16.2, for any triad (X, Y, Z) we have |{X, Y, Z}^{⊥}| = s + 1. If X′, Y′, Z′ are three distinct points in {X, Y, Z}^{⊥} then since (X′, Y′, Z′) is necessarily a triad (by GQ axiom (iii)), we have {X, Y, Z}^{⊥⊥} ⊆ {X′, Y′, Z′}^{⊥} and therefore |{X, Y, Z}^{⊥⊥}| ≤ |{X′, Y′, Z′}^{⊥}| = s + 1. We say a triad (X, Y, Z) is 3-regular provided |{X, Y, Z}^{⊥⊥}| = s + 1. A point X is called 3-regular if and only if each triad containing X is 3-regular.

Result 1.16.3 [67, 1.3.3] Let S be a GQ of order (s, s^{2}), s ≠ 1, and suppose that any triad contained in {X, Y }^{⊥}, X ≁ Y, is 3-regular. Then the incidence structure with pointset {X, Y }^{⊥}, with circleset the sets of elements {Z_{1}, Z_{2}, Z_{3}}^{⊥⊥}, where Z_{1}, Z_{2}, Z_{3} are distinct points in {X, Y }^{⊥}, and with natural incidence, is an inversive plane of order s.

We include a proof of Result 1.16.3 to clarify this case for later reference. Proof: First note that by Result 1.16.1, |{X, Y }^{⊥}| = s^{2} + 1 and so our incidence structure has s^{2} + 1 points. For distinct points Z_{1}, Z_{2}, Z_{3} ∈ {X, Y }^{⊥}, (Z_{1}, Z_{2}, Z_{3}) is a triad (by GQ axiom (iii)) and X, Y ∈ {Z_{1}, Z_{2}, Z_{3}}^{⊥}. It follows that {Z_{1}, Z_{2}, Z_{3}}^{⊥⊥} ⊆ {X, Y }^{⊥} and by the 3-regularity, each circle of our incidence structure is incident with exactly s + 1 points. It also follows that any three distinct points in {X, Y }^{⊥} determine a circle. We now verify each property in Definition 1.14.1: (I1): Let c_{1} = {Z_{1}, Z_{2}, Z_{3}}^{⊥⊥} and c_{2} = {Z′_{1}, Z′_{2}, Z′_{3}}^{⊥⊥} be two distinct circles. Suppose X_{1}, X_{2}, X_{3} are three distinct points incident with both circles. Note that {Z_{1}, Z_{2}, Z_{3}}^{⊥⊥} is determined uniquely by any three distinct points X′_{1}, X′_{2}, X′_{3} in {Z_{1}, Z_{2}, Z_{3}}^{⊥} since,

{Z_{1}, Z_{2}, Z_{3}}^{⊥⊥} ⊆ {X′_{1}, X′_{2}, X′_{3}}^{⊥}

and these two sets have the same cardinality s + 1. It follows that since c_{1} and c_{2} are distinct circles, the sets {Z_{1}, Z_{2}, Z_{3}}^{⊥} and {Z′_{1}, Z′_{2}, Z′_{3}}^{⊥} have at most two points in common. Points X_{1}, X_{2}, X_{3} are each collinear to every point in {Z_{1}, Z_{2}, Z_{3}}^{⊥} and to every point in {Z′_{1}, Z′_{2}, Z′_{3}}^{⊥}, therefore

|{X_{1}, X_{2}, X_{3}}^{⊥}| ≥ (s + 1) + (s + 1) − 2
= 2s
> s + 1 since s > 1,

a contradiction, since for the triad (X_{1}, X_{2}, X_{3}) we have |{X_{1}, X_{2}, X_{3}}^{⊥}| = s + 1. Therefore three distinct points determine a unique circle in our structure. (I2): Distinct points P, Q ∈ {X, Y }^{⊥} are contained in circles {P, Q, Z}^{⊥⊥}, where Z is any point in \{X, Y \}^{⊥} distinct from P and Q. By (I1), there are \frac{s^{2} − 1}{s − 1} = s + 1 choices for Z and therefore there exist s + 1 circles incident with P and Q, with no further point in common. By counting, we obtain that each point of the structure, distinct from P and Q, is incident with (exactly) one circle which contains both P and Q. By (I1) it follows that P is incident with \frac{s^{2}(s^{2} − 1)}{s(s − 1)} = s^{2} + s circles. So there exist circles containing P and not Q, let ℓ_{P} be such a circle. Each circle which contains P and Q intersects ℓ_{P} in at most one point besides P. There are s points of ℓ_{P} besides P and from above, each such point lies in some circle containing P and Q. Therefore there is one remaining circle, which contains P and Q but intersects ℓ_{P} only in the point P. (I3): There are s^{2} +1 points in the structure and s^{2} +1 ≥ s+1 since s > 1 and therefore there exist 4 points not on a common circle. By Definition 1.14.1, the incidence structure is an inversive plane of order s. □

Result 1.16.4 [67, 3.1.2] For each ovoid O in P G(3, q) there is a GQ to be called T_{3}(O) constructed as follows: Let O be an ovoid in P G(3, q). Further, let P G(3, q) = Σ_{∞} be embedded as a hyperplane in P G(4, q). Define points as the following three types:

Type (i) the points P G(4, q)\Σ_{∞},
Type (ii) the hyperplanes Π_{3} of P G(4, q) for which |Π_{3} ∩ O| = 1,
Type (iii) one new symbol (∞).

Lines are defined as the following two types: Type (a) the lines of P G(4, q) which are not contained in Σ_{∞} and meet O

(necessarily in a unique point),

Type (b) the points of O.

Incidence is defined as follows: A point of type (i) is incident only with lines of type (a); here the incidence is that of P G(4, q). A point of type (ii) is incident with all lines of type (a) contained in it and with the unique element of O in it. The point (∞) is incident with no line of type (a) and all lines of type (b). T_{3}(O) is a GQ of order (q, q^{2}). □

Result 1.16.5 [67, 3.3.2(ii)] The point (∞) of the GQ T_{3}(O) is 3-regular. □

Definition of Property (G) 1.16.6 [66] In a generalized quadrangle S of order (s, s^{2}), s ≠ 1, let X, Y be distinct collinear points. We say that the pair {X, Y } has property (G) if every triad (X, X_{1}, X_{2}), with Y ∈ {X, X_{1}, X_{2}}^{⊥}, is 3-regular. Then also every triad (Y, Y_{1}, Y_{2}), with X ∈ {Y, Y_{1}, Y_{2}}^{⊥}, is 3-regular. We say that the generalized quadrangle S has property (G) at a flag (X, ℓ), where X I ℓ, if every pair {X, Y }, X ≠ Y, Y I ℓ has property (G). We say the generalized quadrangle S has property (G) at the line ℓ, or the line ℓ has property (G), if each pair of points {X, Y }, X ≠ Y, and X I ℓ I Y, has property (G).

Result 1.16.7 [86, Section 2.4] The point X of the generalized quadrangle S, of order (s, s^{2}), s ≠ 1, is 3-regular if and only if each flag (X, ℓ), X I ℓ, has property (G). □

The Bruck and Bose representation

in PG(4, q)

In this chapter we examine the Bruck and Bose representation of translation planes π_{q^{2}} of order q^{2} with kernel of order q; that is, translation planes of order q^{2} which are described by 1−spreads of P G(3, q) in the 4−dimensional Bruck and Bose representation ([33, 5.1.11]). In particular we consider the 4−dimensional Bruck and Bose representation of the Desarguesian plane P G(2, q^{2}). In Section 2.1 we recall this special case of the general Bruck and Bose representation and establish notation for the chapter. When we wish to work with this representation we shall refer to it as the Bruck-Bose setting, or simply Bruck-Bose. Note that this Bruck-Bose representation of a translation plane is equivalent to the André group theoretic representation of a translation plane given in [1]. In this chapter we determine the representation in Bruck-Bose of Baer subplanes of P G(2, q^{2}) and present characterisations of these structures. We also determine the representation in Bruck-Bose of conics contained in Baer subplanes of P G(2, q^{2}); this work leads to results concerning the existence of certain 4−dimensional caps which contain many normal rational curves.

2.1 Bruck-Bose in PG(4, q)

Recall from Section 1.10 the following representation of P G(2, q^{2}) in P G(4, q) due to André [1] and Bruck and Bose [21] and [22]; Let ℓ_{∞} denote a fixed line P G(2, q^{2}) and call this line the line at infinity of P G(2, q^{2}). The plane AG(2, q^{2}) = P G(2, q^{2})\ℓ_{∞} is the Desarguesian affine plane of order q^{2}. Embed Σ_{∞} = P G(3, q) as a hyperplane in P G(4, q). Let S be a fixed regular spread of Σ_{∞}. The affine plane AG(2, q^{2}) is represented by the following incidence structure: the points are the points of P G(4, q)\Σ_{∞}, the lines are the planes of P G(4, q), not contained in Σ_{∞} and which meet Σ_{∞} in a line of S and incidence is induced by the incidence in P G(4, q). AG(2, q^{2}) can be completed to the projective plane P G(2, q^{2}) by the addition of ℓ_{∞} whose points are the elements of the spread S. We shall use the phrase a subspace of P G(4, q)\Σ_{∞} to mean a subspace of P G(4, q) which is not contained in Σ_{∞}. The points P G(4, q)\Σ_{∞} shall be referred to as affine points. Also if a line l of P G(2, q^{2}) intersects a Baer subplane B of P G(2, q^{2}) in a Baer subline m, we call l a line of B. Note that P G(2, q^{2}) is a translation plane with respect to any of its lines and therefore there is choice involved in fixing the line at infinity. Moreover, by Theorem 1.10.1.3, any regular 1−spread in P G(3, q) corresponds to a Bruck-Bose representation of P G(2, q^{2}). Unless stated otherwise in this chapter, the Bruck-Bose representation of P G(2, q^{2}) is the representation given above for a fixed line ℓ_{∞} of P G(2, q^{2}) and a fixed regular 1−spread S of Σ_{∞} = P G(3, q). Let π_{q^{2}} denote a translation plane of order q^{2} with kernel of order q and with translation line ℓ_{∞}. Then by [33, 5.1.11] π_{q^{2}} is described by a 1−spread S_{π} of Σ_{∞} = P G(3, q) in the Bruck-Bose setting in P G(4, q). For our discussion, we shall use the expression the representation in P G(4, q) to mean the corresponding Bruck and Bose representation of the projective translation plane being discussed; moreover, the two representations coincide, that is π_{q^{2}} = P G(2, q^{2}), if and only if S_{π} = S is a regular spread of Σ_{∞}. If X denotes a substructure in π_{q^{2}}, it will be convenient at times to denote by X^{∗} the substructure in P G(4, q) which is the Bruck-Bose representation of X. Conversely, if X^{∗} is a substructure of P G(4, q), we shall denote by X the subset of points and lines of π_{q^{2}} which is represented by X^{∗} in Bruck-Bose.

2.2 Known Bruck-Bose representations of Baer sub-

structures

The representation in P G(4, q) of Baer subplanes of P G(2, q^{2}) is of particular importance in our discussion. In this section we review the known results. A transversal plane in P G(4, q) is a plane in P G(4, q)\Σ_{∞} which contains no line of S_{π}. Let B^{∗} be a transversal plane in P G(4, q), then B^{∗} is incident with q + 1 distinct elements of S_{π}; denote these spread elements by ℓ_{1}, ℓ_{2}, …, ℓ_{q+1}. The affine points of B^{∗} together with these q + 1 distinct elements of S_{π} incident with B^{∗} correspond to a set B in π_{q^{2}} of q^{2} + q + 1 points. For each i = 1, 2, …, q + 1, there exist q planes in P G(4, q)\Σ_{∞} containing ℓ_{i} and such that each intersects B^{∗} in q affine points. By the Bruck-Bose correspondence, each such plane represents a line of π_{q^{2}} incident with q + 1 points of B. Furthermore the line at infinity intersects B in q + 1 points. If we call these q^{2} + q + 1 lines, lines of B, then it follows that B satisfies the definition of a finite projective plane and is therefore a Baer subplane of π_{q^{2}}. In this way, the transversal planes of P G(4, q) represent Baer subplanes of π_{q^{2}} which contain the line at infinity as a line; a Baer subplane of π_{q^{2}} which contains the line at infinity as a line will be called an affine Baer subplane of π_{q^{2}}.

Theorem 2.2.1 [21, Section 9] If B^{∗} is transversal plane in P G(4, q) then B^{∗} is the Bruck-Bose representation of an affine Baer subplane B of π_{q^{2}}. □

Since the number of transversal planes in P G(4, q) equals the number of Baer subplanes of P G(2, q^{2}) which contain the line at infinity as a line, we have:

Corollary 2.2.2 [21] B is an affine Baer subplane of P G(2, q^{2}) if and only if in Bruck-Bose, B^{∗} is a transversal plane of P G(4, q).

Let B be an affine Baer subplane of P G(2, q^{2}) and let ℓ ≠ ℓ_{∞} be a line of B. In P G(4, q), the plane ℓ^{∗} intersects the transversal plane B^{∗} in a line of P G(4, q)\Σ_{∞} (which is not contained in Σ_{∞}). Therefore, by Corollary 2.2.2, any Baer subline of P G(2, q^{2}) which intersects ℓ_{∞} in a unique point is represented in P G(4, q) by a line of P G(4, q)\Σ_{∞}; conversely, each line of P G(4, q)\Σ_{∞} is the Bruck-Bose representation of a Baer subline of P G(2, q^{2}) which contains a unique point of ℓ_{∞}.

We now present the representations in P G(4, q) of Baer subplanes of P G(2, q^{2}) which intersect ℓ_{∞} in a unique point, and Baer sublines which are disjoint from ℓ_{∞}. The following result is well known and is a consequence of the example given in Section 1.14.

Lemma 2.2.3 A Baer subline b, containing no point on ℓ_{∞}, of a line a in P G(2, q^{2}), is represented in P G(4, q) by a non-degenerate conic C^{∗} in the plane α representing a. □

Definition 2.2.4 The conics in P G(4, q)\Σ_{∞} which represent Baer sublines of P G(2, q^{2}) shall be called Baer conics.

A Baer conic in P G(4, q) is necessarily disjoint from Σ_{∞}. Note that for the fixed regular spread S in Σ_{∞} there exist non-degenerate conics disjoint from Σ_{∞}, in planes of P G(4, q)\Σ_{∞} about spread elements, but which do not represent Baer sublines of P G(2, q^{2}); that is, there exist non-Baer conics P G(4, q). This result was proved by Metz [60] who showed that the number of Baer sublines of a line ℓ of P G(2, q^{2}) disjoint from a fixed point P ∈ ℓ is strictly less than the number of non-degenerate conics in P G(2, q) disjoint from a fixed line m in P G(2, q). For later reference, we consider the Bruck-Bose representation of some well known configurations of Baer sublines in P G(2, q^{2}).

Lemma 2.2.5 Let L_{1} and L_{2} be distinct affine points of a line a in P G(2, q^{2}). Let M = a ∩ ℓ_{∞}. There are q Baer sublines of a which contain L_{1}, L_{2} and not M.

Proof: The result follows from the fact that in P G(2, q^{2}) there are (q^{2}−1)/(q−1) = q+1 Baer sublines containing L_{1} and L_{2} and there is a unique Baer subline of a containing the three distinct points L_{1}, L_{2} and M (see Theorem 1.2.1 and the subsequent remarks). □ By interpreting the results of Lemma 2.2.3 and Lemma 2.2.5 in Bruck-Bose we obtain:

Lemma 2.2.6 If L_{1}^{∗} and L_{2}^{∗} are distinct affine points in a plane α in P G(4, q)\Σ_{∞} with α ∩ Σ_{∞} = m, where m is an element of the spread S of Σ_{∞}, then there exist q Baer conics in α incident with both L_{1}^{∗} and L_{2}^{∗}. □

We shall also make use of the following:

Lemma 2.2.7 [19] [89] [44] In a projective plane π_{q^{2}} of order q^{2} the number of points common to two Baer subplanes B_{1} and B_{2} of π_{q^{2}} is equal to the number of lines shared by B_{1} and B_{2}. □

Lemma 2.2.8 [72] In P G(2, q^{2}) two distinct Baer subplanes intersect in one of the following configurations:

1. The empty set;

2. One point and one line : the point is either incident or non-incident with the line.

3. Two points and two lines: the point of intersection of the two lines plus a second point on one of the lines;

4. Three points and three lines forming a triangle configuration;

5. q + 1 points and q + 1 lines: the q + 1 points are collinear on one of the lines and the remaining q lines form a pencil through one of the points;

6. q +2 points and q +2 lines: q +1 of the points are collinear on one of the lines and the remaining q + 1 lines form a pencil concurrent in the remaining point; each line contains 2 or q + 1 of the points. □

Above we recalled the representation in P G(4, q) of the affine Baer subplanes of P G(2, q^{2}), that is the Baer subplanes for which ℓ_{∞} is a secant line. We now provide an alternative direct proof of a result obtained in [19] and also in [90], which determines the representation of Baer subplanes of P G(2, q^{2}) which intersect ℓ_{∞} in a unique point. The variety we call a ruled cubic surface V_{2}^{3} is called a twisted ladder in [19]; its structure will be derived in the proof of the following Lemma, and will be used in the proof of Theorem 5.0.3 in a later chapter. (See Section 1.7 for more information on ruled cubic surfaces.)

Lemma 2.2.9 Let B be a Baer subplane in P G(2, q^{2}) such that B intersects ℓ_{∞} in the unique point P. Then B corresponds to a ruled cubic surface B in P G(4, q)\Σ_{∞} with B ∩ Σ_{∞} = p in P G(4, q), where p is a line of the spread S of Σ_{∞}.

Proof: Let B denote the structure in P G(4, q) representing B. As B intersects ℓ_{∞} in a unique point P, in P G(4, q) B intersects Σ_{∞} only in point(s) of the line p of S which represents P. The q + 1 lines of B through the point P ∈ ℓ_{∞} correspond to q + 1 planes in P G(4, q)\Σ_{∞} about p; each of these planes contain a line ℓ_{i}^{∗} of P G(4, q)\Σ_{∞} which represents a Baer subline of B incident with P. It follows that B contains q + 1 lines l_{1}^{∗}, …, l_{q+1}^{∗} in P G(4, q)\Σ_{∞} each incident with p and no two in a plane about p. It follows that no two of these lines intersect in a point not incident with p. Call the lines l_{1}^{∗}, …, l_{q+1}^{∗} generators of B; we now prove that these lines are mutually skew. Suppose l_{i}^{∗} ∩ l_{j}^{∗} ∈ p with 1 ≤ i < j ≤ q + 1. Then ⟨l_{i}^{∗}, l_{j}^{∗}⟩ is a transversal plane which represents a Baer subplane, distinct from B, and sharing with B a non-degenerate quadrangle, which by Theorem 1.2.1 is a contradiction. Therefore through each point of p there passes a unique generator of B. It follows that points of P G(4, q) on these generators are all the points of B. The spread line p is therefore contained in B and p is called the line directrix of B. Let Q be any point of B, distinct from P; let Q^{∗} ∈ B be its representative in P G(4, q). In P G(2, q^{2}), of the Baer sublines in B through Q, one is a subline of the line QP and the remaining q are disjoint from ℓ_{∞}. Therefore, by Lemma 2.2.3, the points of B lie on q distinct Baer conics C_{1}^{∗}, …, C_{q}^{∗} and one generator, l_{q+1}^{∗} say, each through Q^{∗}. Each conic C_{i}^{∗} (i = 1, …, q) lies in a plane α_{i} which intersects Σ_{∞} in a line m_{i} of the spread S. Let α_{q+1} denote the plane ⟨Q^{∗}, p⟩. In P G(2, q^{2}), each subline of B through P intersects each subline of B through Q, and therefore each conic C_{i}^{∗} intersects each generator of B in a unique point. Consider the plane ⟨Q^{∗}, l_{1}^{∗}⟩; since Q^{∗} ∉ l_{1}^{∗}, the plane ⟨Q^{∗}, l_{1}^{∗}⟩ is a transversal plane and therefore represents a Baer subplane B′ of P G(2, q^{2}), distinct from B. Now |B ∩ B′| ≥ q + 2. It follows from Lemma 2.2.8 that the Baer subplanes B and B′ intersect in q + 1 lines of P G(2, q^{2}) through Q and the line represented by ⟨l_{1}^{∗}, p⟩. If l_{1}^{∗} represents the Baer subline u of B, then the lines of P G(2, q^{2}) joining Q to the q + 1 points of u, common to B and B′, are represented by the planes α_{1}, …, α_{q+1}. As a transversal plane intersects Σ_{∞} in a transversal line of a regulus in the regular spread S, it follows that the spread lines m_{1}, …, m_{q}, p, each of which intersects ⟨Q^{∗}, l_{1}^{∗}⟩, are generators of a hyperbolic quadric Q_{2}^{2} of Σ_{∞}. Thus the q + 1 planes α_{1}, …, α_{q+1} constitute a quadric cone V_{3}^{2} of P G(4, q), with the point Q^{∗} as vertex, and the quadric Q_{2}^{2} as base. Let X_{1}^{∗}, …, X_{q+1}^{∗} be the points of the Baer conic C_{1}^{∗}. Then the q + 1 planes ⟨X_{i}^{∗}, p⟩ constitute a quadric cone V′_{3}^{2} of P G(4, q), with the line vertex p, and base C_{1}^{∗}. Note that Q^{∗} = X_{i}^{∗} for some i. These two quadric cones V_{3}^{2} and V′_{3}^{2} have the plane ⟨Q^{∗}, p⟩ in common, and therefore residually intersect in a ruled cubic surface V_{2}^{3}. The planes of the two quadric cones represent lines of B, and therefore, by considering their intersection, it follows that B is precisely the ruled cubic surface V_{2}^{3}. □ Since there exist conics in P G(4, q)\Σ_{∞} which do not represent Baer sublines, it follows that there exist ruled cubic surfaces, with directrix a line of S, which do not represent Baer subplanes B of P G(2, q^{2}) intersecting ℓ_{∞} in a unique point.

Definition 2.2.10 The ruled cubic surfaces in P G(4, q)\Σ_{∞}, with line directrix a line of S, which represent Baer subplanes of P G(2, q^{2}) shall be called Baer ruled cubics.

It is well known that in P G(2, q^{2}) there exist q + 1 Baer subplanes containing a given point P and a Baer subline c of a line a not through P; if we let P ∈ ℓ_{∞} and c be disjoint from ℓ_{∞}, then together with Lemma 2.2.3 and Lemma 2.2.9 this implies the following result, which we shall need in a later chapter.

Lemma 2.2.11 Let α be a plane in P G(4, q)\Σ_{∞}, with α ∩Σ_{∞} = m a line of the spread S of Σ_{∞}. If p is a line in S distinct from m and if C^{∗} is a Baer conic in α, then there exist q + 1 Baer ruled cubics containing p and C^{∗}. □

The representation in Bruck-Bose of Baer subplanes of P G(2, q^{2}) is therefore completely determined. For translation planes π_{q^{2}}, with kernel of order q, the problem of determining the representation of Baer subplanes of π_{q^{2}} in Bruck-Bose is not completely solved. Freeman [40] gives examples of affine Baer subplanes of a translation plane π_{q^{4}}, of order q^{4}, which have a representation in 4−dimensional Bruck-Bose which is distinct from those obtained above for the Desarguesian case (see also Foulser [38] for other examples.)

2.3 The Bruck-Bose representation of Conics in

Baer subplanes of PG(2, q^{2})

For later work we shall need a classification of the possible intersections of a hyperplane of P G(4, q) and a Baer ruled cubic surface in P G(4, q). This is given in the next theorem. Note: A Baer ruled cubic surface in P G(4, q) is a variety of order 3 and dimension 2 properly contained in the 4-dimensional space. By the results of Section 1.6 a hyperplane of P G(4, q) intersects the Baer ruled cubic surface in a cubic curve (see also Theorem 1.7.2); a cubic curve on the Baer ruled cubic surface is one of the following:

(a) One line counted triply;
(b) Two lines, one counted doubly;
(c) Three lines;
(d) A conic and a line;
(e) A twisted cubic curve.

The only lines on a Baer ruled cubic surface are the generators and the line directrix. Apart from points and lines the ruled cubic surface contains no linear subspaces. Also note that the intersection of a hyperplane with a ruled cubic surface may have components in some extension of the base field.

Theorem 2.3.1 Let B be a Baer ruled cubic surface in P G(4, q). Let p ∈ S denote the line directrix of B, so that {p} = B ∩ Σ_{∞}. Denote by Π_{3} a hyperplane of P G(4, q). The intersection B ∩ Π_{3} in P G(4, q) is one of the following:

B ∩ Π_{3}

The number of hyperplanes of P G(4, q) which intersect B in such a configuration:

(a) The line directrix p of B (q^{2} − q)/2

(Note: Σ_{∞} is an example of such a hyperplane)

(b) The union of a (unique) generator of B and the line directrix p of B

q + 1

(c) The union of two generators of B and the line directrix p

(q^{2} + q)/2

(d) The union of a Baer conic and a generator of B (Note: The Baer conic and generator intersect in a unique point)

q^{3} + q^{2}

(e) A twisted cubic curve (Note: such a curve intersects the line directrix p in a unique point)

q^{4} − q^{2}

Proof: By generating hyperplanes Π_{3} from subsets of points of B, the intersection sets Π_{3} ∩ B are determined. We proceed with this method until all q^{4} + q^{3} + q^{2} + q + 1 hyperplanes of P G(4, q) have been considered. Let P_{1}^{∗}, …, P_{q+1}^{∗} denote the points of p and let g_{1}^{∗}, …, g_{q+1}^{∗} denote the generators of B, such that g_{i}^{∗} ∩ p = {P_{i}^{∗}}, i = 1, …, q + 1. Let C^{∗} be a Baer conic of B and let π_{C^{∗}} be the plane in P G(4, q) containing C^{∗}. Hyperplanes ⟨π_{C^{∗}}, g_{i}^{∗}⟩, i = 1, …, q + 1, are the q + 1 hyperplanes of P G(4, q) about the plane π_{C^{∗}}; each hyperplane contains both the Baer conic C^{∗} and generator g_{i}^{∗} respectively. The Baer conic C^{∗} and the generator g_{i}^{∗} constitute a cubic curve in the hyperplane ⟨π_{C^{∗}}, g_{i}^{∗}⟩ hence by the note preceding this theorem each hyperplane ⟨π_{C^{∗}}, g_{i}^{∗}⟩ contains exactly the Baer conic C^{∗} and generator g_{i}^{∗} of B for i = 1, …, q + 1 respectively. There are q^{2} Baer conics of B and q+1 generators of B, thus there exist q^{2}(q+1) = q^{3}+q^{2} hyperplanes of P G(4, q) which intersect B in the union of a Baer conic and a generator of B. There exist q^{2} + q + 1 hyperplanes about the line p, q^{2} + q distinct from Σ_{∞}. If ℓ is a line of π_{C^{∗}}, a plane of P G(4, q)Σ_{∞} which contains a Baer conic C^{∗} of B, then ⟨ℓ, p⟩ is a hyperplane containing the line directrix p. Depending on whether ℓ is an external, tangent or secant line of C^{∗} in π_{C^{∗}}, the hyperplane ⟨ℓ, p⟩ intersects B in p plus 2, 1, or 0 generators of B in P G(4, q) respectively. We consider these cases separately. Two generators of B span a hyperplane about p. Such a hyperplane contains three lines of the Baer ruled cubic surface hence no further point of B. Thus there exist q(q + 1)/2 hyperplanes of P G(4, q) which intersect B in the union of two generators of B and the line directrix p of B. About a plane ⟨p, g_{i}^{∗}⟩, for fixed i, there exist q + 1 hyperplanes; q contain a second generator of B and one intersects B in no further point; here the hyperplane intersects the ruled cubic surface doubly at g_{i}^{∗}. By considering the q + 1 generators in turn, we have that there exist q + 1 hyperplanes of P G(4, q) which intersect B in the union of a generator and the line directrix p of B. The q(q − 1)/2 remaining hyperplanes about p therefore each intersect B in exactly the line directrix p; each such hyperplane intersects the ruled cubic surface at p and two complex conjugate generators of the cubic surface in a quadratic extension extension of the base field. Next consider a spread element m distinct from p. About m there is a unique plane π_{m} containing a Baer conic of B. Hyperplanes ⟨π_{m}, P_{i}^{∗}⟩ i = 1, …, q + 1 each intersect B in the union of the Baer conic in π_{m} and the generator g_{i}^{∗} respectively; these hyperplanes have been counted above. About the plane ⟨m, P_{i}^{∗}⟩, for fixed i, there are q−1 hyperplanes distinct from both Σ_{∞} and ⟨π_{m}, P_{i}^{∗}⟩; let Σ be one of these q − 1 hyperlanes. Σ contains no Baer conic or generator of B, and Σ does not contain the line directrix p of B. The hyperplane Σ intersects each generator line of B in a unique point. As a hyperplane intersects a ruled cubic surface in a cubic curve, we conclude that Σ intersects B in an irreducible cubic curve, namely a twisted cubic curve. The number of spread elements besides p is q^{2}; the number of points of p is q + 1; from above, for a spread element m ≠ p and a point P_{i}^{∗} of p there are q − 1 hyperplanes about the plane ⟨m, P_{i}^{∗}⟩ which each intersect B in a (distinct) twisted cubic curve. Thus there exist q^{2}(q + 1)(q − 1) = q^{4} − q^{2} hyperplanes of P G(4, q) which intersect B in a twisted cubic curve. We have considered (q^{3}+q^{2})+(q^{2}+q)/2+(q+1)+(q^{2}−q)/2+q^{4}−q^{2} = q^{4}+q^{3}+q^{2}+q+1 distinct hyperplanes of P G(4, q), namely all hyperplanes of P G(4, q). □

In Theorem 2.3.1 the intersection sets (a), (b), (c) and (d) can be described in B, the Baer subplane of P G(2, q^{2}) represented by B, as respectively: a unique point P at infinity on B, a Baer subline in B containing P, the union of two distinct Baer sublines in B containing P and the union of a Baer subline in B through P and a Baer subline in B not through P. As a subset of points of B, the intersection set (e) has properties which are not so readily recognised. We now show that an intersection set of type (e) in B is a non-degenerate conic in B.

Lemma 2.3.2 Let B be a Baer ruled cubic surface in the Bruck-Bose representation of P G(2, q^{2}). If ζ^{∗} is a twisted cubic curve on B then ζ^{∗} is the Bruck-Bose representation of an oval ζ in the corresponding Baer subplane B of P G(2, q^{2}).

Proof: A twisted cubic curve ζ^{∗} lies in a hyperplane Σ_{ζ^{∗}} of P G(4, q). Since ζ^{∗} is contained in B and since Σ_{ζ^{∗}} ∩B is a cubic curve, we have Σ_{ζ^{∗}} ∩B = ζ^{∗}. By Theorem 2.3.1 and its proof we have the following:

1. The hyperplane Σ_{ζ^{∗}}, which contains the twisted cubic curve ζ^{∗}, contains a unique spread element m distinct from the line directrix p of B;

2. The planes about m in Σ_{ζ^{∗}} each contain a unique point of ζ^{∗};

3. ζ^{∗} has exactly q + 1 points, one on each generator line of B;

4. ζ^{∗} contains a unique point of the line directrix p of B.

Now we consider ζ^{∗} as a set of points ζ in B, the Baer subplane of P G(2, q^{2}) represented by B, and show that no line of B contains three points of ζ. Interpreting the properties (1) − (4) in the Baer subplane B we have by (4) ζ contains the point {P } = B ∩ l_{∞}. By (3) each line of B through P intersects ζ in at most one further point. Now suppose there exists a line l of B containing three distinct points of ζ; by the previous statement P ∉ l. Also note that l ≠ l_{∞} since l_{∞} is not a line of B. In Bruck-Bose, l is a plane α_{l} containing 3 distinct points of the twisted cubic curve ζ^{∗} and therefore by (1), and since no three points of a twisted cubic curve are collinear, the plane α_{l} is contained in the hyperplane Σ_{ζ^{∗}}. Since α_{l} is necessarily a plane about a spread element (α_{l} is a Bruck-Bose representation of a line of P G(2, q^{2})) α_{l} contains the unique spread element m in Σ_{ζ^{∗}}. By (2) α_{l} therefore intersects ζ^{∗} in a unique point, a contradiction to our assumption that α_{l} contains three distinct points of ζ^{∗}. Thus in P G(2, q^{2}) there exists no Baer subline in B which intersects ζ in more than two points. □

Lemma 2.3.3 Let C be a non-degenerate conic in P G(2, q). Embed P G(2, q) as a Baer subplane in P G(2, q^{2}). Let C_{q^{2}} be the conic obtained by extending C to a conic in P G(2, q^{2}). Let M be a point of C_{q^{2}}\C. The q + 1 lines joining M to the points of C are lines of a Baer subplane containing M.

Proof: Without loss of generality let C be the conic with equation xy = z^{2} and let M have coordinates (θ^{2}, 1, θ) where θ ∈ GF (q^{2})\GF (q). Note that (0, 1, 0) and (1, 0, 0) are points of C. The lines in P G(2, q^{2}) joining M to the points of C have line coordinates given by:

{[1, 0, −θ] + ϕ[0, θ, −1] | ϕ ∈ GF (q) ∪ {∞}}.

The pencil of lines {[1, 0, 1] + ϕ[0, 1, 1] | ϕ ∈ GF (q) ∪ {∞}} in P G(2, q) is the pre-image of the above set of lines under the projectivity of P G(2, q^{2}) given by the non-singular

matrix [ \begin{matrix}1 & 0 & 0 \\ 0 & θ & 0 \\ 0 & −1 + θ & −θ \end{matrix} ] ∈ P GL(3, q^{2}). We have therefore that the q + 1 lines joining

M to the points of C are q + 1 lines of a Baer subplane containing M. □

Theorem 2.3.4 The q^{4} − q^{2} twisted cubic curves on a Baer ruled cubic B with line directrix p in P G(4, q) are the Bruck-Bose representations of the q^{4} − q^{2} non-degenerate conics in B on the point P, where B is the Baer subplane of P G(2, q^{2}) represented by B and P the point of B at infinity represented by p.

Proof: For q odd the result follows from Lemma 2.3.2 and Segre’s Theorem 1.11.1. For q even, let C be a non-degenerate conic in a Baer subplane B of P G(2, q^{2}) such that B ∩ l_{∞} = {P }, a unique point, and let P ∈ C. Conic C is a subconic of a conic C_{q^{2}} of P G(2, q^{2}) and since l_{∞} is not the tangent to C_{q^{2}} at the point P, l_{∞} is a secant to the conic C_{q^{2}}. Let M be the point of C_{q^{2}} distinct from P on the line l_{∞}. In B the points of C besides P lie on distinct lines of B on P. Thus in Bruck-Bose B is a Baer ruled cubic surface B with line directrix p (representing P) and the points C^{∗} of the conic in Bruck-Bose besides p lie on distinct generator lines of B; the point M is represented in Bruck-Bose by a spread element m. We now show that these points C^{∗} lie in a hyperplane of P G(4, q) so that by Theorem 2.3.1 the points of C^{∗} are points of a twisted cubic curve on B. In P G(2, q^{2}), by Result 2.3.3, the q + 1 lines on M joining M to the points of conic C are a pencil of lines in a Baer subplane B′ of P G(2, q^{2}) containing M. Since MP is a line of B′, the line at infinity ℓ_{∞} is secant to B′ and therefore B′ is represented in Bruck-Bose by a transversal plane B^{′∗}. The hyperplane ⟨m, B^{′∗}⟩ of P G(4, q) therefore contains the points C^{∗} representing the conic C. Since the hyperplane ⟨m, B^{′∗}⟩ of P G(4, q) contains the q affine points of C^{∗} on B together with a unique point of p ⊆ B and since ⟨m, B^{′∗}⟩ contains no Baer conic on B, by Theorem 2.3.1 ⟨m, B^{′∗}⟩ intersects B in a twisted cubic curve. It follows that the non-degenerate conic C is represented in Bruck-Bose by a twisted cubic curve on the Baer ruled cubic B. □

Corollary 2.3.5 Let B be a Baer subplane of P G(2, q^{2}) such that |B ∩ l_{∞}| = 1; let the unique point at infinity of B be P. The non-degenerate conics in B are represented in Bruck-Bose by either a twisted cubic curve (when the conic contains the point P) or a 4-dimensional normal rational curve (when the conic does not contain the point P) on the Baer ruled cubic B.

We shall give two proofs; the first is quite short but does not cover all cases and the second is a proof valid for all prime powers q ≥ 3. Proof of Corollary 2.3.5 (cases q even and q odd, q ≤ 7 or (10.25)^{2} ≤ q): By Theorem 2.3.4 it remains to prove that a non-degenerate conic in B which does not contain the point at infinity P, is represented in Bruck-Bose by a 4-dimensional normal rational curve. Let C be a non-degenerate conic in B which does not contain the point P. Each line of B intersects C in at most two points. Since non-degenerate conics on P in B are represented in Bruck-Bose by twisted cubic curves (see Theorem 2.3.4) and since a distinct nondegenerate conic in B intersects C in at most four points, by Theorem 2.3.1 a hyperplane of P G(4, q) intersects the Bruck Bose representation of conic C in at most four points. We have therefore that the Bruck-Bose representation of conic C is a set of q + 1 points C^{∗} in P G(4, q) with the property that no hyperplane intersects the set in more than four points; in other words we have a (q + 1)_{4}−arc C^{∗} in P G(4, q) and by Theorems 1.5.1, 1.5.2 and 1.5.3, this arc is a 4−dimensional normal rational curve for q even and q odd, q ≤ 7 or (10.25)^{2} ≤ q. □

Lemma 2.3.6 There exists a, b ∈ GF (q^{2})\GF (q), q ≥ 3, with the following properties:

\begin{matrix}(i) & a ≠ b, −b, \\ (ii) & ab^{−1} + a^{−1}b ∈ GF (q^{2})\GF (q), \\ (iii) & ab^{−1} ∈ GF (q^{2})\GF (q), \end{matrix}

and for such a, b we have ab ≠ 0, a^{2} ≠ 0, b^{2} ≠ 0.

Proof: First we prove that there exists x ∈ GF (q^{2})\GF (q), q ≥ 3, such that x + x^{−1} ∉ GF (q): For q = 3, GF (3) = {0, 1, 2} and GF (9) = {0, 1, ω, ω^{2}, 2, ω^{5}, ω^{6}, ω^{7}} where ω^{2}−ω−1 = 0. Here ω + ω^{7} = ω and ω^{2} + ω^{5} = ω^{2} as required. For q > 3 consider

\begin{aligned}& x + x^{−1} & = & λ \\ ⟺ & x^{2} − λx + 1 & = & 0 \end{aligned}

for λ ∈ GF (q) and x ≠ 0. For each λ ∈ GF (q), there exist at most two solutions x, x^{−1} ∈ GF (q^{2})\GF (q) to this quadratic equation. Therefore there exist at least

q^{2} − 2q − q = q(q − 3) > 0

elements x ∈ GF (q^{2})\GF (q) for which x + x^{−1} ∉ GF (q). It remains to show that for x ∈ GF (q^{2})\GF (q) for which x + x^{−1} ∉ GF (q), there exists a, b ∈ GF (q^{2})\GF (q) such that a ≠ b, −b and bx = a: By considering all b ∈ GF (q^{2})\GF (q), we obtain q^{2} − q distinct elements bx and bx ≠ 0 as neither x nor b is equal to 0. Since

q^{2} − q > q − 1 = |GF (q)\{0}|

there exists a choice of b ∈ GF (q^{2})\GF (q) for which bx = a ∉ GF (q). If a = b then b(x − 1) = 0 implies that x = 1 ∈ GF (q), a contradiction. If a = −b then b(x + 1) = 0 implies that x = −1 ∈ GF (q), a contradiction. □ Proof of Corollary 2.3.5 (case q ≥ 3): We investigate the representation in Bruck-Bose of a particular^{1} non-degenerate conic C in a Baer subplane B of P G(2, q^{2}) with

|B ∩ ℓ_{∞}| = 1 and C ∩ ℓ_{∞} = ∅. Let Cq^{2}^{′} be the conic {(θ^{2}, 1, θ); θ ∈ GF (q^{2}) ∪ {∞}}, that

is, the conic in P G(2, q^{2}) with equation z^{2} = xy. The conic Cq^{2}^{′} has nucleus N(0, 0, 1) if q is even. Cq^{2}^{′} is fixed by projectivities of the plane defined by a matrix of the form:

H = [ \begin{matrix}a^{2} & b^{2} & 2ab \\ c^{2} & d^{2} & 2cd \\ ac & bd & bc + ad \end{matrix} ]

such that ad − bc ≠ 0 (as |H| = (ad − bc)^{3}) (see [52, Theorem 2.37]). The action of such a projectivity on the points of the conic Cq^{2}^{′} is the map:

\begin{matrix}(θ^{2}, 1, θ) & ⟼ & ((\frac{aθ+b}{cθ+d})^{2}, 1, \frac{aθ+b}{cθ+d}) \\ (1, 0, 0) & ⟼ & ((\frac{a}{c})^{2}, 1, \frac{a}{c}) \\ ( N & ⟼ & N, for q even ). \end{matrix}

1For a non-degenerate conic C_{1} in a Baer subplane B_{1}, B_{1}^{σ} = B for some collineation σ and C_{1}^{σ} is a non-degenerate conic in B and is therefore projectively equivalent to C via a collineation in B.

Let C′ ⊆ Cq^{2}^{′} be the points of Cq^{2}^{′} in the Baer subplane P G(2, q); so C′ = {(θ^{2}, 1, θ); θ ∈ GF (q) ∪ {∞}}. We now find a projectivity with matrix H of the above form which maps P G(2, q) to a Baer subplane B and maps C′ to a conic C in B, with |B ∩ ℓ_{∞}| = 1, and such that B ∩ ℓ_{∞} ≠ (0, 1, 0), (1, 0, 0), that is, such that the unique point of B on ℓ_{∞} is not a point of the conic. For our coordinate representation of P G(2, q^{2}) the line at infinity ℓ_{∞} is the line of P G(2, q^{2}) with equation z = 0. We will then represent C via coordinates in the Bruck-Bose setting and determine C as a set of points of a normal rational curve in P G(4, q). Consider the projectivity H defined by matrix:

[ \begin{matrix}a^{2} & b^{2} & 2ab \\ b^{2} & a^{2} & 2ab \\ ab & ab & a^{2} + b^{2} \end{matrix} ]

where a ≠ b, −b, a, b, ab^{−1}, ab^{−1} + a^{−1}b ∈ GF (q^{2})\GF (q) (refer to Lemma 2.3.6). The Baer subplane P G(2, q) is mapped by H as follows: For a point (x, y, z) ∈ P G(2, q),

H ( \begin{matrix}x \\ y \\ z \end{matrix} ) = [ \begin{matrix}a^{2}x + b^{2}y + 2abz \\ b^{2}x + a^{2}y + 2abz \\ abx + aby + (a^{2} + b^{2})z \end{matrix} ] .  (2.1)

The resulting set of points constitute a Baer subplane B whose intersection with ℓ_{∞} is the set of points (2.1) with third coordinate zero, that is with

abx + aby + (a^{2} + b^{2})z = 0,
that is, x + y + (ab^{−1} + a^{−1}b)z = 0.  (2.2)

Since ab^{−1}+a^{−1}b ∈ GF (q^{2})\GF (q), equation (2.2) is the equation of a line not in P G(2, q) and therefore the line (2.2) intersects P G(2, q) in a unique point; that is, there exists a unique point X = (x′, y′, z′) in P G(2, q) for which x′ + y′ + (ab^{−1} + a^{−1}b)z′ = 0. Thus the Baer subplane B intersects ℓ_{∞} in a unique point, namely the point with coordinates,

H ( \begin{matrix}x′ \\ y′ \\ z′ \end{matrix} ) = [ \begin{matrix}a^{2}x′ + b^{2}y′ + 2abz′ \\ b^{2}x′ + a^{2}y′ + 2abz′ \\ 0 \end{matrix} ] = X^{H}.  (2.3)

Also we need to show that X^{H} is not a point of the conic C. The only points of Cq^{2}^{′} on the line at infinity are (1, 0, 0) and (0, 1, 0) thus if X^{H} ⊆ C ∩ ℓ_{∞} then X^{H} is (1, 0, 0) or (0, 1, 0). Now X^{H} = (0, 1, 0) or X^{H} = (1, 0, 0) if and only if a^{2}x′ + b^{2}y′ + 2abz′ = 0 or b^{2}x′ + a^{2}y′ + 2abz′ = 0 respectively. We need to show a^{2}x′ + b^{2}y′ + 2abz′ ≠ 0 and b^{2}x′ + a^{2}y′ + 2abz′ ≠ 0. Consider the lines [a^{2}, b^{2}, 2ab], [b^{2}, a^{2}, 2ab], [1, 1, ab^{−1} + a^{−1}b]. The point of intersection of lines [a^{2}, b^{2}, 2ab] and [1, 1, ab^{−1} + a^{−1}b] is (b^{2}, a^{2}, −ab) ≡ (a^{−1}b, ab^{−1}, −1). The point of intersection of lines [b^{2}, a^{2}, 2ab] and [1, 1, ab^{−1} + a^{−1}b] is (a^{2}, b^{2}, −ab) ≡ (ab^{−1}, a^{−1}b, −1). Since X ∈ P G(2, q) and since −1 ∈ GF (q) and ab^{−1}, a^{−1}b ∈ GF (q^{2})\GF (q), it follows that X ≠ (a^{−1}b, ab^{−1}, −1) and X ≠ (ab^{−1}, a^{−1}b, −1). Hence we conclude that X^{H} is not a point of C. Hence B = P G(2, q)^{H} is a Baer subplane with unique point X^{H} on ℓ_{∞} and the Baer subplane B contains the non-degenerate conic C = C′^{H} for which C ∩ ℓ_{∞} = ∅. The coordinates of the points of C = C′^{H} are given by:

H ( \begin{matrix}θ^{2} \\ 1 \\ θ \end{matrix} ) = ( 1_{\frac{aθ+b}{bθ+a}}^{(\frac{aθ+b}{bθ+a})2} ) ≡ ( \frac{aθ+b}{bθ+a} ; \frac{bθ+a}{aθ+b} ; 1 )

where θ ∈ GF (q) ∪ {∞}. (Note that if aθ + b = 0 then θ = −a^{−1}b is not an element of GF (q), a contradiction. Hence aθ + b ≠ 0 and similarly bθ + a ≠ 0.) We now transform these plane coordinates to coordinates in P G(4, q), that is the coordinates of the points C^{∗} representing C in the Bruck-Bose setting in P G(4, q)\Σ_{∞} using the results of Section 1.10.4. Let α be an element of GF (q^{2})\GF (q) with minimal polynomial

x^{2} − λx − μ,

where λ, μ ∈ GF (q). Using x ⟼ x̄ = x^{q} to denote the Fröbenius map, we have

\begin{aligned}α + ᾱ & = & λ \\ αᾱ & = & −μ. \end{aligned}

Each element x ∈ GF (q^{2}) can be written uniquely in the form x = x_{1} + αx_{2} where x_{1}, x_{2} ∈ GF (q).

We have therefore:

a = a_{1} + αa_{2}, a_{1}, a_{2} ∈ GF (q),
b = b_{1} + αb_{2}, b_{1}, b_{2} ∈ GF (q)
and θ = θ̄ since θ ∈ GF (q) for points in C′.

Also ā = a_{1} + ᾱa_{2} = a_{1} + (λ − α)a_{2} = (a_{1} + λa_{2}) − αa_{2} and similarly b̄ = (b_{1} + λb_{2}) − αb_{2}. For a point P in C, with the coordinates (\frac{aθ+b}{bθ+a}, \frac{bθ+a}{aθ+b}, 1) of P written as a row vector, and using the above representation we obtain:

(\frac{aθ + b}{bθ + a}, \frac{bθ + a}{aθ + b}, 1) ≡ (\frac{aθ + b}{bθ + a} × \frac{b̄θ + ā}{b̄θ + ā}, \frac{bθ + a}{aθ + b} × \frac{āθ + b̄}{āθ + b̄}, 1) .

Now

\frac{aθ + b}{bθ + a} × \frac{b̄θ + ā}{b̄θ + ā} = \frac{ab̄θ^{2} + θ(bb̄ + aā) + bā}{bb̄θ^{2} + θ(ab̄ + bā) + aā}

and note that the denominator is an element of GF (q). Each term in the numerator is an element of GF (q) except for ab̄ and bā, which we can write as follows,

ab̄ = (a_{1} + αa_{2})(b_{1} + ᾱb_{2})
= a_{1}b_{1} + αa_{2}b_{1} + ᾱa_{1}b_{2} + αᾱa_{2}b_{2}
= a_{1}b_{1} + αa_{2}b_{1} + (λ − α)a_{1}b_{2} − μa_{2}b_{2}
= a_{1}b_{1} − μa_{2}b_{2} + λa_{1}b_{2} + α(a_{2}b_{1} − a_{1}b_{2})
= Q + αL (for ease of notation)
and bā = a_{1}b_{1} − μa_{2}b_{2} + λb_{1}a_{2} + α(b_{2}a_{1} − b_{1}a_{2})
= Q̃ + αL̃ (for ease of notation).

(Note that Q, Q̃, L, L̃ are all elements of GF (q).) In Bruck-Bose the point P therefore corresponds to the point P^{∗} in P G(4, q)\Σ_{∞} with homogeneous coordinates:

( \frac{Qθ^{2} + θ(bb̄ + aā) + Q̃}{bb̄θ^{2} + θ(ab̄ + bā) + aā} ; \frac{Lθ^{2} + L̃}{bb̄θ^{2} + θ(ab̄ + bā) + aā} ; \frac{Q̃θ^{2} + θ(bb̄ + aā) + Q}{aāθ^{2} + θ(ab̄ + bā) + bb̄} ; \frac{L̃θ^{2} + L}{aāθ^{2} + θ(ab̄ + bā) + bb̄} ; 1 )

(in [6] Buekenhout applied a similar transformation to unitals in P G(2, q^{2}) to find their image in Bruck-Bose via coordinates in P G(4, q)). If we multiply through by

(bb̄θ^{2} + θ(ab̄ + bā) + aā) (aāθ^{2} + θ(ab̄ + bā) + bb̄) ,

which is a non-zero element of GF (q), we obtain each component of the coordinate vector as a degree 4 polynomial in θ over GF (q), where θ ∈ GF (q) ∪ {∞}. On simplification, the points of C represented in Bruck-Bose are the points with coordinates given by:

[ \begin{matrix}aāQ & \begin{matrix}(ab̄ + bā)Q \\ +aā(bb̄ + aā) \end{matrix} & \begin{matrix}bb̄Q + aāQ̃ \\ +(bb̄ + aā)(ab̄ + bā) \end{matrix} & \begin{matrix}(ab̄ + bā)Q̃ \\ +bb̄(aā + bb̄) \end{matrix} & bb̄Q̃ \\ aāL & (ab̄ + bā)L & bb̄L + aāL̃ & (ab̄ + bā)L̃ & bb̄L̃ \\ bb̄Q̃ & \begin{matrix}(ab̄ + bā)Q̃ \\ +bb̄(bb̄ + aā) \end{matrix} & \begin{matrix}bb̄Q + aāQ̃ \\ +(bb̄ + aā)(ab̄ + bā) \end{matrix} & \begin{matrix}(ab̄ + bā)Q \\ +aā(aā + bb̄) \end{matrix} & aāQ \\ bb̄L̃ & (ab̄ + bā)L̃ & bb̄L + aāL̃ & (ab̄ + bā)L & aāL \\ aābb̄ & (bb̄ + aā)(ab̄ + bā) & \begin{matrix}(aā)^{2} + (bb̄)^{2} \\ (ab̄ + bā)^{2} \end{matrix} & (aā + bb̄)(ab̄ + bā) & aābb̄ \end{matrix} ]( \begin{matrix}θ^{4} \\ θ^{3} \\ θ^{2} \\ θ \\ 1 \end{matrix} )(2.4)

for θ ∈ GF (q) ∪ {∞}. Denote by M the coefficient matrix in (2.4) and note that M ∈ GL(5, q) . Here we have shown the set of points of C in Bruck-Bose is the set of images of points of a normal rational curve in P G(4, q), in particular the image of {(θ^{4}, θ^{3}, θ^{2}, θ, 1); θ ∈ GF (q) ∪ {∞}} under the projectivity determined by the matrix M. Matrix M is necessarily non-singular as C^{∗} is a (q + 1)_{4}-arc in P G(4, q) (see the first proof of corollary 2.3.5) and hence C^{∗} is not contained in a hyperplane of P G(4, q). □

It remains to consider the representation in Bruck-Bose of non-degenerate conics which lie in Baer subplanes B of P G(2, q^{2}) for which ℓ_{∞} is a line of B. Let C be such a conic in a Baer subplane B of P G(2, q^{2}) for which ℓ_{∞} is a line of B. Coordinatise the plane so that ℓ_{∞} is the line with equation z = 0 and B is the subplane P G(2, q), then the conic C is defined by a homogeneous quadratic equation Q(x, y, z) in variables x, y, z and with coefficients in GF (q), that is

\begin{aligned}C & = & \{(x, y, z)|Q(x, y, z) = 0\} \\ & = & \{(x, y, 1)|Q(x, y, 1) = 0\} ∪ \{(x, y, 0)|Q(x, y, 0) = 0\}. \end{aligned}

In Bruck-Bose, B is therefore the transversal plane B^{∗} defined by the equations x_{2} = y_{2} = 0 and C is represented by a non-degenerate conic C^{∗}, where

C^{∗} = {(x_{1}, 0, y_{1}, 0, 1)|Q(x_{1}, y_{1}, 1) = 0}

∪{spread elements corresponding to the points of C at infinity}.

Thus C in Bruck-Bose is essentially a non-degenerate conic C^{∗} in the transversal plane B^{∗}.

We have therefore determined the representation in Bruck-Bose of any non-degenerate conic C in a Baer subplane of P G(2, q^{2}). In summary, the Bruck-Bose representation C^{∗} of C is determined and is one of the following:

• a non-degenerate conic in a (transversal) plane of P G(4, q);

• a twisted cubic curve on a Baer ruled cubic surface of P G(4, q);

• a 4-dimensional normal rational curve on a Baer ruled cubic surface of P G(4, q).

2.4 A characterisation of Baer ruled cubic surfaces

In Section 2.2 we reviewed the Bruck-Bose representation of non-affine Baer subplanes of P G(2, q^{2}); such Baer subplanes are represented in Bruck-Bose by certain ruled cubic surfaces which we call Baer ruled cubics. For a given Bruck-Bose representation of P G(2, q^{2}), that is for a fixed regular spread S of Σ_{∞}, let R be the set of all ruled cubic surfaces V_{2}^{3} in P G(4, q) with the property that the intersection V_{2}^{3} ∩ Σ_{∞} in P G(4, q) is a line which is an element of the spread S. There exist more ruled cubic surfaces in R than there exist non-affine Baer subplanes of P G(2, q^{2}). That is, the Baer ruled cubic surfaces in Bruck-Bose constitute a proper subset of all the ruled cubic surfaces in R. In this section we characterise the Baer ruled cubic surfaces, amongst all ruled cubic surfaces in R, for a given regular spread in the Bruck-Bose representation of P G(2, q^{2}) in P G(4, q).

2.4.1 The extended ruled cubic surface

Consider a ruled cubic surface V_{2}^{3} in P G(4, q) as defined in Section 1.7 with the notation introduced there. So V_{2}^{3} has base conic C, line directrix ℓ and associated projectivity ϕ ∈ P GL(2, q). Embed P G(4, q) as a Baer subspace in P G(4, q^{2}). Consider the ruled cubic surface V_{2}^{3} over the extended field. Let ℓ_{∼} be the (unique) line of P G(4, q^{2}) such that ℓ_{∼}∩P G(4, q) = ℓ. Similarly let C_{∼} be the (unique) conic in P G(4, q^{2}) such that C_{∼} ∩ P G(4, q) = C. The plane of C_{∼} is denoted by S_{2}_{∼} and S_{2}_{∼} ∩ P G(4, q) = S_{2}, where S_{2} is the plane of the conic C in P G(4, q). Note that in P G(4, q^{2}) the line ℓ_{∼} is skew to the plane S_{2}_{∼}. Since if not then Σ_{∼} = ⟨ℓ_{∼}, S_{2}_{∼}⟩ is at most a hyperplane of P G(4, q^{2}) and by Theorem 1.3.1, the intersection Σ = Σ_{∼}∩P G(4, q) is either a hyperplane or a plane of P G(4, q). Since the line directrix ℓ and the base conic C of V_{2}^{3} are contained in Σ, we have that the entire ruled cubic V_{2}^{3} is contained in Σ. Recall from Section 1.7 that a ruled cubic surface V_{2}^{3} in P G(4, q) is not contained in any hyperplane of P G(4, q) and so we obtain a contradiction. The associated projectivity of V_{2}^{3} between ℓ and C can be applied to the non-homogeneous coordinates λ_{∼} (λ_{∼} ∈ GF (q^{2})∪{∞}) of points on ℓ_{∼}. We denote this action by ϕ_{∼} and obtain a projective correspondence between points P (λ_{∼}) on ℓ_{∼} and points P (θ_{∼}) on C_{∼} given by,

θ_{∼} = ϕ_{∼}(λ_{∼}), λ_{∼} ∈ GF (q^{2}) ∪ {∞}.

Note that the projectivity ϕ_{∼} restricted to points of ℓ (in P G(4, q)) is simply the projectivity ϕ ∈ P GL(2, q), that is

ϕ_{∼}|_{ℓ} = ϕ,

hence ϕ_{∼} and ϕ are defined by the same 2 × 2 matrix over GF (q). In other words ϕ_{∼} is an element of P GL(2, q). Let G_{∼} be the set of lines P P ϕ_{∼} where the point P ranges over ℓ_{∼}. In this way we obtain a ruled cubic surface in P G(4, q^{2}), which we shall denote by V_{2}^{3}_{∼}, with line directrix ℓ_{∼}, base conic C_{∼} and associated projectivity ϕ_{∼} ∈ P GL(2, q) (see Section 1.7). Since ϕ_{∼}|_{ℓ} = ϕ, we note that

V_{2}^{3}_{∼}|_{P G(4,q)} = V_{2}^{3}

and we call V_{2}^{3}_{∼} the extended ruled cubic of V_{2}^{3}. We denote by the Fröbenius automorphism of GF (q^{2}),

\begin{aligned}: & GF (q^{2}) & ⟶ & GF (q^{2}) \\ & x & ⟼ & x^{q}. \end{aligned}

We also denote by the automorphic collineation of P G(4, q^{2}) induced by the Fröbenius automorphism. The context in which this is done should make the meaning clear.

\begin{aligned}: & P G(4, q^{2}) & ⟶ & P G(4, q^{2}) \\ & P = P (x_{0}, x_{1}, x_{2}, x_{3}, x_{4}) & ⟼ & \overline{P} = \overline{P} (x_{0}^{q}, x_{1}^{q}, x_{2}^{q}, x_{3}^{q}, x_{4}^{q}). \end{aligned}

Note that this collineation of P G(4, q^{2}), which we call the Fröbenius collineation, fixes the Baer subspace P G(4, q) pointwise and hence the ruled cubic surface V_{2}^{3} is fixed pointwise also. Since ϕ_{∼} ∈ P GL(2, q), the ruled cubic surface V_{2}^{3}_{∼} in P G(4, q^{2}) is fixed by the Fröbenius collineation in the following way,

V_{2}^{3}_{∼} = {P P ϕ_{∼}|P ∈ ℓ_{∼}}
= {P (λ_{∼})P (ϕ_{∼}(λ_{∼}))|λ_{∼} ∈ GF (q^{2}) ∪ {∞} (P ∈ ℓ_{∼})}
= {P (λ_{∼}^{q})P (ϕ_{∼}(λ_{∼})^{q})|λ_{∼} ∈ GF (q^{2}) ∪ {∞} (P ∈ ℓ_{∼})}

(The Fröbenius automorphism permutes the elements of GF (q^{2}))

= {P (λ_{∼}^{q})P (ϕ_{∼}(λ_{∼}^{q}))|λ_{∼} ∈ GF (q^{2}) ∪ {∞} (P ∈ ℓ_{∼})}

(since ϕ_{∼} ∈ P GL(2, q))

= \{\overline{P} \overline{P ϕ_{∼}}|P ∈ ℓ_{∼}\}
= \overline{V_{2}^{3}_{∼}}.

Note that since ϕ ∈ P GL(2, q) the generators of V_{2}^{3}, and in particular the generator containing the points of ℓ with non-homogeneous coordinate ∞, are fixed as a set under the Fröbenius collineation.

We now use P G(4, q^{2}) as our setting for a new proof of a result originally proved by Bernasconi and Vincenti [15, Theorem 2.4].

Theorem 2.4.1 [15] A ruled cubic surface V_{2}^{3} of P G(4, q) represents a non-affine Baer subplane of a translation plane π(S), defined by a 1−spread S in a hyperplane Σ_{∞} of P G(4, q) in the usual way, if and only if π(S) is Desarguesian.

Proof: (⇐) The necessary result is Theorem 2.2.9 (see also [19] [90]). (⇒) To begin with, we establish the existence of a 1−spread S in a hyperplane Σ_{∞} of P G(4, q), that is, the Bruck-Bose representation in of a translation plane. This is done using the method of Bernasconi and Vincenti [15], but the proof that the spread S is regular is new. Let V = V_{2}^{3} be a ruled cubic surface in P G(4, q) with line directrix ℓ, base conic C and associated projectivity ϕ ∈ P GL(2, q) as defined in Section 1.7 with the notation introduced there. The conic C lies in a plane of P G(4, q) which we denote by S_{2}. Let t be any external line of C in S_{2}. Put Σ_{∞} = ⟨t, ℓ⟩, that is Σ_{∞} is the hyperplane of P G(4, q) spanned by the pair of skew lines t and ℓ. Since each generator of V joins a point of ℓ and a point of C, each generator g of V intersects Σ_{∞} in a point of ℓ. thus in P G(4, q) the ruled cubic V intersects the hyperplane Σ_{∞} precisely in its line directrix ℓ. By Theorem 1.7.2, since no three generators of V are contained in a hyperplane of P G(4, q), the planes of two distinct conics on V are not contained in a hyperplane. Also, since any two distinct conics on V intersect in a unique point, the planes containing the q^{2} conics on V meet the hyperplane Σ_{∞} in q^{2} distinct and pairwise skew lines. Denote the conics on V by C = C_{1}, C_{2}, …, C_{q^{2}} and denote the planes of these conics by S_{2} = α_{1}, α_{2}, …, α_{q^{2}} respectively. Let α_{i} ∩ Σ_{∞} = ℓ_{i} for i = 1, …, q^{2} and note that the set of lines,

S = {t = ℓ_{1}, ℓ_{2}, …, ℓ_{q^{2}}} ∪ {ℓ}

is a set of q^{2} +1 pairwise skew lines which partition the points of Σ_{∞}. Thus S is a spread of Σ_{∞} and it remains to show that S is a regular spread. Embed P G(4, q) as a Baer subspace in P G(4, q^{2}). Consider the base conic C of V in the plane S_{2}. The spread element t in S_{2} is an external line of C and so intersects C in two points X, \overline{X} in the quadratic extension. The points X, \overline{X} are conjugate with respect to the quadratic extension in the sense that the set \{X, \overline{X}\} is fixed by the Fröbenius collineation. Note that X, \overline{X} are points on the base conic of the extended ruled cubic V_{2}^{3}_{∼} in P G(4, q^{2}). Consider the points on ℓ_{∼} which correspond to X = P (θ), \overline{X} = P (\overline{θ}) via the associated projectivity ϕ_{∼} of V_{2}^{3}_{∼}. For some fixed λ ∈ GF (q^{2}) ∪ {∞}, we have

θ = ϕ(λ)
and therefore \overline{θ} = \overline{ϕ(λ)}
= ϕ(\overline{λ}) since ϕ ∈ P GL(2, q).

Hence the points \{X, \overline{X}\} on C_{∼} are in projective correspondence with points \{A = P (λ), \overline{A} = P (\overline{λ})\} on ℓ_{∼} and A, \overline{A} are conjugate with respect to the quadratic extension. Let g, \overline{g} denote the pair of generators XA and \overline{X}\overline{A} of the ruled cubic surface V_{2}^{3}_{∼}. The lines g, \overline{g} lie in the quadratic extension Σ_{∞}_{∼} of the hyperplane Σ_{∞} and are disjoint from Σ_{∞}. By Theorem 1.9.6, g, \overline{g} determine a unique regular spread of Σ_{∞} consisting of the q^{2} +1 lines of Σ_{∞} obtained by joining each point of g with its conjugate on \overline{g}. We denote this regular spread by S_{g\overline{g}}. Note that t = X\overline{X} and ℓ = A\overline{A} are elements of the regular spread S_{g\overline{g}}. For i ≠ 1, consider the conic C_{i} on V. The conic lies in a plane α_{i} which contains the spread element ℓ_{i} of S. Since ℓ_{i} is an external line of C_{i} in the plane of the conic, the intersection ℓ_{i} ∩ C_{i} is a pair X_{i}, \overline{X_{i}} of points conjugate with respect to the quadratic extension. Thus in P G(4, q^{2}) the spread line ℓ_{i} contains the points X_{i}, \overline{X_{i}} of the extended ruled cubic V_{2}^{3}_{∼} and the conic C_{i} extends uniquely to a conic C_{i}_{∼} contained in V_{2}^{3}_{∼}. By Theorem 1.7.2, the conic C_{i}_{∼} intersects each generator of the extended ruled cubic, in particular C_{i}_{∼} contains a point of g and a point of \overline{g}. But since the plane of the conic C_{i}_{∼} is not contained in Σ_{∞}_{∼} (as α_{i} is not contained in Σ_{∞}) it follows that

\{g, \overline{g}\} ∩ C_{i} = \{g, \overline{g}\} ∩ ℓ_{i} = \{X_{i}, \overline{X_{i}}\}.

Thus the spread S is the unique regular spread S_{g\overline{g}} of Σ_{∞} determined by lines g, \overline{g} in the quadratic extension of Σ_{∞}. The Bruck-Bose incidence structure π(S) is therefore a Desarguesian plane of order q^{2} and the ruled cubic surface V is a Baer ruled cubic with respect to the regular spread S. □

In the following characterisation of Baer ruled cubic surfaces, S_{g\overline{g}} denotes a regular 1−spread of Σ_{∞} = P G(3, q) determined in the usual way (see Theorem 1.9.6) by a pair of lines g, \overline{g} in the quadratic extension of Σ_{∞}. Recall that a Baer ruled cubic surface in Bruck-Bose is a ruled cubic surface which represents via Bruck-Bose a non-affine Baer subplane of the corresponding Desarguesian plane.

Theorem 2.4.2 A characterisation of Baer ruled cubic surfaces: Let P G(2, q^{2}) have Bruck-Bose representation π(S) ⊆ P G(4, q), for a fixed regular spread S = S_{g\overline{g}}. A ruled cubic V_{2}^{3} in P G(4, q) is a Baer ruled cubic surface if and only if V_{2}^{3} has line directrix an element of S and such that the extended ruled cubic of V_{2}^{3} in P G(4, q^{2}) contains the lines g and \overline{g} as generators.

Proof: (⇒) The necessary result follows from the proof of Theorem 2.4.1 and Theorem 1.9.6 which states that a regular spread of P G(3, q) is determined by a unique pair of conjugate lines in the quadratic extension. (⇐) We count the number of ruled cubic surfaces in P G(4, q) which have line directrix an element of S and such that the extended ruled cubic in P G(4, q^{2}) contains g and \overline{g} as generators. We show that the number of such ruled cubics equals the number of non-affine Baer subplanes of P G(2, q^{2}). Consider the Bruck-Bose representation π(S) of P G(2, q^{2}), where S = S_{g\overline{g}}. Let ℓ and t be two distinct elements of the spread S. Let \{X\} = g ∩ t, and \{\overline{X}\} = \overline{g} ∩ t denote the points of t on lines g and \overline{g} respectively, in the quadratic extension. Let α be any plane of P G(4, q)\Σ_{∞} which contains the line t. Let C be a non-degenerate conic in α such that C ∩ t = \{X, \overline{X}\}. In particular, note that t is an external line of C in the plane α in P G(4, q). Let m be a line of P G(4, q) joining a point of ℓ with a point of C. Consider the situation in the quadratic extension P G(4, q^{2}): we have a line ℓ_{∼} and a conic C_{∼} such that the plane α_{∼} of the conic is skew to the line ℓ_{∼}. The three lines g, \overline{g} and m associate three distinct points of ℓ_{∼} with three distinct points of C_{∼} and so define a unique projectivity ϕ_{∼} of P GL(2, q^{2}) between ℓ_{∼} and C_{∼}. By Section 1.7 we have a ruled cubic surface V_{2}^{3}_{∼} in P G(4, q^{2}) with line directrix ℓ_{∼}, base conic C_{∼} and associated projectivity ϕ_{∼} ∈ P GL(2, q^{2}). Under the Fröbenius collineation ℓ_{∼}, C_{∼} and the generator m of V_{2}^{3}_{∼} are fixed, since ℓ, C and m are contained in P G(4, q). Also the pair of generators \{g, \overline{g}\} of V_{2}^{3}_{∼} are fixed as a set. Thus V_{2}^{3}_{∼} and its image \overline{V_{2}^{3}_{∼}} under the Fröbenius collineation are a pair of ruled cubic surfaces in P G(4, q^{2}) with the same line directrix, base conic and which share three distinct generators. Since the projectivity ϕ_{∼} is determined uniquely by the three generators g, \overline{g} and m, it follows that V_{2}^{3}_{∼} and \overline{V_{2}^{3}_{∼}} both have ϕ_{∼} as associated projectivity. Hence,

V_{2}^{3}_{∼} = \overline{V_{2}^{3}_{∼}}

and therefore the points of ℓ (in P G(4, q)) are associated by ϕ_{∼} to points of C (in P G(4, q)). Thus ϕ_{∼} restricted to points of ℓ (in P G(4, q)) defines a ruled cubic surface V_{2}^{3} in P G(4, q) with line directrix ℓ and base conic C. We have therefore determined that the projectivity ϕ_{∼} is an element of P GL(2, q) ⊆ P GL(2, q^{2}). Moreover the ruled cubic V_{2}^{3} in P G(4, q) determined above has a line directrix an element of the spread S and has extended ruled cubic V_{2}^{3}_{∼} which contains lines g and \overline{g} as generators. Since V_{2}^{3} is determined uniquely by the choice of line directrix ℓ, base conic C and generator m, we count the number of such ruled cubic surfaces in P G(4, q) as follows. In the following C denotes a non-degenerate conic in AG(2, q) containing a fixed pair of special points, conjugate with respect to the quadratic extension.

|S| × \frac{(|S|−1)×|\{ planes of P G(4, q)\Σ_{∞} containing a spread element\}|×|\{C\}|}{|\{conics on a ruled cubic surface in P G(4, q)\}|} × \frac{(q+1)^{2}}{q+1}

= (q^{2} + 1) × \frac{(q^{2})(q^{2})(q^{3}−q^{2})}{q^{2}} × (q + 1)
= q^{4}(q^{4} − 1).

Now P G(2, q^{2}) = π(S) contains precisely this many non-affine Baer subplanes since P G(2, q^{2}) contains (q^{2} −q +1)(q^{2} +1)q^{3}(q +1) Baer subplanes of which q^{3}(q^{3} +q^{2} +q +1) contain the line at infinity as a line. The result now follows. □

Corollary 2.4.3 A characterisation of Baer conics: Let P G(2, q^{2}) have Bruck-Bose representation π(S) ⊆ P G(4, q), for a fixed regular spread S = S_{g\overline{g}}. A non-degenerate conic C in P G(4, q) is a Baer conic if and only if C is disjoint from Σ_{∞} in P G(4, q) and such that in the quadratic extension C contains a pair of conjugate points, X, \overline{X} say, on the lines g and \overline{g}. □

We have now completely determined the Bruck-Bose representation of the Baer substructures of P G(2, q^{2}). The motivation for this work came from a paper by Jef Thas in which the plane model of a Miquelian inversive plane of order q is given. Consider the Miquelian inversive plane I with points the points of a line P G(1, q^{2}) and with circles the Baer sublines of P G(1, q^{2}). For a fixed point P of P G(1, q^{2}) consider the internal plane I_{P} ≅ AG(2, q). By Section 1.14, the circles in I which do not contain P correspond precisely to the non-degenerate conics in I_{P} which contain a fixed pair X, \overline{X} of of special points, conjugate with respect to the quadratic extension. By the above characterisation of Baer conics, this plane model of Miquelian inversive planes is evident in the Bruck-Bose representation of P G(2, q^{2}) for each affine line ℓ (≅ P G(1, q^{2})) of P G(2, q^{2}) and letting {P } = ℓ ∩ ℓ_{∞}.

2.5 Additional properties

In this section we present a result, valid for q even, which determines some properties of the q^{2} nuclei associated with the q^{2} Baer conics on a Baer ruled cubic surface in the Bruck-Bose representation of P G(2, q^{2}) in P G(4, q).

Result 2.5.1 Let B be a Baer ruled cubic in the Bruck-Bose representation of P G(2, q^{2}), q even. Let B be the Baer subplane of P G(2, q^{2}) which is represented by B in Bruck-Bose. The q^{2} nuclei associated with the q^{2} Baer conics of B are distinct and lie in a plane of P G(4, q)\Σ_{∞} about the line directrix p of B.

Proof: Let p denote the line directrix of B. Let C_{1}^{∗} be a Baer conic on B in plane α_{1} of P G(4, q). Let m_{1} denote the unique spread element contained in α_{1}. Since q is even, we denote the nucleus of C_{1}^{∗} in α_{1} by N_{1}^{∗}. Note that as m_{1} is an external line to the conic C_{1}^{∗}, the nucleus N_{1}^{∗} is not incident with m_{1}. We have therefore that the nucleus of a Baer conic is an affine point of P G(4, q), that is, a point in P G(4, q)\Σ_{∞}. Let Q^{∗} be a point on C_{1}^{∗}; Q^{∗} represents a point Q of B in P G(2, q^{2}). In the proof of Theorem 2.2.9 it is shown that the Baer ruled cubic surface B is contained in the intersection of two quadric cones, namely V_{3}^{2}, with line vertex p and base the conic C_{1}^{∗}, and V_{3}^{′2}, with point vertex Q^{∗} and base the hyperbolic quadric in Σ_{∞} determined by the regulus of spread elements which represent the points at infinity in P G(2, q^{2}) of the lines of B incident with Q. Let P_{1}^{∗}, P_{2}^{∗}, …, P_{q+1}^{∗} denote the q + 1 distinct points of the line directrix p of B and denote by g_{1}^{∗}, g_{2}^{∗}, … g_{q+1}^{∗} the generators of B, labelled so that P_{i}^{∗} is incident with g_{i}^{∗}, i = 1, 2, …, q + 1. Then P_{1}^{∗}C_{1}^{∗} is a conic cone in a hyperplane, Π_{3}^{1} say, and by Theorem 2.3.1 the intersection Π_{3}^{1} ∩ B is the union of the conic C_{1}^{∗} and the generator g_{1}^{∗} of B. In Π_{3}^{1}, the line P_{1}^{∗}N_{1}^{∗} is the nuclear line of the conic cone P_{1}^{∗}C_{1}^{∗}, that is, every non-degenerate conic C in a plane α and such that C is the plane section α ∩ P_{1}^{∗}C_{1}^{∗}, has nucleus the point α ∩ {P_{1}^{∗}N_{1}^{∗}}. By considering the conic cones P_{i}^{∗}C_{1}^{∗}, i = 1, 2, …, q + 1, which are all contained in the quadric V_{3}^{2}, and by repeating the above argument, we have that each non-degenerate conic in V_{3}^{2} has a nucleus incident with the plane ⟨N_{1}^{∗}, p⟩ about p in P G(4, q). Moreover, no such nucleus is incident with p. Since B ⊆ V_{3}^{2} ∩ V_{3}^{′2} and since B contains q^{2} distinct Baer conics, the q^{2} associated nuclei of these conics lie in the plane ⟨N_{1}^{∗}, p⟩. It remains to show that these nuclei are distinct, so that they constitute all q^{2} points of ⟨N_{1}^{∗}, p⟩ not incident with p. Consider two distinct Baer conics C^{∗} and C^{′∗} of B in planes α and α′ respectively. Since α and α′ represent lines of B in P G(2, q^{2}) and since B is not contained in a hyperplane of P G(4, q), the planes α and α′ intersect in a unique point and this point of intersection is in B (see Theorem 1.7.2). Hence α, α′ have no point in common which is not a point of B, hence the Baer conics C^{∗}, C^{′∗} have distinct nuclei. □

2.6 An Alternative Approach

The Ruled Cubic Surface R_{2}^{3} as a model for PG(2, q)

In this section we discuss the ruled cubic surface obtained as the projection of the Veronese Surface V_{2}^{4} from any one of its points. The Veronese Surface is the variety

V_{2}^{4} = {P (x^{2}, xy, y^{2}, xz, yz, z^{2}) | (x, y, z) a point of P G(2, q)}

of P G(5, q). It is of order 4 and dimension 2. (In [50, Section 25.1], the Veronese Surface is referred to as the quadric Veronesean of P G(2, q).) If we write (x_{0}, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}) for the coordinates of a general point in P G(5, q), then V_{2}^{4} is the complete intersection of the quadrics

\begin{aligned}x_{1}^{2} − x_{0}x_{2} & = & 0, & x_{0}x_{4} − x_{1}x_{3} & = & 0, \\ x_{3}^{2} − x_{0}x_{5} & = & 0, & x_{1}x_{5} − x_{3}x_{4} & = & 0, \\ x_{4}^{2} − x_{2}x_{5} & = & 0, & x_{2}x_{3} − x_{1}x_{4} & = & 0. \end{aligned}

Moreover, the Veronese Surface contains no lines, that is, V_{2}^{4} is a cap in P G(5, q).

The map

ζ : P G(2, q) ⟶ P G(5, q)
defined by (x, y, z) ⟼ (x^{2}, xy, y^{2}, xz, yz, z^{2})

is a bijection of P G(2, q) onto the Veronese Surface V_{2}^{4}; hence, V_{2}^{4} contains exactly |V_{2}^{4}| = q^{2} + q + 1 points. Also, under ζ, the points of the conic

ax^{2} + by^{2} + cz^{2} + fyz + gzx + hxy = 0  (2.5)

in P G(2, q) correspond to the points of intersection of the hyperplane of P G(5, q) with coordinates [a, h, b, g, f, c] and the Veronese Surface. Since a curve C^{r} of order r in P G(2, q) intersects a conic in 2r points (see Theorem 1.6.3), it follows that a curve C^{r} of P G(2, q) maps by ζ into a curve of degree 2r on V_{2}^{4}. In particular a line of P G(2, q) maps into an irreducible conic on V_{2}^{4} and an irreducible conic of P G(2, q) maps into an irreducible curve of order 4. By considering lines of P G(2, q), we have

Theorem 2.6.1 Properties of the Veronese Surface:

1. [50, Theorems 25.1.7, 25.1.9] Let ℓ be any line in P G(2, q). Then ζ(ℓ) is a nondegenerate conic on V_{2}^{4}. Moreover, each non-degenerate conic contained in V_{2}^{4} is of the form ζ(ℓ) for some line ℓ in P G(2, q). Each plane in P G(5, q) which contains a non-degenerate conic on V_{2}^{4} is called a conic plane of V_{2}^{4}.

2. [50, Theorem 25.1.11] Any two conic planes of V_{2}^{4} have exactly one point in common and this common point belongs to V_{2}^{4}.

Thus V_{2}^{4} contains q^{2}+q+1 non-degenerate conics, two distinct points of V_{2}^{4} are contained in a unique non-degenerate conic on V_{2}^{4} and two distinct non-degenerate conics on V_{2}^{4} intersect in a unique point. A degenerate conic which is a repeated line or two distinct lines in P G(2, q) corresponds to a hyperplane section of V_{2}^{4}, where the hyperplane meets V_{2}^{4} in a non-degenerate conic (counted doubly), or two conics with exactly one point in common, respectively. At each point P of V_{2}^{4}, the q + 1 tangent lines to the q + 1 irreducible conics of V_{2}^{4} at P span a plane π(P); π(P) is called the tangent plane of V_{2}^{4} at P, and π(P) ∩ V_{2}^{4} = {P }. Also we note that by [50, Lemma 25.1.6 and Theorem 25.1.10] a projectivity of P G(2, q) induces a permutation of the pointset of V_{2}^{4} which is induced by a unique projectivity of P G(5, q) which fixes V_{2}^{4}. Consult [50, Section 25.1] for further detail regarding the Veronese Surface V_{2}^{4}. We now project V_{2}^{4} from a point of V_{2}^{4} to obtain a ruled cubic surface in a hyperplane of P G(5, q) (see also [71, Section 3.22]). Consider the point P (0, 0, 0, 0, 0, 1) on V_{2}^{4} which is the image under ζ of the point P′(0, 0, 1) of P G(2, q). The tangent plane π(P) to V_{2}^{4} at P is given by the equations x_{0} = x_{1} = x_{2} = 0. Project V_{2}^{4} from P onto the hyperplane Π_{4} with equation x_{5} = 0. Since P is incident with q + 1 conic planes of V_{2}^{4}, which pairwise meet in P, the given projection of V_{2}^{4} from P yields q points of each of q + 1 distinct lines g_{1}, g_{2}, …, g_{q+1} in Π_{4}. The q + 1 remaining points on g_{1}, g_{2}, …, g_{q+1} (one on each line) are collinear in a line ℓ of Π_{4}, where ℓ is the projection from P of the tangent plane π(P) of V_{2}^{4} at P. The projection of V_{2}^{4} from P onto the hyperplane Π_{4} with equation x_{5} = 0, is then the set

{(x^{2}, xy, y^{2}, xz, xy, 0) | (x^{2}, xy, y^{2}, xz, yz, z^{2}) is a point of V_{2}^{4}}

of q^{2} + q points of Π_{4}. By Section 1.7, these points are q^{2} + q points of a ruled cubic surface with line directrix ℓ in Π_{4}. For the following discussion, we recall that a ruled cubic surface is defined as follows (see Section 1.7).

Definition: 2.6.1 In Π_{4} = P G(4, q), consider a conic C^{2} and a line ℓ skew to the plane of C^{2}. Set up a projective correspondence between them, and join corresponding points by lines. The ruled surface so obtained is of order 3, and is denoted R_{2}^{3}.

By choosing the coordinate system in P G(4, q), let point (0, 0, 0, x, y) (where x, y ∈ GF (q), (x, y) ≠ (0, 0)) on ℓ correspond to point (x^{2}, xy, y^{2}, 0, 0) on C^{2}. Thus R_{2}^{3} is

{(x^{2}, xy, y^{2}, zx, zy); x, y ∈ GF (q), (x, y) ≠ (0, 0), z ∈ GF (q) ∪ {∞} }

Define

σ : R_{2}^{3} → P G(2, q)

by

\begin{matrix}(x^{2}, xy, y^{2}, zx, zy) & ⟼ & (x, y, z) \\ ℓ (z = ∞) & ⟼ & (0, 0, 1) \end{matrix}

Thus σ contracts ℓ into the point (0, 0, 1), and

σ : R_{2}^{3}\ℓ → P G(2, q)\{(0, 0, 1)}

is a bijection, by definition of σ. In an abuse of notation, we shall use σ to denote the map between R_{2}^{3}\ℓ, in P G(4, q), and P G(2, q)\{(0, 0, 1)}, in both directions. First note that under σ our original conic

C^{2} = {(x^{2}, xy, y^{2}, 0, 0)| x, y ∈ GF (q), (x, y) ≠ (0, 0)}

on R_{2}^{3} is mapped to the line z = 0 of P G(2, q). We now consider the image in P G(4, q), under σ, of lines and conics in P G(2, q). One method is to use the bijection ζ of P G(2, q) onto the Veronese Surface V_{2}^{4} and then project V_{2}^{4} from P (0, 0, 0, 0, 0, 1) onto the ruled cubic R_{2}^{3}; for clarity we explicitly determine these images using our bijection σ of P G(2, q)\{(0, 0, 1)} onto R_{2}^{3}\ℓ. First consider a line ax + by + cz = 0 (a, b, c ∈ GF (q), c ≠ 0) in P G(2, q), not through (0, 0, 1). A parametric form of this line is

(x, y, z) ≡ (1, t, \frac{−a − bt}{c}),

where t ∈ GF (q) ∪ {∞}. Using the map σ we have

(x^{2}, xy, y^{2}, zx, zy) = (x_{0}, x_{1}, x_{2}, x_{3}, x_{4})  (2.6)
= (1, t, t^{2}, \frac{−a − bt}{c}, \frac{−at − bt^{2}}{c} ).  (2.7)

Thus the image in P G(4, q) of the line of P G(2, q) is the set of points (x_{0}, x_{1}, x_{2}, x_{3}, x_{4}) on R_{2}^{3} with the parametrisation (2.7) in quadratic functions of t. These points (2.7) also satisfy,

ax_{0} + bx_{1} + cx_{3} = 0 = x(ax + by + cz)
and ax_{1} + bx_{2} + cx_{4} = 0 = y(ax + by + cz);  (2.8)

the equations of two distinct hyperplanes, which intersect in a plane of P G(4, q). By Section 1.5, we have therefore that the image of the line ax + by + cz = 0 (c ≠ 0) is the set of points (x_{0}, x_{1}, x_{2}, x_{3}, x_{4}) (2.7) of a conic on R_{2}^{3} lying in the plane of P G(4, q) defined by equations (2.8). Furthermore, the conic (2.7) is non-degenerate since it is the pre-image of the non-degenerate conic {(1, t, t^{2}, 0, 0)| t ∈ GF (q) ∪ {∞}}, in the plane x_{3} = x_{4} = 0 of P G(4, q), under the projectivity of P G(4, q) defined by the matrix,

[ \begin{matrix}1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ a & b & 0 & c & 0 \\ 0 & a & b & 0 & c \end{matrix} ] .

Note: 2.6.2 In P G(2, q), projectivities fixing (0, 0, 1) induce a transformation on (x^{2}, xy, y^{2}, zx, zy) in P G(4, q). As a consequence, the “conics” in P G(4, q), which are the image under σ of lines ax + by + cz = 0 (a, b, c ∈ GF (q), c ≠ 0), are projectively equivalent.

Proof: Consider two lines ℓ_{1} and ℓ_{2} in P G(2, q) with equations ax + by + cz = 0, c ≠ 0, and a′x + b′y + c′z = 0, c′ ≠ 0, respectively; note that (0, 0, 1) is not incident with either of these two lines. A projectivity fixing (0, 0, 1) in P G(2, q) and which maps ℓ_{1} to ℓ_{2} is given by

[ 1 ; t ; \frac{−a′−b′t}{c′} ] = [ \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ \frac{a}{c} − \frac{a′}{c′} & \frac{b}{c} − \frac{b′}{c′} & 1 \end{matrix} ] [ 1 ; t ; \frac{−a−bt}{c} ] .

The corresponding projectivity in P G(4, q) is therefore given by

[ 1 ; t ; t^{2} ; \frac{−a′−b′t}{c′} ; \frac{−a′t−b′t^{2}}{c′} ] = [ \begin{matrix}1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ \frac{a}{c} − \frac{a′}{c′} & \frac{b}{c} − \frac{b′}{c′} & 0 & 1 & 0 \\ 0 & \frac{a}{c} − \frac{a′}{c′} & \frac{b}{c} − \frac{b′}{c′} & 0 & 1 \end{matrix} ] [ 1 ; t ; t^{2} ; \frac{−a−bt}{c} ; \frac{−at−bt^{2}}{c} ] .

□

Now consider a line ax + by = 0 (a, b ∈ GF (q), a, b not both zero) in P G(2, q), that is, a line incident with (0, 0, 1). Imposing the condition ax + by = 0 on (2.6) gives

ax_{0} + bx_{1} = 0,
ax_{1} + bx_{2} = 0,
and ax_{3} + bx_{4} = 0  (2.9)

which define the equation of a line in P G(4, q) contained in R_{2}^{3} and therefore incident with ℓ. Thus the image of the line ax + by = 0 under σ is a (generator) line of R_{2}^{3}. Consider the following linear combination of the equations (2.9),

\begin{aligned}& a(ax_{0} + bx_{1}) + b(ax_{1} + bx_{2}) + c(ax_{3} + bx_{4}) & = & 0, \\ that is & a(ax_{0} + bx_{1} + cx_{3}) + b(ax_{1} + bx_{2} + cx_{4}) & = & 0, \end{aligned}

which is a linear combination of the equations (2.8) and is therefore the equation of the hyperplane of P G(4, q) which meets R_{2}^{3} in the union of the conic (2.7) and the generator line (2.9). A pencil of lines in P G(2, q) through a fixed point (x, y, z) ≠ (0, 0, 1) corresponds to a collection of q conics on the surface R_{2}^{3} together with a generator line of R_{2}^{3}. The planes containing these q conics generate a quadric cone in P G(4, q) with point vertex (x^{2}, xy, y^{2}, zx, zy) in R_{2}^{3}; this is proved as follows, the plane (2.8), where a, b, c satisfy ax + by + cz = 0 for some fixed (x, y, z), passes through the point (x_{0}, x_{1}, x_{2}, x_{3}, x_{4}) of P G(4, q) if and only if

| \begin{matrix}x_{0} & x_{1} & x_{3} \\ x_{1} & x_{2} & x_{4} \\ x & y & z \end{matrix} | = 0

that is, if and only if x(x_{1}x_{4} − x_{2}x_{3}) + y(x_{1}x_{3} − x_{0}x_{4}) + z(x_{0}x_{2} − x_{1}^{2}) = 0. (2.10)

It is easy to show that the quadric with equation (2.10) has the (fixed) point (x^{2}, xy, y^{2}, zx, zy) as a singular point, by showing the first partial derivatives of the defining polynomial are identically zero at (x^{2}, xy, y^{2}, zx, zy), hence by considering all possible quadrics in P G(4, q), the quadric (2.10) in P G(4, q) is a quadric cone with point vertex (x^{2}, xy, y^{2}, zx, zy) and base a hyperbolic quadric. It also follows that R_{2}^{3} is the complete intersection of the three quadrics given by,

x_{1}x_{4} − x_{2}x_{3} = 0, x_{1}x_{3} − x_{0}x_{4} = 0, and x_{0}x_{2} − x_{1}^{2} = 0.  (2.11)

Through a point of P G(4, q) not contained on the surface R_{2}^{3}, that is, not satisfying (2.11), there passes a unique plane of the system (2.8), namely the plane such that

a : b : c = x_{1}x_{4} − x_{2}x_{3} : x_{1}x_{5} − x_{0}x_{4} : x_{0}x_{2} − x_{1}^{2}.  (2.12)

This is a plane of an irreducible conic unless x_{0}x_{2} − x_{1}^{2} = 0, in which case, the conic is the union of the line directrix ℓ and a generator line of R_{2}^{3}.

Hyperplane sections of R_{2}^{3}

As stated in Theorem 1.7.2 and in Section 1.6, the intersection of a hyperplane of P G(4, q) with the ruled cubic surface R_{2}^{3} is a cubic curve; such a cubic curve is possibly reducible, in which case the intersection is either the union of a conic and a generator line of R_{2}^{3} or is the union of three lines of R_{2}^{3} (not necessarily distinct or belonging to GF (q)). By considering R_{2}^{3} as the projection of the Veronese Surface V_{2}^{4} in P G(5, q) from a point of V_{2}^{4}, it is possible to determine the nature of the hyperplane sections of R_{2}^{3} in P G(4, q). We work through this explicitly using our bijection σ between P G(2, q)\{(0, 0, 1)} and R_{2}^{3}\{ℓ}. Let λ̄= [λ_{0}, λ_{1}, λ_{2}, λ_{3}, λ_{4}], λ_{i} ∈ GF (q) not all zero, be the coordinates of a hyperplane of P G(4, q). The hyperplane intersects R_{2}^{3} in points (x^{2}, xy, y^{2}, xz, yz) satisfying

λ_{0}x^{2} + λ_{1}xy + λ_{2}y^{2} + λ_{3}zx + λ_{4}zy = 0  (2.13)

which corresponds to the points of a conic through (0, 0, 1) in P G(2, q). The number of conics in P G(2, q) through (0, 0, 1) equals

q^{4} − q^{2} + \binom{q + 1}{2} + \binom{q}{2} + q^{2}(q + 1) + q + 1 = q^{4} + q^{3} + q^{2} + q + 1,

counting non-degenerate conics and the four types of degenerate conics incident with (0, 0, 1). Hence there is a one-to-one correspondence between the conics incident with (0, 0, 1) in P G(2, q) and the hyperplane sections of the cubic surface R_{2}^{3} in P G(4, q). Thus, R_{2}^{3} is the projective model of the system of conics incident with (0, 0, 1) in the plane. We now consider each case in more detail. Case 1: (2.13) is the equation of a non-degenerate conic in P G(2, q). By considering the first polar of (0, 0, 1) with respect to (2.13), the tangent line to the conic at (0, 0, 1) is given by

λ_{3}x + λ_{4}y = 0.

The condition for the hyperplane λ̄ to intersect ℓ in (0, 0, 0, 1, m), m ∈ GF (q) ∪ {∞}, is

\begin{aligned}λ_{3} + λ_{4}m & = & 0 \\ that is, m & = & −\frac{λ_{3}}{λ_{4}} \end{aligned}

and so in P G(2, q), from above, y = mx is the tangent line to the conic (2.13) at (0, 0, 1).

Case 2: (2.13) is degenerate and λ_{3} = λ_{4} = 0. The hyperplane λ̄= [λ_{0}, λ_{1}, λ_{2}, 0, 0] in P G(4, q) contains the line ℓ of R_{2}^{3}. The equation (2.13) becomes

\begin{aligned}λ_{0}x^{2} + λ_{1}xy + λ_{2}y^{2} & = & 0 \\ that is, (y − m_{1}x)(y − m_{2}x) & = & 0 \end{aligned}

and the lines y = m_{1}x and y = m_{2}x may be distinct or coincident in P G(2, q) or lie in an extension of the base field. \underline{Case 2a: m_{1} = m_{2} ∈ GF (q) ∪ \{∞\}} The hyperplane λ̄ intersects R_{2}^{3} in ℓ together with a unique generator line counted twice. In P G(2, q), we have a degenerate conic on (0, 0, 1) consisting of a (repeated) line y = m_{1}x. \underline{Case 2b: m_{1} ≠ m_{2}, m_{1}, m_{2} ∈ GF (q) ∪ \{∞\}} The hyperplane λ̄ intersects R_{2}^{3} in ℓ together with two generator lines. In P G(2, q), we have a degenerate conic on (0, 0, 1) consisting of two distinct lines y = m_{1}x and y = m_{2}x. Case 2c: m_{1} and m_{2} are conjugate elements of GF (q^{2})\GF (q). The hyperplane λ̄ intersects R_{2}^{3} in ℓ together with two generator lines in the quadratic extension, that is, ℓ together with two generators of the extended ruled cubic surface (as discussed in Section 2.4.1). In P G(2, q), we have a degenerate conic on (0, 0, 1) consisting of two conjugate lines in the quadratic extension. Case 3: (2.13) is degenerate and λ̄= [−ma, a, b, −m, 1]. The equation (2.13) degenerates to become

(y − mx)(ax + by + z) = 0

for some m ∈ GF (q) ∪ {∞}, a, b ∈ GF (q). From above, the hyperplane λ̄ intersects R_{2}^{3} in a generator line and a conic on R_{2}^{3}. In P G(2, q), we have a degenerate conic on (0, 0, 1) consisting of two lines, y = mx (incident with (0, 0, 1)) and ax + by + z = 0 (not incident with (0, 0, 1)).

We now consider in more detail a non-degenerate conic through P′(0, 0, 1) in the plane P G(2, q). By Section 1.5, such a conic has points with parametric coordinates of the form (f_{0}(t), f_{1}(t), f_{2}(t)) = (a_{1} + b_{1}t, a_{2} + b_{2}t, a_{3} + b_{3}t + c_{3}t^{2}), a_{i}, b_{i} ∈ GF (q), where P′ is the point of the conic associated with the parameter t = ∞. Since the conic is nondegenerate it is a normal rational curve in the plane and therefore the polynomials f_{i} are linearly independent and have non non-trivial common factor. When the coordinates of the points of the conic are substituted into (2.6) we obtain a parametrisation of the corresponding points (x_{0}, x_{1}, x_{2}, x_{3}, x_{4}) on R_{2}^{3} in which x_{0}, x_{1}, x_{2} are quadratic in t and x_{3}, x_{4} are cubic in t, namely

\begin{aligned}x_{0} & = x^{2} = & (a_{1} + b_{1}t)^{2}, \\ x_{1} & = xy = & (a_{1} + b_{1}t)(a_{2} + b_{2}t), \\ x_{2} & = y^{2} = & (a_{2} + b_{2}t)^{2}, \\ x_{3} & = zx = & (a_{1} + b_{1}t)(a_{3} + b_{3}t + c_{3}t^{2}), \\ x_{4} & = zy = & (a_{2} + b_{2}t)(a_{3} + b_{3}t + c_{3}t^{2}). \end{aligned}

Thus the conic corresponds to a rational cubic curve on the surface R_{2}^{3}. By case 1, these q+1 points lie in a hyperplane section of R_{2}^{3}. Note that all possible reducible cubic curves on R_{2}^{3} have been considered in cases 2 and 3, with each such cubic curve obtained as a hyperplane section of R_{2}^{3}. Therefore by Section 1.5 and Theorem 1.7.2, the rational cubic curve above is an (irreducible) twisted cubic curve. Note that the point with parameter t = ∞ of this twisted cubic lies on the line directrix ℓ of R_{2}^{3}. Alternatively, by the known results concerning the Veronese Surface V_{2}^{4} in P G(5, q) quoted above, we note that such a non-degenerate conic containing P′(0, 0, 1) in P G(2, q^{2}) corresponds to a rational quartic curve containing the point P (0, 0, 0, 0, 0, 1) with points

(x_{0}, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}) = (f_{0}^{2}(t), f_{0}(t)f_{1}(t), f_{1}^{2}(t), f_{0}(t)f_{2}(t), f_{1}(t)f_{2}(t), f_{2}^{2}(t))

and which constitute a hyperplane section of the Veronese Surface V_{2}^{4} of P G(5, q) containing P. On projection from P onto R_{2}^{3}, this quartic curve containing P projects to a cubic curve contained in a 3−space. Consider a non-degenerate conic in P G(2, q) which does not contain P′(0, 0, 1). For example z^{2} = xy has points with coordinates (1, t^{2}, t), where t ∈ GF (q) ∪ {∞}. When we substitute these coordinates into (2.6) we obtain the coordinates

(x_{0}, x_{1}, x_{2}, x_{3}, x_{4}) = (1, t^{2}, t^{4}, t, t^{3})

of points of a 4−dimensional normal rational curve contained on the surface R_{2}^{3} and disjoint from the line directrix ℓ. We now show that every non-degenerate conic in P G(2, q) which does not contain the point P′(0, 0, 1) is mapped by σ to a 4−dimensional normal rational curve on R_{2}^{3} disjoint from the line directrix ℓ.

Curves on R_{2}^{3}

By Section 1.6, every curve C^{2m} in P G(2, q), of order 2m, with m-fold point at (0, 0, 1) has equation

z^{m}u_{m} + z^{m−1}u_{m+1} + … + u_{2m} = 0  (2.14)

where u_{i} ≡ u_{i}(x, y) (i = m, …, 2m) is homogeneous of degree i in x and y. As we shall show, the equation (2.14) can be expressed as a monomial of degree m in x^{2}, xy, y^{2}, zx, yz, so that the equation of the curve transforms by (2.6) into that of a primal (or hypersurface) V_{3}^{m} of order m in P G(4, q). The image of the curve C^{2m} in P G(2, q) under σ is V_{3}^{m} ∩ R_{2}^{3} = C_{1}^{3m}, a curve of order 3m in P G(4, q) on the ruled cubic R_{2}^{3}. For example, the equation of C^{2m} with m-fold point at P′(0, 0, 1) is

z^{m}u_{m} + … + z^{0}u_{2m} = 0

where z^{m−i}u_{m+i} = z^{m−i}(ρ_{0}x^{m+i} + ρ_{1}x^{m+i−1}y + … + ρ_{j}x^{m+i−j}y^{j} + … + ρ_{m+i}y^{m+i}) for i = 0, …, m and j = 0, …, m + i for a fixed i and some ρ_{j} ∈ GF (q). Now

\begin{aligned}z^{m−i}x^{m+i−j}y^{j} & = & (zx)^{m−i}x^{2(i−j)}(xy)^{j} & for i ≥ j \\ & = & (zx)^{m−j}(zy)^{j−i}(xy)^{i} & for i < j \end{aligned}

hence (2.14) can be expressed as a monomial of degree m in x^{2}, xy, y^{2}, zx, yz. If a curve C^{n} in P G(2, q) has an m-fold point at (0, 0, 1), but is of order n < 2m, it can be turned into a curve of order n′ = 2m with an m-fold point at (0, 0, 1) by adding 2m−n lines not through (0, 0, 1). We then obtain a curve C^{2m} in P G(2, q) with an equation given by the product of the equation defining C^{n} and the equations of the 2m − n lines. Since the lines do not pass through (0, 0, 1), the multiplicity of P′(0, 0, 1) does not change. As stated above, the equation of C^{2m} can be expressed as a monomial of degree m in x^{2}, xy, y^{2}, zx, zy, that is, a polynomial which defines the variety of intersection V_{3}^{m} ∩ R_{2}^{3} of a hypersurface V_{3}^{m} in P G(4, q) and the ruled cubic surface R_{2}^{3}. Thus C^{2m} has image in P G(4, q) given by this intersection of R_{2}^{3} with the hypersurface V_{3}^{m}, namely, a variety V_{1}^{3m} which consists of the image of C^{n} together with 2m − n (Baer) conics. If a curve C^{n} has a m-fold point at (0, 0, 1), but is of order n > 2m, the addition of n − 2m lines through (0, 0, 1) makes it of order n + n − 2m = 2(n − m) with a (n − 2m + m = n − m)-fold point at (0, 0, 1). Thus yielding a curve C^{2(n−m)} of order 2(n − m) with a (n − m)-fold point at (0, 0, 1). In P G(4, q), the curve C^{2(n−m)} has image V_{3}^{n−m} ∩ R_{2}^{3} = C_{1}^{3(n−m)} which degenerates into the image of C^{n} and n − 2m generator lines. We summarise these results in a theorem.

Theorem 2.6.3 A curve C^{n} of order n in P G(2, q) with m−fold point at (0, 0, 1) has image in P G(4, q) on the surface R_{2}^{3}, given by σ as follows

σ(C^{n}) = \{ \begin{matrix}C_{1}^{3m} & if n = 2m, \\ C_{1}^{2n−m} & if n < 2m, \\ C_{1}^{2n−m} & if n > 2m. \end{matrix}

□

In particular, we note the following examples:

1. An irreducible conic C^{2} through (0, 0, 1), with n = 2, m = 1 and n = 2m in this case, has as image V_{3}^{1} ∩ R_{2}^{3} = V_{1}^{3}, a twisted cubic curve on the surface R_{2}^{3}, as discussed above.

2. An irreducible conic C^{2} not through (0, 0, 1), with n = 2, m = 0 and n > 2m in this case, together with two lines through (0, 0, 1) forms a quartic curve C_{1}^{4} with 2-fold point at (0, 0, 1). In P G(4, q), the image is V_{3}^{2} ∩ R_{2}^{3} = V_{1}^{6} = V_{1}^{1} ∪ V_{1}^{1} ∪ V_{1}^{4}, that is two generator lines of R_{2}^{3} together with a quartic curve on R_{2}^{3}. Thus σ(C^{2}) is a quartic curve C_{1}^{4} with q + 1 points and this quartic curve C_{1}^{4} is a normal rational curve for the following reasons:

(a) C_{1}^{4}, being the image of a conic not through (0, 0, 1), has no point on the directrix ℓ of R_{2}^{3}.

(b) C_{1}^{4} therefore cannot have a linear component n, since any line on R_{2}^{3} has at least one point in common with ℓ.

(c) If C_{1}^{4} is reducible, then by (b) it can only be reducible to a pair of conics C_{1}, C_{2}. But these conics on R_{2}^{3} are the image under σ of lines σ(C_{1}) and σ(C_{2}) of P G(2, q); a contradiction, since our original conic C^{2} in P G(2, q) is irreducible.

(d) Consider any hyperplane S_{3}^{1} of P G(4, q). Then C_{1}^{4} ∩ S_{3}^{1} ⊆ R_{2}^{3} ∩ S_{3}^{1}. But R_{2}^{3} ∩ S_{3}^{1} is the image of a conic C in P G(2, q) through (0, 0, 1) (by our results above). Since |C ∩ C^{2}| ≤ 4, we have |S_{3}^{1} ∩ C_{1}^{4}| ≤ 4. Hence C_{1}^{4} is properly contained in P G(4, q) for all q > 3.

(e) If q = 2 or 3, the result follows.

We now prove that every 4−dimensional normal rational curve contained in R_{2}^{3} and disjoint from the line directrix ℓ is the image under σ of a non-degenerate conic in P G(2, q) which contains P′(0, 0, 1) Consider a normal rational curve C^{4} of P G(4, q). Then, by Section 1.5, the curve C^{4} is given by

{P (t) = P (f_{0}(t), f_{1}(t), …, f_{4}(t)) | t ∈ GF (q) ∪ {∞} }

where

(i) each polynomial f_{i} has degree at most 4, i = 0, 1, 2, 3, 4, with at least one of the polynomials having degree 4.

(ii) the polynomials f_{0}, f_{1}, f_{2}, f_{3}, f_{4} are linearly independent with no non-trivial common factor.

(iii) C^{4} is projectively equivalent to {(t^{4}, t^{3}, t^{2}, t, 1) | t ∈ GF (q) ∪ {∞}}.

Suppose that C^{4} is a normal rational curve of P G(4, q), lying on the ruled cubic R_{2}^{3}. Therefore C^{4} is given by

{(f(t)^{2}, f(t)g(t), g(t)^{2}, f(t)h(t), g(t)h(t)); t ∈ GF (q) ∪ {∞} }

where

(i) f, g are of degree at most 2 and h is of degree at most 3, since at least one of f, g is non-constant.

(ii) f^{2}, fg, g^{2}, fh, gh are linearly independent quartic polynomials, with at least one having degree 4, and have no non-constant polynomial as common divisor.

Note that f and g have no non-constant polynomial as a common divisor since otherwise f^{2}, fg, g^{2}, fh, gh have a common non-trivial divisor.

Define C in P G(2, q) to be the image under σ of C^{4} as follows

σ(C^{4}) = C = {(f(t), g(t), h(t)); t ∈ GF (q) ∪ {∞} }

Suppose C contains P′(0, 0, 1). We consider each case separately. \underline{Possibility 1}: There exists t_{1} ∈ GF (q) such that f(t_{1}) = g(t_{1}) = 0. In that case p = t−t_{1} divides both f and g, a contradiction, since by condition (ii) above, the polynomials f^{2}, fg, g^{2}, fh, gh have no non-trivial common factor. \underline{Possibility 2}: t = ∞ corresponds to (0,0,1), in which case the degree of h is strictly greater than the degree of f and strictly greater than the degree of g. By considering properties (i), (ii) above, this implies that h has degree 3, the degree of f is 0 or 1 and the degree of g is 1 or 0. In this case σ(C^{4}) = C is not a conic since it is not a normal rational curve (of order 2) in P G(2, q). Example: f(t) = 1, g(t) = t and h(t) = t^{3} so that

C^{4} = {(1, t, t^{2}, t^{3}, t^{4})| t ∈ GF (q) ∪ {∞} }

in which case C = {(1, t, t^{3})| t ∈ GF (q) ∪ {∞} }. We have:

Theorem 2.6.4 If a normal rational curve C^{4} lies on R_{2}^{3} and if, using the above notation for C^{4}, the degree of h is less than or equal to 2, then there is no point of C^{4} on the line directrix ℓ of R_{2}^{3}.

Suppose C^{4} is the normal rational curve contained in R_{2}^{3} with the above notation, and suppose h has degree less than or equal to 2. It follows that at least one of f, g has degree 2. Moreover, f, g, h are polynomials of degree at most 2, with at least one of degree 2. The polynomials have no non-trivial common factor, since f^{2}, fg, g^{2}, fh, gh have no nontrivial common factor, and therefore C = {(f(t), g(t), h(t)) | t ∈ GF (q) ∪ {∞} }, by the definition of a normal rational curve in Section 1.5, is a non-degenerate conic in P G(2, q). Hence,

Theorem 2.6.5 Every normal rational curve C^{4} lying on R_{2}^{3} and disjoint from the line directrix ℓ is mapped by σ to an irreducible conic of P G(2, q) which does not contain (0, 0, 1).

Summarising our results in one theorem we obtain the following bijective correspondence.

Theorem 2.6.6 There are as many normal rational curves of P G(4, q) on R_{2}^{3} which are disjoint from ℓ as there are irreducible conics of P G(2, q) not through (0, 0, 1). This number is q^{5} − q^{4}.

Proof: An irreducible conic in P G(2, q) not containing (0, 0, 1) maps under σ to a normal rational curve on R_{2}^{3} with no point on ℓ (by the above Note 2.) By Theorem 2.6.5 the converse is true. The number of irreducible conics in P G(2, q) is q^{5} −q^{2}. The number of irreducible conics in P G(2, q) on (0, 0, 1) is q^{4} − q^{2}; therefore the number of irreducible conics in P G(2, q) not through (0, 0, 1) is q^{5} − q^{4}. □

2.7 A look at PG(2, q^{4}) in Bruck-Bose

In this section we investigate more closely the Bruck-Bose representation of P G(2, q^{4}). The plane P G(2, q^{4}) has a 4-dimensional Bruck-Bose representation over GF (q^{2}), which we shall denote by Π_{4,q^{2}}, and an 8-dimensional Bruck-Bose representation over GF (q), which we denote by Π_{8,q}. Consider a line ℓ, distinct from ℓ_{∞}, of P G(2, q^{4}). As we have noted earlier in this chapter, a Baer subline b_{q^{2}+1} of ℓ which contains no point on ℓ_{∞} is represented in Π_{4,q^{2}} by a nondegenerate conic b_{q^{2}+1}^{∗} in the plane ℓ^{∗}, which represents ℓ. Furthermore, the conic b_{q^{2}+1}^{∗} is disjoint from Σ_{∞}, the hyperplane at infinity of Π_{4,q^{2}}. In this section we show that the Baer sublines b_{q+1} of b_{q^{2}+1} are each represented in Π_{4,q^{2}} by a subconic of b_{q^{2}+1}^{∗} and each such subconic b_{q+1}^{∗} is contained in a Baer subplane of the plane ℓ^{∗} in Π_{4,q^{2}}. Each Baer subplane of this type intersects Σ_{∞} in a unique point in Π_{4,q^{2}}. Using this result together with Corollary 2.3.5 we obtain that such a Baer subline b_{q^{2}+1} in the 8-dimensional Bruck-Bose representation is a set of q^{2} +1 points in a 4-space ℓ^{∗∗} (the representation in Π_{8,q} of the line ℓ) of Π_{8,q}, which contains at least q^{3} + q 4-dimensional normal rational curves. We note that in 8-dimensional Bruck-Bose, b_{q^{2}+1} is in fact a (q^{2} + 1)−cap in the 4−space ℓ^{∗∗} of Π_{8,q} and since b_{q^{2}+1} is disjoint from ℓ_{∞} in P G(2, q^{4}) the corresponding (q^{2} + 1)−cap is disjoint from the hyperplane at infinity in Π_{8,q}.

We begin by establishing a coordinate system for P G(2, q^{4}) in 4-dimensional Bruck-Bose as in Section 1.10.4. Let ℓ_{∞} be the line of P G(2, q^{4}) with equation z = 0. Let ℓ be the line y = 0 and denote by P = P (1, 0, 0) the point of intersection of the lines ℓ and ℓ_{∞}. Each point of ℓ − {P } has coordinates of the form (a, 0, 1) where a ∈ GF (q^{4}). Let GF (q^{4}) = GF (q^{2})(α), where α ∈ GF (q^{4})\GF (q^{2}) has minimal polynomial x^{2} +x+c for some fixed element c in GF (q^{2}). Then every element b in GF (q^{4}) can be uniquely expressed in the form b = b_{0} + αb_{1} where the b_{i} are in GF (q^{2}). Following Section 1.10.4, in Bruck-Bose ℓ is the plane, which we denote by ℓ^{∗}, with affine points (a_{0}, a_{1}, 0, 0, 1), a_{i} ∈ GF (q^{2}), where each point (a = a_{0} + αa_{1}, 0, 1) is a point of ℓ. The spread element in ℓ^{∗} is J(∞) = ⟨1, α⟩ ≡ ⟨(1, 0, 0, 0, 0), (0, 1, 0, 0, 0)⟩. For convenience, from now on we shall represent the coordinates of points of ℓ and ℓ^{∗} as follows:

ℓ = {P (1, 0)} ∪ {(a, 1) | a ∈ GF (q^{4})}
ℓ^{∗} = {J(∞) = ⟨(1, 0, 0), (0, 1, 0)⟩} ∪ {(a_{0}, a_{1}, 1) | a_{0}, a_{1} ∈ GF (q^{2})}.

The Baer sublines of ℓ which contain the point P (1, 0) are represented in Bruck-Bose by the lines of ℓ^{∗} distinct from J(∞). We may divide these Baer sublines into two classes as follows.

Baer sublines of ℓ which contain P  Lines of ℓ^{∗} − {J(∞)}

(i) {θ(1, 0) + (a, 1)} ∪ {(1, 0)}, where a ∈ GF (q^{4}) is fixed and θ ∈ GF (q^{2}).

(i) The lines y = a_{1}, where a_{1} ∈ GF (q^{2}) is fixed.

(ii) {θ(b+α, 0)+(a, 1)}∪{(b+α, 0)}, where b ∈ GF (q^{2}) is fixed, a ∈ GF (q^{4}) is fixed and θ ∈ GF (q^{2}).

(ii) The lines x = by + (a_{0} − ba_{1}), for
all b, a_{0}, a_{1} ∈ GF (q^{2})

Note that there are q^{2} distinct lines of type (i) and q^{4} distinct lines of type (ii). Let b_{q^{2}+1} denote the Baer subline of ℓ with the following points.

\{(1, 1)\} ∪ \{[ \begin{matrix}α & 0 \\ α & 1 \end{matrix} ] ( \begin{matrix}θ \\ 1 \end{matrix} ) | θ ∈ GF (q^{2})\}.

P (1, 0) is not a point of b_{q^{2}+1} since if (1, 0) ≡ (αθ, αθ + 1), then αθ + 1 = 0 implies α is an element of GF (q^{2}), a contradiction.

Consider a point (αθ, αθ + 1) ≡ (αθ(αθ + 1)^{−1}, 1) of b_{q^{2}+1}. Suppose that (αθ + 1)^{−1} = b_{0} + αb_{1} for a necessarily unique pair b_{0}, b_{1} ∈ GF (q^{2}). Then,

(b_{0} + αb_{1})(1 + αθ) = 1

and on solving the system,

\begin{aligned}b_{0} & − & cθb_{1} & = & 1 \\ θb_{0} & + & b_{1}(1 − θ) & = & 0 \end{aligned}

we obtain b_{0} = (θ − 1)(−cθ^{2} + θ − 1)^{−1} and b_{1} = θ(−cθ^{2} + θ − 1)^{−1}. Hence we may write the coordinates (αθ(b_{0} + αb_{1}), 1) as

\begin{aligned}& (αθ((θ − 1)(−cθ^{2} + θ − 1)^{−1} + αθ(−cθ^{2} + θ − 1)^{−1}), 1) \\ = & (−cθ^{2}(−cθ^{2} + θ − 1)^{−1} − αθ(−cθ^{2} + θ − 1)^{−1}, 1) \end{aligned}

Therefore in Bruck-Bose, the Baer subline b_{q^{2}+1} is the set b_{q^{2}+1}^{∗} of points of ℓ^{∗} with coordinates,

{(1, 0, 1)} ∪ {(−cθ^{2}(−cθ^{2} + θ − 1)^{−1}, −θ(−cθ^{2} + θ − 1)^{−1}, 1) | θ ∈ GF (q^{2})}.

Therefore b_{q^{2}+1}^{∗} is the image of the conic y^{2} = xz in ℓ^{∗} under a projectivity of ℓ^{∗} since b_{q^{2}+1}^{∗} is given by,

\{(1, 0, 1)\} ∪ \{[ \begin{matrix}−c & 0 & 0 \\ 0 & −1 & 0 \\ −c & 1 & −1 \end{matrix} ] ( \begin{matrix}θ^{2} \\ θ \\ 1 \end{matrix} ) | θ ∈ GF (q^{2})\}.

If we now let b_{q+1} be the subset of b_{q^{2}+1} of points with parameter θ ∈ GF (q) ∪ {∞}, then b_{q+1} is a Baer subline of b_{q^{2}+1}. From the above calculations we have that b_{q+1} in Bruck-Bose is a subconic b_{q+1}^{∗} of the conic b_{q^{2}+1}^{∗}. Moreover, b_{q+1}^{∗} is projectively related to the conic {(θ^{2}, θ, 1) | θ ∈ GF (q) ∪ {∞} } in the real Baer subplane P G(2, q) of ℓ^{∗}. Since b_{q^{2}+1} does not contain P, b_{q^{2}+1}^{∗} and hence b_{q+1}^{∗} has no point in Σ_{∞}. Suppose the Baer subplane B of ℓ^{∗} which contains b_{q+1}^{∗} intersects Σ_{∞} in q + 1 points. Then since ℓ^{∗} is a quadratic extension of B, the conic b_{q^{2}+1}^{∗} would necessarily intersect Σ_{∞} in two distinct and conjugate points, a contradiction. Thus, the Baer subplane B intersects Σ_{∞} in a unique point. Another way to see this is to consider the projectivity which maps P G(2, q) on to the Baer subplane B. A point (x, y, z) of P G(2, q) is mapped by the projectivity to a point of Σ_{∞} if and only if −cx + y − z = 0. Since c is an element of GF (q^{2}), the line [−c, 1, −1] is not a line of P G(2, q) and so intersects P G(2, q) in a unique point. Thus B contains a unique point on Σ_{∞}. Now consider another Baer subline b^{1} ≠ b_{q+1} of b_{q^{2}+1}. Any Baer subline of ℓ has stabiliser a subgroup of P GL(2, q^{4}) isomorphic to P GL(2, q^{2}). In particular the real Baer subline,

b^{0} = {(1, 0)} ∪ {(θ, 1) | θ ∈ GF (q^{2})}

of ℓ has stabiliser P GL(2, q^{2}). P GL(2, q^{2}) acts triply transitively on the points of b^{0} and so acts transitively on the Baer sublines of b^{0}. Hence any Baer subline of b^{0} is given by,

( \begin{matrix}θ′ \\ 1 \end{matrix} ) = ( 1^{\frac{aθ+b}{cθ+d}} )

for some choice of a, b, c, d in GF (q^{2}) such that ad−bc ≠ 0, where the parameter θ ranges over GF (q) ∪ {∞}. This Baer subline is projectively related to the Baer subline of b_{q^{2}+1} given by

b^{1} = [ \begin{matrix}α & 0 \\ α & 1 \end{matrix} ] ( \begin{matrix}θ′ \\ 1 \end{matrix} ) ,

where θ′ ∈ GF (q^{2}) ∪ {∞}. In Bruck-Bose, this Baer subline is given by,

[ \begin{matrix}−c & 0 & 0 \\ 0 & −1 & 0 \\ −c & 1 & −1 \end{matrix} ] ( \begin{matrix}θ^{′2} \\ θ′ \\ 1 \end{matrix} ) = [ \begin{matrix}−c & 0 & 0 \\ 0 & −1 & 0 \\ −c & 1 & −1 \end{matrix} ] [ \begin{matrix}a^{2} & ac & c^{2} \\ 2ab & ad + bc & 2cd \\ b^{2} & bd & d^{2} \end{matrix} ] ( \begin{matrix}θ^{2} \\ θ \\ 1 \end{matrix} )

(See [52, Theorem 2.37]). Hence in Bruck-Bose, b^{1} is the image b^{1∗} of the subconic b_{q+1}^{∗} under a projectivity of ℓ^{∗} and so b^{1∗} is a subconic of b_{q^{2}+1}^{∗} contained in a Baer subplane of ℓ^{∗}. Again this Baer subplane intersects Σ_{∞} in a unique point. Hence every Baer subline of b_{q^{2}+1} in Bruck-Bose is a subconic of b_{q^{2}+1}^{∗} contained in some Baer subplane of ℓ^{∗}, and the Baer subplane necessarily intersects Σ_{∞} in a unique point. We have concentrated our attention on a specific Baer subline b_{q^{2}+1} of ℓ, but any Baer subline of ℓ is the image of b_{q^{2}+1} under an element of P GL(2, q^{4}). Let bq^{2}+1^{′} ≠ b_{q^{2}+1} be a Baer subline of ℓ which does not contain the point P, then

bq^{2}+1^{′} = \{H [ \begin{matrix}α & 0 \\ α & 1 \end{matrix} ] ( \begin{matrix}θ \\ 1 \end{matrix} ) | θ ∈ GF (q^{2}) ∪ \{∞\} \}

where H is some element of P GL(2, q^{4}). Then H′ = H [ \begin{matrix}α & 0 \\ α & 1 \end{matrix} ] is an element of

P GL(2, q^{4}) and in Bruck-Bose, by repeating the arguments used for b_{q^{2}+1} above, b_{q^{2}+1}^{′∗} is a set of points of ℓ^{∗} given by,

H^{′∗} ( \begin{matrix}θ^{2} \\ θ \\ 1 \end{matrix} )

where H^{′∗} is an element of P GL(3, q^{2}) since b_{q^{2}+1}^{′∗} is a non-degenerate (Baer) conic in ℓ^{∗}. Thus each Baer subline of bq^{2}+1^{′} in Bruck-Bose is projectively related to a subconic of b_{q^{2}+1}^{∗}, which represents a Baer subline of b_{q^{2}+1}. We therefore have the following theorem valid for any Baer subline b_{q^{2}+1} of an affine line ℓ of P G(2, q^{4}) whose Bruck-Bose representation in P G(4, q^{2}) is a non-degenerate (Baer) conic b_{q^{2}+1}^{∗} in the plane ℓ^{∗} which represents ℓ.

Theorem 2.7.1 If b_{q+1} is a Baer subline of b_{q^{2}+1}, then in 4-dimensional Bruck-Bose b_{q+1} is a subconic b_{q+1}^{∗} of the conic b_{q^{2}+1}^{∗} such that b_{q+1}^{∗} lies in a Baer subplane B of ℓ^{∗}. Moreover, B intersects Σ_{∞} in a unique point. □

Note that since the plane ℓ^{∗} is isomorphic to P G(2, q^{2}), we can represent ℓ^{∗} in 4-dimensional Bruck-Bose over GF (q), using ℓ^{∗} ∩ Σ_{∞} as the line at infinity of ℓ^{∗}. Since b_{q^{2}+1}^{∗} is a non-degenerate conic in ℓ^{∗} disjoint from the line at infinity of ℓ^{∗}, in Bruck-Bose b_{q^{2}+1}^{∗} is a (q^{2} +1)-cap. Moreover this cap contains the Bruck-Bose image of the subconics of b_{q^{2}+1}^{∗} which each lie in non-affine Baer subplanes of ℓ^{∗}. Since the subconics have no point on the line at infinity of ℓ^{∗} and by Section 2.3 we have,

Theorem 2.7.2 If b_{q^{2}+1} is a Baer subline of a line ℓ of P G(2, q^{4}) such that b_{q^{2}+1} is disjoint from the line at infinity of P G(2, q^{4}), then in 8-dimensional Bruck-Bose over GF (q), b_{q^{2}+1} is a (q^{2} + 1)-cap in the 4-space which represents ℓ. Moreover, this (q^{2} + 1)-cap contains at least q^{3} + q 4-dimensional normal rational curves. □

Note that a cap of the type of Theorem 2.7.2 is not a 3-dimensional ovoid. Since if the cap is contained in a hyperplane Σ of the 4-space of Π_{8} which contains the cap, then Σ and the hyperplane at infinity intersect in a plane. It then follows that the cap contains points of Σ_{∞}, a contradiction. Hence such a cap is not contained in any hyperplane of the 4-space which represents ℓ in Π_{8,q}. Furthermore, such a cap is disjoint from at least one hyperplane of the 4−space in which it lies.

The Bruck and Bose representation

in PG(n, q), n > 4

In this chapter we consider the Bruck-Bose representation in projective space of dimension greater than 4. In other words we consider the Bruck-Bose representation defined by spreads other than 1−spreads of P G(3, q).

3.1 Some properties concerning (h − 1)−spreads of

PG(2h − 1, q)

In [50, page 206] a method for constructing spreads is given; a particular case of which is the following. Note that by Theorem 1.9.1 a (2h − 1)−spread S_{2h−1,q^{2}} exists in P G(4h − 1, q^{2}) and since S_{2h−1,q^{2}} has more elements than there are points in P G(4h − 1, q), there exists an element of S_{2h−1,q^{2}} which is disjoint from P G(4h − 1, q); therefore, it is possible to embed P G(2h − 1, q^{2}) in the extension P G(4h − 1, q^{2}) of P G(4h−1, q) in such a way that P G(2h−1, q^{2}) does not contain a point of P G(4h−1, q).

Construction 3.1.1 A construction of a (2h − 1)−spread of P G(4h − 1, q) from a (h − 1)−spread of P G(2h − 1, q^{2}): Consider a projective space P G(2h − 1, q^{2}), h ≥ 1. By Theorem 1.9.1, there exists an (h − 1)−spread S′ of P G(2h − 1, q^{2}) and S′ contains q^{2h} + 1 spread elements Π_{h−1,q^{2}}^{j}, j = 1, …, q^{2h} + 1, of dimension h − 1 over GF (q^{2}). Embed P G(2h − 1, q^{2}) in the extension P G(4h − 1, q^{2}) of P G(4h − 1, q) so that P G(2h − 1, q^{2}) does not contain a point of P G(4h − 1, q). The (h − 1)−space Π_{h−1,q^{2}}^{j} and its conjugate \overline{Π}_{h−1,q^{2}}^{j} generate a (2h−1)−space Π_{2h−1,q^{2}}^{j} of P G(4h−1, q^{2}) and Π_{2h−1,q^{2}}^{j} ∩P G(4h−1, q) is a (2h−1)−space Π_{2h−1,q}^{j} of P G(4h − 1, q). The q^{2h} + 1 spaces Π_{2h−1,q}^{j} form a partition of P G(4h − 1, q) and we denote this (2h − 1)−spread of P G(4h − 1, q) by S.

We now prove:

Theorem 3.1.1 In the Construction 3.1.1, if the (h − 1)−spread S′ of P G(2h − 1, q^{2}) is regular, then the (2h − 1)−spread S of P G(4h − 1, q) is regular.

Proof: Let Π_{h−1,q^{2}}^{1}, Π_{h−1,q^{2}}^{2}, Π_{h−1,q^{2}}^{3} be three distinct elements of S′. Denote by R′ = R(Π_{h−1,q^{2}}^{1}, Π_{h−1,q^{2}}^{2}, Π_{h−1,q^{2}}^{3}) the unique (h−1)−regulus of P G(2h−1, q^{2}) containing these three spread elements. Let Π_{2h−1,q}^{1}, Π_{2h−1,q}^{2}, Π_{2h−1,q}^{3} be the three distinct elements of S corresponding to Π_{h−1,q^{2}}^{1}, Π_{h−1,q^{2}}^{2}, Π_{h−1,q^{2}}^{3} respectively in the given construction. Let R = R(Π_{2h−1,q}^{1}, Π_{2h−1,q}^{2}, Π_{2h−1,q}^{3}) denote the unique (2h − 1)−regulus of P G(4h − 1, q) containing Π_{2h−1,q}^{1}, Π_{2h−1,q}^{2} and Π_{2h−1,q}^{3}. So R is a system of maximal (2h − 1)−spaces of a Segre variety ζ_{1,2h−1} in P G(4h − 1, q). Over GF (q^{2}), R becomes a (2h − 1)−regulus R_{q^{2}} of P G(4h − 1, q^{2}). Due to the above construction of the spread S we have for j = 1, 2, 3, Π_{h−1,q^{2}}^{j} is contained in Π_{2h−1,q^{2}}^{j} where Π_{2h−1,q^{2}}^{j} is the unique element of the regulus R_{q^{2}} which contains Π_{2h−1,q}^{j}. Thus the line transversals of R′ in P G(2h − 1, q^{2}) are line transversals of R_{q^{2}} and therefore R′ is a Segre subvariety ζ_{1,h−1} of R_{q^{2}} and by Theorem 1.8.10, the regulus R′ is precisely the intersection R_{q^{2}} ∩ P G(2h − 1, q^{2}). It now follows that for any (2h − 1)−space Π_{2h−1,q}^{j} in R, where Π_{2h−1,q}^{j} is distinct from Π_{2h−1,q}^{1}, Π_{2h−1,q}^{2} and Π_{2h−1,q}^{3}, the unique element Π_{2h−1,q^{2}}^{j} of R_{q^{2}} which contains Π_{2h−1,q}^{j} has the property Π_{2h−1,q^{2}}^{j} ∩ P G(2h − 1, q^{2}) = Π_{h−1,q^{2}}^{j}, for some element Π_{h−1,q^{2}}^{j} of R′. By the construction of S from S′, if Π_{2h−1,q}^{j} (∈ R) is an element of S, then Π_{h−1,q^{2}}^{j} (∈ R′) is an element of S′. The converse of the preceding statement is true if Π_{h−1,q^{2}}^{j} (∈ R′) is one of the q + 1 elements of R′ associated to the elements of R via the construction of the spread S. (Note that R′ has q^{2} + 1 elements and R has q + 1 elements). If S′ is a regular spread, then the regulus R′ defined by Π_{h−1,q^{2}}^{1}, Π_{h−1,q^{2}}^{2}, Π_{h−1,q^{2}}^{3} is contained in S′ and therefore, by the preceding argument, the regulus R of P G(4h − 1, q) defined by Π_{2h−1,q}^{1}, Π_{2h−1,q}^{2}, Π_{2h−1,q}^{3} is contained in S. The result now follows. □

Consider a translation plane π of order q^{2h} defined by a Bruck-Bose construction with a (h − 1)−spread S′ of Σ∞^{′} = P G(2h − 1, q^{2}). We now have a convenient correspondence between this Bruck-Bose representation of π and a second Bruck-Bose representation of π defined by a (2h − 1)−spread S of Σ_{∞} = P G(4h − 1, q), where S′ and S are associated by the above construction. For Desarguesian planes of certain orders which have a Bruck-Bose representation, the above Construction 3.1.1 and Theorem 3.1.1 provide us with a convenient method to obtain a Bruck-Bose representation of the plane in a space of higher dimension and lower order. To illustrate this, we consider the Desarguesian plane P G(2, q^{4}). The plane P G(2, q^{4}) has a 4-dimensional Bruck-Bose representation defined by a regular line spread S′ of P G(3, q^{2}) and an 8−dimensional Bruck-Bose representation defined by a regular 3−spread S of P G(7, q). In the previous chapter we investigated the 4-dimensional Bruck-Bose representation of the Baer substructures of Desarguesian planes of square order. We now determine properties concerning the 8-dimensional Bruck-Bose representation of the Baer substructures of P G(2, q^{4}) and some generalisations.

Theorem 3.1.2 A regular 3−spread S in P G(7, q) has a well-defined and unique set of induced 1−spreads, one in each element of S.

Proof: By Theorem 1.9.5, the regular 3−spreads of P G(7, q) are projectively equivalent. Therefore, we can assume that S is the regular 3−spread of P G(7, q) obtained from a regular 1−spread S′ of P G(3, q^{2}) by the Construction 3.1.1 with h = 2. We repeat the construction for this special case to establish notation. Embed P G(7, q) in P G(7, q^{2}) and let Σ_{3,q^{2}} be a 3−space over GF (q^{2}) in P G(7, q^{2}) which has no point in common with P G(7, q). Let S′ be a regular 1−spread of Σ_{3,q^{2}}. Consider the conjugate space \overline{Σ}_{3,q^{2}} of Σ_{3,q^{2}}. For each element Π_{1,q^{2}}^{j} in S′, j = 1, …, q^{4} + 1, the 3−space Π_{3,q^{2}}^{j} spanned by Π_{1,q^{2}}^{j} and its conjugate \overline{Π}_{1,q^{2}}^{j} intersects P G(7, q) in a 3−space Π_{3,q}^{j}. These q^{4} +1 3−spaces Π_{3,q}^{j} form a 3−spread S of P G(7, q) which by Theorem 3.1.1 is regular. Each element Π_{3,q}^{j} of S is the intersection ⟨Π_{1,q^{2}}^{j}, \overline{Π}_{1,q^{2}}^{j}⟩ ∩ P G(7, q) for a unique line Π_{1,q^{2}}^{j} of S′. For j fixed, the join of each point P of Π_{1,q^{2}}^{j} to its conjugate \overline{P} yields a line of Π_{3,q}^{j} and the collection of these q^{2} + 1 lines constitutes a regular 1−spread S_{1}^{j} of Π_{3,q}^{j} by Bruck’s result (Theorem 1.9.6). Hence each element Π_{3,q}^{j} of the regular 3−spread S of P G(7, q) has a well defined induced regular 1−spread which we denote by S_{1}^{j}. □

Consider the regular line spread S′ of Σ_{3,q^{2}} ≅ P G(3, q^{2}) and the regular 3−spread S of P G(7, q) associated to S′ by the Construction 3.1.1. By the Bruck-Bose construction of Section 1.10, these spreads correspond to a 4-dimensional Bruck-Bose representation of P G(2, q^{4}) and an 8-dimensional Bruck-Bose representation of P G(2, q^{4}) respectively. Denote these Bruck-Bose incidence structures by Π_{4,q^{2}} and Π_{8,q} respectively. By Theorem 3.1.2 and its proof, there exists a well defined 1-1 correspondence between the points of Σ_{3,q^{2}} and the (line) elements of the induced 1−spreads {S_{1}^{j}} of P G(7, q).

Definition 3.1.3 For S a regular 3−spread of P G(7, q), the (line) elements of the q^{4}+1 induced regular 1−spreads {S_{1}^{j}} shall be called partition lines. That is, for each 3−space Σ_{j} ∈ S, a line ℓ of Σ_{j} is a partition line if ℓ ∈ S_{1}^{j}, otherwise ℓ is a non-partition line. The remaining lines of P G(7, q) are those not contained in any element of S; these shall be called transversal lines.

In Section 2.2, we discussed the representation in 4-dimensional Bruck-Bose of affine Baer subplanes and affine Baer sublines of Desarguesian planes of square order. By Corollary 2.2.2 an affine Baer subplane B of P G(2, q^{4}) is represented in Π_{4,q^{2}} by a plane not contained in Σ∞^{′} = Σ_{3,q^{2}} and which meets Σ∞^{′} in a line ℓ which is not an element of S′. Consider such a line ℓ in Σ_{3,q^{2}}. The line ℓ and its conjugate \overline{ℓ} generate a 3−space ⟨ℓ, \overline{ℓ}⟩ of P G(7, q^{2}) and the intersection ⟨ℓ, \overline{ℓ}⟩∩P G(7, q) is a 3−space Σ_{ℓ} of P G(7, q). Since ℓ is incident with exactly q^{2} + 1 1−spread elements in Σ_{3,q^{2}}, the 3−space Σ_{ℓ} intersects exactly q^{2} + 1 of the 3−spaces in the spread S of P G(7, q), meeting each in a partition line. So in particular Σ_{ℓ} is disjoint from the remaining spread elements in S. Consider the 8−dimensional Bruck-Bose representation, Π_{8,q}, of P G(2, q^{4}) defined by the regular 3−spread S of P G(7, q). Consider a 4−dimensional subspace B^{∗} of Π_{8,q} which intersects P G(7, q) in the 3−space Σ_{ℓ}. Any 4−space l^{∗} in Π_{8,q}, not contained in P G(7, q), and which intersects P G(7, q) in a unique element of S, either intersects B^{∗} in a unique affine point, or the spread element contained in l^{∗} is one of the q^{2} + 1 incident with B^{∗}. It follows by Theorem 1.2.2, that B^{∗} represents an affine Baer subplane of P G(2, q^{4}), since B^{∗} and the q^{2} + 1 3−spread elements incident with B^{∗} constitute a (q^{4} + q^{2} + 1)−blocking set in P G(2, q^{4}). By considering all lines ℓ in Σ_{3,q^{2}} which are not elements of the 1−spread S′ and repeating the above procedure, we obtain the 8−dimensional Bruck-Bose representation of all q^{4}(q^{8} + q^{6} + q^{4} + q^{2}) affine Baer subplanes of P G(2, q^{4}). Intrinsic to this representation is the existence of q^{8} + q^{6} + q^{4} + q^{2} 3−spaces of P G(7, q) which intersect precisely q^{2} + 1 elements of the regular 3−spread S of P G(7, q) and such that the intersection in each case is a unique partition line, namely an element of the induced 1−spread of that 3−space.

Theorem 3.1.4 Let S be a regular 3−spread of P G(7, q). For each 3−space Σ of P G(7, q) one of the following holds:

(1) Σ is an element of S and therefore Σ = Σ_{j} has a induced regular 1−spread S_{1}^{j}. By definition Σ contains exactly q^{2} + 1 partition lines. There are q^{4} + 1 3−spaces Σ of this type in P G(7, q).

(2) Σ intersects exactly q^{2} + 1 elements of S, meeting each in a partition line. This set of q^{2} + 1 partition lines constitutes a regular 1−spread of Σ which we shall call a partition 1−spread. Any two partition lines, contained in distinct elements of S, span such a 3−space. There are q^{8} + q^{6} + q^{4} + q^{2} 3−spaces Σ of this type in P G(7, q).

(3) Σ intersects x elements of S where x > q^{2} + 1. In this case either:

(i) x = q^{3} + 1 and Σ intersects one element of S in a plane (which necessarily contains a partition line) and Σ intersects a further q^{3} elements of S, meeting each in a point, or,

(ii) Σ intersects y elements of S in a line and Σ intersects a further x − y = (q^{3} + q^{2} + q + 1) − y(q + 1) > 0 elements of S meeting each in a point. In this case Σ contains at most one partition line.

If Π_{1}, Π_{2}, Π_{3} are three distinct elements of S which each intersects Σ in a line, then Σ has a non-trivial intersection with each element of S in the 3−regulus R(Π_{1}, Π_{2}, Π_{3}); indeed Σ intersects each such element of S in a line.

Proof: By Theorem 3.1.2 the q^{4} + 1 elements of S constitute the 3−spaces of P G(7, q) of type (1). By the remarks preceding Theorem 3.1.4 there exist q^{8} + q^{6} + q^{4} + q^{2} 3−spaces of P G(7, q) which each intersect q^{2} +1 distinct elements of S and which contain a partition 1−spread. We shall call these 3−spaces partition 3−spaces of P G(7, q). We must show that these are the only 3−spaces of P G(7, q) which intersect exactly q^{2} + 1 distinct elements of S. Π_{8,q} is the 8−dimensional Bruck-Bose representation of P G(2, q^{4}). The line at infinity ℓ_{∞} is the line with “points” the elements of S in P G(7, q). As usual, the Baer subplanes of P G(2, q^{4}) which are secant to ℓ_{∞} are called affine Baer subplanes. There exist precisely q^{4}(q^{8} + q^{6} + q^{4} + q^{2}) affine Baer subplanes of P G(2, q^{4}). Consider a 4−space B^{∗} in P G(8, q) not contained in P G(7, q) and which intersects P G(7, q) in a 3−space Σ where Σ intersects exactly q^{2} + 1 elements of S. Necessarily, Σ intersects each of these q^{2} + 1 elements of S in a line. By the incidence in Π_{8,q}, B^{∗} intersects ℓ_{∞} in exactly q^{2} + 1 points. Each 4−space ℓ of P G(8, q) which represents a line of P G(2, q^{4}) distinct from ℓ_{∞} is not contained in P G(7, q) and meets P G(7, q) in an element of S. Such a 4−space ℓ either intersects B^{∗} in a point of P G(8, q)\P G(7, q) or the element of S incident with ℓ is one of the q^{2} + 1 3−spread elements incident with B^{∗}. It follows that B^{∗} represents a (q^{4} + q^{2} + 1)−blocking set B in P G(2, q^{4}). By Theorem 1.2.2, B is an affine Baer subplane of P G(2, q^{4}). Therefore any 4−space of P G(8, q), not contained in P G(7, q) and which meets P G(7, q) in a partition 3−space represents an affine Baer subplane of P G(2, q^{4}). There are q^{4}(q^{8} + q^{6} + q^{4} + q^{2}) such 4−spaces of P G(8, q). Since this is also the number of affine Baer subplanes of P G(2, q^{4}), there exist no further 3−spaces of P G(7, q) (besides the partition 3−spaces) which intersect exactly q^{2} + 1 elements of S. Let Σ be a 3−space of P G(7, q) spanned by partition lines ℓ_{1} and ℓ_{2} where ℓ_{1} and ℓ_{2} lie in distinct elements of S. In the quadratic extension, the lines ℓ_{1} and ℓ_{2} intersect Σ_{3,q^{2}} in distinct points L_{1} and L_{2} respectively. The 3−space (over GF (q^{2})) spanned by the line L_{1}L_{2} and its conjugate \overline{L_{1}L_{2}} is the quadratic extension Σ_{q^{2}} of Σ. By joining each point P on L_{1}L_{2} to its conjugate \overline{P} on \overline{L_{1}L_{2}} we obtain a set of q^{2} + 1 lines of Σ which by Bruck’s Theorem 1.9.6 constitutes a regular 1−spread of Σ. The elements of this 1−spread are all partition lines and hence by definition, this regular 1−spread is a partition spread. Thus Σ is a partition 3−space. We have that two partition lines from distinct elements of S span a partition 3−space. The 3−spaces of P G(7, q) of type (1) and (2) have now been classified. The type (3) 3−spaces include all possible exceptions. It remains to prove the final remark regarding a 3−space of type (3)(ii). Consider a 3−space Σ of P G(7, q) which intersects strictly greater than q^{2} + 1 elements of S but which meets no element of S in a plane. Suppose Σ intersects the 3−spread elements Π_{1}, Π_{2}, Π_{3} each in a line ℓ_{1}, ℓ_{2}, ℓ_{3} respectively. The lines ℓ_{1}, ℓ_{2}, ℓ_{3} define a unique 1−regulus R_{1} = R(ℓ_{1}, ℓ_{2}, ℓ_{3}) in Σ. The 3−spread elements Π_{1}, Π_{2}, Π_{3} define a unique 3−regulus R_{3} = R(Π_{1}, Π_{2}, Π_{3}) which is contained in S since S is regular. The line transversals of R_{1} are contained in Σ and are necessarily transversals of the regulus R_{3}. Hence each spread element in R_{3} intersects Σ in a line, namely a maximal space of the Segre variety R_{1}. □

Corollary 3.1.5 Let Π_{8,q} denote the Bruck-Bose representation of P G(2, q^{4}) in P G(8, q) and let Σ_{∞} denote the hyperplane at infinity of P G(8, q). B is an affine Baer subplane of P G(2, q^{4}) if and only if in Π_{8,q} B is a 4−space not contained in Σ_{∞} and which intersects Σ_{∞} in a partition 3−space. b_{ℓ} is a Baer subline of P G(2, q^{4}) that contains a point of ℓ_{∞} if and only if in Π_{8,q} b_{ℓ} is a plane not contained in Σ_{∞} and which intersects Σ_{∞} in a partition line.

Proof: The affine Baer subplane structure in Π_{8,q} was determined in the proof of Theorem 3.1.4. A line ℓ, distinct from ℓ_{∞}, of an affine Baer subplane B of P G(2, q^{4}) intersects B in a Baer subline b_{ℓ} that contains a point of ℓ_{∞}. In Π_{8,q}, ℓ is a 4−space which intersects Σ_{∞} in an element Σ_{j} of the 3−spread S and B is a 4−space which intersects Σ_{∞} in a partition 3−space Σ. The 3−spaces Σ_{j} and Σ intersect in a partition line, hence in Π_{8,q}, the intersection ℓ ∩ B is a plane, not contained in Σ_{∞} and which contains a partition line. This plane is then the Bruck-Bose representation of the Baer subline b_{ℓ}. □

Before considering the representation in Π_{8,q} of the non-affine Baer subplanes of P G(2, q^{4}) and the Baer sublines b_{ℓ} of P G(2, q^{4}) which contain no point of ℓ_{∞}, we need to present some extra material. In the next section we recall the Bose representation of a plane P G(2, q^{2}) in the projective space P G(5, q) which was introduced in [18]. We conclude this section with a generalisation of the results determined for P G(2, q^{4}) thus far.

Theorem 3.1.6 Consider the Desarguesian plane P G(2, q^{2^{n}}) (n ≥ 1) and the n Bruck-Bose representations Π_{2^{i+1}} = Π_{2^{i+1},q^{2^{n−i}}} (1 ≤ i ≤ n) which are determined by a

regular 1−spread in P G(3, q^{2^{n−1}}),
regular 3−spread in P G(7, q^{2^{n−2}}),
⋮
regular (2^{i} − 1)−spread in P G(2^{i+1} − 1, q^{2^{n−i}}),
⋮
regular (2^{n} − 1)−spread in P G(2^{n+1} − 1, q) respectively.

Then the regular (2^{i} −1)−spread in the hyperplane P G(2^{i+1} −1, q^{2^{n−i}}) at infinity of Π_{2^{i+1}} has a set of induced regular (2^{i−1}−1)−spreads, one in each element of the (2^{i}−1)−spread. Furthermore, for each such induced regular (2^{i−1}−1)−spread, there exists a set of induced regular (2^{i−2} −1)−spreads, and so on, until finally there exist induced regular 1−spreads.

Proof Let S_{1} be regular 1−spread of P G(3, q^{2^{n−1}}) and embed P G(3, q^{2^{n−1}}) as a subspace in P G(7, q^{2^{n−1}}) in such a way that it is skew to P G(7, q^{2^{n−2}}). By Theorem 3.1.1 and the Construction 3.1.1, S_{1} determines a regular 3−spread S_{3} of P G(7, q^{2^{n−2}}) which has a set of induced regular 1−spreads, one in each element of S_{3}, by Theorem 3.1.2. Embed P G(7, q^{2^{n−2}}) as a subspace in P G(15, q^{2^{n−2}}) in such a way that P G(7, q^{2^{n−2}}) is skew to the Baer subspace P G(15, q^{2^{n−3}}) of P G(15, q^{2^{n−2}}) and recursively repeat the above procedure using Construction 3.1.1. At the final stage we obtain a regular (2^{n} − 1)−spread in P G(2^{n+1} − 1, q) which contains the nested induced regular spreads of each stage. If we stop the procedure before the final stage we have a regular (2^{i} − 1)−spread in P G(2^{i+1} −1, q^{2^{n−i}}) with the nested induced regular spreads obtained up until that stage.

By Theorem 1.9.5, regular (2^{i} −1)−spreads in P G(2^{i+1} −1, q^{2^{n−i}}) are projectively equivalent, and so the regular spread we have constructed, which contains nested induced spreads, is representative. □

Corollary 3.1.7 For each 1 ≤ i ≤ n, embed P G(2^{i+1} − 1, q^{2^{n−i}}) = Σ_{∞}^{2^{i+1}−1} as a hyperplane in P G(2^{i+1}, q^{2^{n−i}}) and let Π_{2^{i+1}} denote the Bruck-Bose representation of P G(2, q^{2^{n}}) in P G(2^{i+1}, q^{2^{n−i}}) determined by the regular (2^{i} − 1)−spread S_{2^{i}−1} of Σ_{∞}^{2^{i+1}−1}, as in Theorem 3.1.6. Then B is an affine Baer subplane of P G(2, q^{2^{n}}) if and only if in Π_{2^{i+1}}, B is a (2^{i})−space B^{∗} of P G(2^{i+1}, q^{2^{n−i}}) not contained in Σ_{∞}^{2^{i+1}−1} and which intersects Σ_{∞}^{2^{i+1}−1} in exactly q^{2^{n−1}} + 1 elements of S_{2^{i}−1}. Furthermore, each element Λ ∈ S_{2^{i}−1} is either disjoint to B^{∗} or intersects B^{∗} in a unique element (a (2^{i−1}−1)−space of order q^{2^{n−i}}) of the induced regular (2^{i−1}−1)−spread S_{j}^{2^{i−1}−1} in Λ. □

Note that the Bruck-Bose representation B^{∗} of an affine Baer subplane B of P G(2, q^{2^{n}}) is determined in Corollary 3.1.7, regardless of which of the n possible Bruck-Bose representations of P G(2, q^{2^{n}}) is being considered. Moreover, implicit to Theorem 3.1.6 and its Corollary 3.1.7 is the Bruck-Bose representation of subplanes of order q^{2^{n−j}} of P G(2, q^{2^{n}}) which contain the line at infinity as a line. Due to the existence of the induced spreads determined in Theorem 3.1.6, in a Bruck-Bose representation Π_{2^{i+1}} of a Desarguesian plane P G(2, q^{2^{n}}) we have the Bruck-Bose representations of the subplanes, which contain the line at infinity as a line, nested in Π_{2^{i+1}} as linear subspaces. Finally, let ℓ be a subline of order q^{2^{n−j}} (1 ≤ j ≤ n) of a line L of P G(2, q^{2^{n}}) such that ℓ contains a unique point on the line at infinity. It follows from the above discussion that the representation of ℓ in any Bruck-Bose representation Π_{2^{i+1}} of P G(2, q^{2^{n}}) is determined.

Corollary 3.1.8 Let ℓ be a subline of order q^{2^{n−j}} (1 ≤ j ≤ n) of a line L of P G(2, q^{2^{n}}) such that ℓ contains a unique point on the line at infinity ℓ_{∞} of P G(2, q^{2^{n}}). Let Π_{2^{i+1}} (1 ≤ i ≤ n) denote the Bruck-Bose representation of P G(2, q^{2^{n}}) defined by a regular (2^{i} − 1)−spread of P G(2^{i+1} − 1, q^{2^{n−i}}). Then the subline ℓ is represented by a (2^{i−j})−subspace ℓ^{∗} of the (2^{i})−space L^{∗}, which represents L, in Π_{2^{i+1}}. Moreover, ℓ^{∗} intersects the hyperplane at infinity P G(2^{i+1} − 1, q^{2^{n−i}}) in exactly a unique induced spread element of dimension 2^{i−j} − 1 and order q^{2^{n−i}}. □

In this way we obtain the Bruck-Bose representations of any (not just a Baer) subplane of P G(2, q^{2^{n}}) which contains the line at infinity as a line and any (not just a Baer) subline of a line of P G(2, q^{2^{n}}) such that the subline contains a unique point on the line at infinity.

3.2 The Bose representation of PG(2, q^{2}) in PG(5, q)

The results of this section are well known and form part of the folklore of finite projective geometry. References are given where possible; however, it has been difficult to locate references for some of the well known results which are presented here. In order to provide a complete discussion of the Bose representation, we deemed it appropriate to prove these results. In [18], Bose calls a 1−spread of P G(5, q) a spread (of lines), and we shall also in this section; that is, a spread of lines of P G(5, q) is a set of lines in P G(5, q) such that each point of P G(5, q) is contained in one and only one line of the set. We shall also define a dual spread of 3−spaces in P G(5, q) to be a set of 3−spaces such that each 4−space of P G(5, q) contains one and only one 3−space of the set. Note that a spread in P G(5, q) is equivalent to a dual spread in the dual space of P G(5, q). Hence, the number of elements in a spread or dual spread of P G(5, q) equals

\frac{q^{6} − 1}{q − 1} × \frac{1}{q + 1} = q^{4} + q^{2} + 1.

The Bose representation of P G(2, q^{2}) in P G(5, q) relies on the existence of a spread S of P G(5, q) of the following type.

Definition 3.2.1 A spread S (of lines) of P G(5, q) is a Bose spread if for any two distinct elements ℓ_{1}, ℓ_{2} of S, the 3−space spanned by ℓ_{1} and ℓ_{2} contains exactly q^{2} + 1 elements of S. For a Bose spread S in P G(5, q), denote by H_{3} the collection of 3−spaces

{Σ_{ℓ_{1},ℓ_{2}} = ⟨ℓ_{1}, ℓ_{2}⟩ | ℓ_{1}, ℓ_{2} ∈ S, ℓ_{1} ≠ ℓ_{2}}.

Note that there are precisely q^{4} + q^{2} + 1 3−spaces in the set H_{3} for a given Bose spread S of P G(5, q). Given a Bose spread S in P G(5, q), let π_{q^{2}}(S) be the incidence structure with: points the elements of S; lines the 3−space elements of H_{3} and incidence given by containment.

Theorem 3.2.2 [18] The incidence structure π_{q^{2}}(S), where S is a Bose spread of P G(5, q), is a projective plane of order q^{2}.

Proof: Two distinct points of π_{q^{2}}(S) correspond to two distinct elements ℓ_{1} and ℓ_{2} of S. The lines ℓ_{1}, ℓ_{2} span a unique 3−space of P G(5, q), which contains q^{2} − 1 further elements of S since S is a Bose spread. Hence two points of π_{q^{2}}(S) are contained in a unique line of π_{q^{2}}(S). Two lines of π_{q^{2}}(S) correspond to two elements of H_{3} which we shall denote by Σ_{1} and Σ_{2}. Suppose the 3−spaces Σ_{1}, Σ_{2} intersect in a plane σ of P G(5, q). Each of Σ_{1} and Σ_{2} contains a subspread of S, denote these subspreads by S_{1} and S_{2} respectively. The plane σ necessarily contains an element of S_{1} (and S_{2} respectively). Since two lines in σ intersect, and the spreads S_{1}, S_{2} ⊆ S, it follows that σ contains an element ℓ of S and ℓ is an element of both S_{1} and S_{2}. The plane σ is incident with the remaining q^{2} elements of S_{1}, meeting each such element in a point. Similarly σ is incident with the remaining q^{2} elements of S_{2}\{ℓ}, meeting each such element in a point. Thus each point of σ\{ℓ} is incident with an element of S_{1} and an element of S_{2}. Since S is a spread of P G(5, q), we have a contradiction. Thus two distinct 3−spaces in H_{3} intersect exactly in a line which is necessarily an element of S. Therefore two distinct lines of π_{q^{2}}(S) intersect in a unique point. Since π_{q^{2}}(S) has q^{4} + q^{2} + 1 points, it follows that π_{q^{2}}(S) is a projective plane of order q^{2}. □

Corollary 3.2.3 If S is a Bose spread of P G(5, q) then the associated collection of 3−spaces H_{3} is a dual spread of P G(5, q).

Proof: In the proof of Theorem 3.2.2 we showed that two distinct elements of H_{3} intersect in a line, a line which is an element of S. Therefore no hyperplane of P G(5, q) contains two elements of H_{3}. Since there are q^{4} + q^{2} + 1 3−spaces in the set H_{3}, and since each 3−space of P G(5, q) is contained in q + 1 distinct hyperplanes of P G(5, q), it follows that each hyperplane of P G(5, q) contains a unique element of H_{3}. □ We now prove a result of Thas which states that a projective plane defined by a Bose spread of P G(5, q), in the manner defined above, is necessarily Desarguesian.

Theorem 3.2.4 [87] The projective plane π_{q^{2}}(S), defined above for a Bose spread S of P G(5, q), is Desarguesian.

Proof: Let S be a Bose spread of P G(5, q) and let π_{q^{2}}(S) be the incidence structure defined as above, which by Theorem 3.2.2 is a projective plane of order q^{2}. Embed P G(5, q) as a hyperplane in P G(6, q). Let Σ_{3,q^{2}} be the incidence structure with: points the points of P G(6, q)\P G(5, q); lines the planes of P G(6, q) not contained in P G(5, q) and which intersect P G(5, q) in a unique element of S and incidence given by containment. It can be shown that Σ_{3,q^{2}} is an affine 3−space and the plane π_{q^{2}}(S) is then the plane at infinity of Σ_{3,q^{2}}. Any projective plane embedded in an affine projective 3−space is Desarguesian, hence π_{q^{2}}(S) is Desarguesian as required. This configuration also provides additional information about the Bose spread S. Let \overline{Σ}_{3,q^{2}} be the projective completion of Σ_{3,q^{2}}. Then the planes of \overline{Σ}_{3,q^{2}} distinct from π_{q^{2}}(S) are given by the 4−spaces of P G(6, q) not contained in P G(5, q) and which intersect P G(5, q) in an element of H_{3}. Each such 4−space is therefore a Bruck-Bose representation of a Desarguesian projective plane of order q^{2} with hyperplane at infinity an element of H_{3}. It follows that for each 3−space in H_{3}, the q^{2} + 1 elements of S contained in this 3−space constitute a regular spread of the space (see Section 1.10 and Theorem 1.10.1.3). □

Corollary 3.2.5 If S is a Bose spread of P G(5, q) and if H_{3} is the following collection of 3−spaces of P G(5, q)

{Σ_{ℓ_{1},ℓ_{2}} = ⟨ℓ_{1}, ℓ_{2}⟩ | ℓ_{1}, ℓ_{2} ∈ S, ℓ_{1} ≠ ℓ_{2}},

then for every element Σ_{ℓ_{1},ℓ_{2}} of H_{3} the subset of q^{2} + 1 elements of S contained in Σ_{ℓ_{1},ℓ_{2}} constitutes a regular 1− spread of Σ_{ℓ_{1},ℓ_{2}}. □

In summary, if S is Bose spread of P G(5, q), then by definition any pair of elements of S spans a 3−space containing q^{2} +1 elements of S. If a 3−space of P G(5, q) contains q^{2} +1 elements of S, then these spread elements necessarily constitute a regular 1−spread of the 3−space. The collection of such 3−spaces is a dual spread H_{3} of P G(5, q) and the Desarguesian plane P G(2, q^{2}) is isomorphic to the incidence structure with: points the elements of S; lines the elements of H_{3} and incidence given by containment. We call such a representation of P G(2, q^{2}) in P G(5, q) a Bose representation of PG(2, q^{2}).

Lemma 3.2.6 [18] For S a Bose spread in P G(5, q): Each \underline{line} m of P G(5, q) is either (I) an element of S, or (II) m is a transversal to a regulus of lines in S and m is contained in the unique element of H_{3} spanned by this regulus. Each \underline{plane} of P G(5, q) is either: (I) contained in an element of H_{3} and contains exactly one element of S, or (II) contained in no element of H_{3} and is incident with q^{2} + q + 1 distinct elements of S.

Proof Let m be a line of P G(5, q) which is not an element of S. Since S is a spread of P G(5, q), the line m is incident with exactly q + 1 elements of S. Any two elements of S incident with m span a 3−space Σ ∈ H_{3} which contains exactly q^{2} − 1 further elements of S. Since m is contained in Σ it follows that each of the elements of S incident with m is contained in Σ. By Corollary 3.2.5, the elements of S in Σ form a regular 1−spread and therefore m is a transversal to a regulus of elements of S. Each element ℓ of S is contained in q^{3} +q^{2} +q +1 planes of P G(5, q) and each such plane is spanned by ℓ and a point on a distinct element of S. Therefore, by Corollary 3.2.5, such a plane is contained in a unique element of H_{3}. Moreover, in an element of H_{3} the q^{2} + 1 elements of S constitute a 1−spread, therefore any plane contained in an element of H_{3} necessarily contains an element of S. There are (q^{4} + q^{2} + 1)(q^{3} + q^{2} + q + 1) planes of P G(5, q) of this type and the remaining planes of P G(5, q) therefore contain no element of S. Let σ be a plane in P G(5, q) which contains no element of S; σ is therefore not contained in any element of H_{3}. As S is a spread, each point of σ is incident with an element of S and since σ contains no line which is an element of S, σ is incident with q^{2} + q + 1 distinct elements of S. Each line in σ is a line of type (II) which is therefore contained in an element of H_{3}. Since σ is not contained in any element of H_{3}, σ is incident with q^{2} + q + 1 elements of H_{3}, meeting each in a line. □

Theorem 3.2.7 If S is a Bose spread of P G(5, q) and π_{q^{2}}(S) is the Bose representation of P G(2, q^{2}) defined by S, then B is a Baer subplane of P G(2, q^{2}) if and only if in the Bose representation, B is a 2−regulus (or Segre variety ρ_{1;2}) whose q^{2} +q +1 transversal lines are elements of S.

Proof Let σ be a plane in P G(5, q) of type (II) in the sense of Lemma 3.2.6. Let B be the set containing the q^{2} +q +1 elements of S incident with σ and the q^{2} +q +1 elements of H_{3} which each intersect σ in a (distinct) line. Since σ is a projective plane of order q, two elements ℓ_{1}, ℓ_{2} of S in B define a line of type (II) in σ which is contained in a unique element of H_{3} in B and which contains ℓ_{1} and ℓ_{2}. Conversely, two elements of H_{3} in B intersect in a unique element of S in B. Therefore B is a Baer subplane of P G(2, q^{2}) in the Bose representation of P G(2, q^{2}) in P G(5, q) determined by S. Counting shows there exists q^{3}(q^{3} + 1)(q^{2} + 1)(q + 1) planes σ of type (II) in P G(5, q) and from above, each such plane corresponds to a Baer subplane of P G(2, q^{2}). Since P G(2, q^{2}) contains q^{3}(q^{2} − q + 1)(q^{2} + 1)(q + 1) distinct Baer subplanes, there exists a set B, as above, of q^{2}+q+1 elements of S which contains at least 3 planes of type (II). These planes must be pairwise disjoint, otherwise two intersecting planes of type (II) span at most a 4−space of P G(5, q) which would necessarily contain more than one element of H_{3}; a contradiction. By Theorem 1.9.2, B is then a 2−regulus in P G(5, q). We continue in this way until all planes of type (II) have been considered. Each plane of type (II) therefore defines a 2−regulus of planes of type (II) in P G(5, q) with transversals all elements of S; each such 2−regulus corresponds to a Baer subplane of P G(2, q^{2}) in the Bose representation and every Baer subplane of P G(2, q^{2}) in the Bose representation is obtained in this way. □

We now prove the existence of Bose spreads in P G(5, q).

Lemma 3.2.8 It is possible to embed Π_{2,q^{2}} = P G(2, q^{2}) in P G(5, q^{2}) in such a way that Π_{2,q^{2}} = P G(2, q^{2}) is disjoint from P G(5, q), the real Baer 5−space of P G(5, q^{2}).

Proof By Sved’s result 1.3.1 a hyperplane of P G(5, q^{2}) intersects P G(5, q) in either a 4−space or a 3−space of P G(5, q). Each subspace S_{n} of P G(5, q) of dimension n extends uniquely to an n−space S_{n,q^{2}} over GF (q^{2}), since a basis for S_{n} as a vector space over GF (q) is a basis for S_{n,q^{2}} as a vector space over GF (q^{2}). Furthermore, there are more hyperplanes in P G(5, q^{2}) than in P G(5, q). It follows that there exist hyperplanes of P G(5, q^{2}) which intersect P G(5, q) in a 3−space of P G(5, q).

Let Σ_{4,q^{2}} be a hyperplane of P G(5, q^{2}) such that the intersection Σ_{4,q^{2}} ∩ P G(5, q) is a 3−space; denote this 3−space by Σ_{3}. In Σ_{4,q^{2}} the 3−space Σ_{3} extends uniquely to a 3−space Σ_{3,q^{2}} over GF (q^{2}). Let ℓ be a line of Σ_{3,q^{2}} which is skew to Σ_{3}. Let m be a line of Σ_{4,q^{2}}\Σ_{3,q^{2}} which intersects Σ_{3,q^{2}} in a unique point of ℓ. Note that m is therefore disjoint from P G(5, q). The plane spanned by ℓ and m is disjoint from Σ_{3} and is therefore disjoint from P G(5, q). □ By Lemma 3.2.8 we can embed Π_{2,q^{2}} = P G(2, q^{2}) in P G(5, q^{2}) in such a way that Π_{2,q^{2}} is disjoint from the Baer 5−space P G(5, q) of P G(5, q^{2}) and therefore no point of Π_{2,q^{2}} is fixed by the Fröbenius automorphism. The conjugate plane \overline{Π}_{2,q^{2}} is then disjoint from Π_{2,q^{2}} and disjoint from P G(5, q). The join of each point P in Π_{2,q^{2}} to its conjugate point \overline{P}, with respect to the extension GF (q^{2}) of GF (q), is a line P \overline{P} which intersects P G(5, q) in a Baer subline of P \overline{P}. In this way we obtain a set S of q^{4} + q^{2} + 1 lines of P G(5, q), one through each point P of Π_{2,q^{2}} and its conjugate \overline{P} in \overline{Π}_{2,q^{2}}. Let ℓ_{1} and ℓ_{2} be two distinct lines of S in P G(5, q). If ℓ_{1} and ℓ_{2} intersect, then since the plane spanned by ℓ_{1} and ℓ_{2} contains a line in Π_{2,q^{2}} and a line in \overline{Π}_{2,q^{2}}, the planes Π_{2,q^{2}} and \overline{Π}_{2,q^{2}} intersect in a point; a contradiction. The lines of S are therefore pairwise disjoint and since S contains q^{4} + q^{2} + 1 elements we have that S is a spread of lines in P G(5, q). It remains to prove that S is a Bose spread, that is, to prove that every 3−space of P G(5, q) spanned by two distinct elements of S contains exactly q^{2} + 1 elements of S. For each point P_{i} of Π_{2,q^{2}} denote by ℓ_{i} the line in S incident with P_{i}. Let ℓ_{i} and ℓ_{j} be distinct elements of S in P G(5, q). The 3−space Σ_{3} of P G(5, q) spanned by ℓ_{i} and ℓ_{j} extends uniquely to a 3−space Σ_{3,q^{2}} over GF (q^{2}). The lines P_{i}P_{j} of Π_{2,q^{2}} and \overline{P_{i}P_{j}} of \overline{Π}_{2,q^{2}} are contained in Σ_{3,q^{2}} and the regular spread of Σ_{3} determined by P_{i}P_{j} and \overline{P_{i}P_{j}} consists of elements of S (see Theorem 1.9.6). Hence Σ_{3} = ⟨ℓ_{i}, ℓ_{j}⟩ contains exactly q^{2} +1 elements of S. We have therefore shown,

Lemma 3.2.9 If Π_{2,q^{2}} = P G(2, q^{2}) is embedded in P G(5, q^{2}) in such a way that Π_{2,q^{2}} is disjoint from the Baer 5−space P G(5, q) of P G(5, q^{2}), then if each point P of Π_{2,q^{2}} is joined to its conjugate point \overline{P} with respect to the extension GF (q^{2}) of GF (q) we obtain a collection S of q^{4} + q^{2} + 1 lines of P G(5, q).

The set S of lines of P G(5, q) so constructed is a Bose spread of P G(5, q) which we shall call a canonical Bose spread. □

By this construction of a Bose spread S, the isomorphism between Π_{2,q^{2}} = P G(2, q^{2}) and the incidence structure π_{q^{2}}(S), determined in Theorem 3.2.4, arises in a very natural way. Furthermore, for the Bose representation of P G(2, q^{2}) defined by this construction of a Bose spread of P G(5, q), the representation of the Baer subplanes of P G(2, q^{2}) also arises in a natural way and is determined in the following theorem; a special case of [50, Lemma 25.6.8].

Theorem 3.2.10 [50, Lemma 25.6.8] Let P G(5, q^{2}) be an extension of the projective space P G(5, q). In P G(5, q^{2}), let Π_{2,q} be a 2−space over GF (q) skew to P G(5, q). If P ∈ Π_{2,q} and if \overline{P} is the conjugate point of P with respect to the extension GF (q^{2}) of GF (q), then the intersection of a line P \overline{P} and the space P G(5, q) is a line ℓ of P G(5, q). These lines ℓ form a system of maximal spaces of a Segre variety ρ_{1;2} of P G(5, q).

Finally we show that all Bose spreads of P G(5, q) are equivalent to a Bose spread constructed as in Lemma 3.2.9.

Theorem 3.2.11 If S is a Bose spread of P G(5, q), then S is a canonical Bose spread of P G(5, q).

Proof Let S be a Bose spread of P G(5, q) and let π_{q^{2}}(S) be the Bose representation of P G(2, q^{2}) in P G(5, q) defined by S. Embed P G(5, q) as a Baer subspace in P G(5, q^{2}). By Theorem 3.2.7 each Baer subplane of π_{q^{2}}(S) is a 2−regulus in P G(5, q) with the property that the transversals of the 2−regulus are all elements of S. By Theorem 1.9.5, the 2−reguli in P G(5, q) are projectively equivalent. Choose a Baer subplane B in π_{q^{2}}(S) and let ρ_{1;2} be the corresponding 2−regulus in P G(5, q). The 2−regulus ρ_{1;2} in P G(5, q) extends uniquely to a 2−regulus in P G(5, q^{2}) which we shall denote by ρ_{(q^{2})1;2}. The q^{2} + q + 1 transversals of ρ_{1;2} in P G(5, q) are therefore all elements of S and are transversals of ρ_{(q^{2})1;2} when considered as lines over GF (q^{2}). The Baer sublines of B are represented by the reguli of elements of S contained in ρ_{1;2}. Over GF (q^{2}), these reguli have transversals, one in each axis plane of ρ_{(q^{2})1;2}. Each element ℓ_{i} of S not in B is contained in a unique element Σ_{i} of H_{3} such that Σ_{i} is an element (a line) of B. In Σ_{i}, the line ℓ_{i} is disjoint from the regulus of elements of S which are the points of B in Σ_{i}. Therefore, over GF (q^{2}), ℓ_{i} intersects this regulus, which is a 3−dimensional hyperbolic quadric, in two conjugate points P_{i} and \overline{P}_{i}, where P_{i} and \overline{P_{i}} lie in distinct and conjugate axis planes of ρ_{(q^{2})1;2}. Note that P_{i} and \overline{P}_{i} are the only points of ℓ_{i} incident with the 2−regulus ρ_{(q^{2})1;2}, since otherwise ℓ_{i} would be a transversal line of ρ_{(q^{2})1;2} and would then either be a point of B or be disjoint to P G(5, q); in each case we have a contradiction. Each of the q^{2} + q + 1 elements of H_{3} in B contain a regular spread of elements of S and each such regular spread is defined by a line (and its conjugate line) over GF (q^{2}) which is contained in an axis plane (and the conjugate axis plane respectively) of ρ_{(q^{2})1;2}; denote these lines by m_{1}, m_{2}, …, m_{q^{2}+q+1} and the corresponding conjugate lines by \overline{m}_{1}, \overline{m}_{2}, …, \overline{m}_{q^{2}+q+1}. Note that every element ℓ_{i} in S is incident with at least one line m_{j}; if ℓ_{i} ∈ B, then ℓ_{i} as a line over GF (q^{2}) is incident with q+1 lines m_{j} and if ℓ_{i} ∈ S\B, then ℓ_{i} is incident with a unique line m_{j}. Since there are (q^{2} −q)/2 pairs of conjugate axis planes of ρ_{(q^{2})1;2} which are disjoint from P G(5, q), at least one of these axis planes contains two of the lines m_{j}; denote this plane by Π_{2,q^{2}} and suppose m_{1} and m_{2} are contained in Π_{2,q^{2}}. Consider four distinct points of Π_{2,q^{2}}, two incident with m_{1}, two incident with m_{2} and such that the four points form a quadrangle in Π_{2,q^{2}}. The elements of S incident with these four points correspond to a quadrangle of points in π_{q^{2}}(S) which therefore defines a unique Baer subplane B′ of π_{q^{2}}(S). By Theorem 3.2.7, B′ corresponds to a 2−regulus ρ of elements of S. Over GF (q^{2}) the 2−regulus ρ extends uniquely to a 2−regulus ρ_{q^{2}} and since Π_{2,q^{2}} intersects ρ_{q^{2}} in four distinct points, no three collinear, the plane Π_{2,q^{2}} is an axis plane of ρ_{q^{2}}. Therefore every element of S which is a point of B′ is incident with Π_{2,q^{2}} and incident with the conjugate plane \overline{Π}_{2,q^{2}}. By considering all choices of a quadrangle of points incident with the lines m_{1} and m_{2} in Π_{2,q^{2}} and since every point of Π_{2,q^{2}} lies in at least one Baer subplane of Π_{2,q^{2}} which contains the lines m_{1} and m_{2}, we obtain that every element of S is incident with Π_{2,q^{2}} and incident with the conjugate plane \overline{Π}_{2,q^{2}}. Hence every element of the spread S of P G(5, q) is obtained by joining a point P of Π_{2,q^{2}} to its conjugate \overline{P}, where Π_{2,q^{2}} is a plane of P G(5, q^{2}) skew to P G(5, q). The spread S is therefore a canonical Bose spread of P G(5, q). □

The original method of Bose in [18] was to obtain a coordinate representation of P G(2, q^{2}) in P G(5, q). Since we shall require this coordinate representation for some later calculations, we briefly present this work of Bose. Let α be a primitive root of GF (q^{2}). Then α satisfies an equation

α^{2} = α + γ

where x^{2} − x − γ is irreducible over GF (q). Any point of P G(2, q^{2}) has coordinates (x, y, z) ≠ (0, 0, 0), x, y, z ∈ GF (q^{2}) where (x, y, z) and (ρx, ρy, ρz), ρ ∈ GF (q^{2})\{0}, represent the same point in homogeneous coordinates. Since {1, α} is a basis for GF (q^{2}) as a vector space over GF (q) we can write x = x_{0} + αx_{1}, y = y_{0} + αy_{1}, z = z_{0} + αz_{1} for a unique choice of x_{i}, y_{i}, z_{i} ∈ GF (q). We let the coordinates (x, y, z) in P G(2, q^{2}) correspond to the coordinates (x_{0}, x_{1}, y_{0}, y_{1}, z_{0}, z_{1}) in P G(5, q). Note that (x_{0}, x_{1}, y_{0}, y_{1}, z_{0}, z_{1}) and (rx_{0}, rx_{1}, ry_{0}, ry_{1}, rz_{0}, rz_{1}), r ∈ GF (q)\{0}, represent the same point in P G(5, q), and they correspond to the points (x, y, z), (rx, ry, rz) respectively; the same point of P G(2, q^{2}). Consider three distinct elements a, b, c ∈ GF (q^{2})\{0}. As GF (q^{2}) is a 2−dimensional vector space over GF (q), the three elements a, b, c are linearly dependent over GF (q). Therefore there exist λ_{1}, λ_{2}, λ_{3} ∈ GF (q) not all zero and such that

λ_{1}a + λ_{2}b + λ_{3}c = 0.  (3.1)

If a = a_{0} + αa_{1}, b = b_{0} + αb_{1}, c = c_{0} + αc_{1}, where a_{i}, b_{i}, c_{i} ∈ GF (q), then from (3.1) we have,

λ_{1}a_{0} + λ_{2}b_{0} + λ_{3}c_{0} = 0
and λ_{1}a_{1} + λ_{2}b_{1} + λ_{3}c_{1} = 0  (3.2)

Consider the triplets (ax, ay, az), (bx, by, bz), (cx, cy, cz) which represent the same point (x, y, z) of P G(2, q^{2}). Since for example,

(a_{0} + αa_{1})(x_{0} + αx_{1}) = a_{0}x_{0} + γa_{1}x_{1} + α(a_{1}x_{0} + a_{0}x_{1} + a_{1}x_{1}),

the triplets correspond to the sextuplets,

(a_{0}x_{0} + γa_{1}x_{1}, a_{1}x_{0} + a_{0}x_{1} + a_{1}x_{1}, a_{0}y_{0} + γa_{1}y_{1}, a_{1}y_{0} + a_{0}y_{1} + a_{1}y_{1}, a_{0}z_{0} + γa_{1}z_{1}, a_{1}z_{0} + a_{0}z_{1} + a_{1}z_{1}) (b_{0}x_{0} + γb_{1}x_{1}, b_{1}x_{0} + b_{0}x_{1} + b_{1}x_{1}, b_{0}y_{0} + γb_{1}y_{1}, b_{1}y_{0} + b_{0}y_{1} + b_{1}y_{1}, b_{0}z_{0} + γb_{1}z_{1}, b_{1}z_{0} + b_{0}z_{1} + b_{1}z_{1}) (c_{0}x_{0} + γc_{1}x_{1}, c_{1}x_{0} + c_{0}x_{1} + c_{1}x_{1}, c_{0}y_{0} + γc_{1}y_{1}, c_{1}y_{0} + c_{0}y_{1} + c_{1}y_{1}, c_{0}z_{0} + γc_{1}z_{1}, c_{1}z_{0} + c_{0}z_{1} + c_{1}z_{1})

respectively. By (3.2), these sextuplets are linearly dependent over GF (q) and are therefore the coordinates of three collinear points in P G(5, q) whenever a, b, c are from distinct cosets of GF (q)\{0} in the multiplicative group GF (q^{2})\{0}. Therefore as ρ varies over the q^{2} −1 non-zero elements of GF (q^{2}) the sextuplets corresponding to (ρx, ρy, ρz) represent q + 1 collinear points of P G(5, q). In this way, the q^{4} + q^{2} + 1 points of P G(2, q^{2}) can be made to correspond to a set of q^{4} + q^{2} + 1 lines S_{B} of P G(5, q), each point corresponding to one line.

Theorem 3.2.12 [18, Theorem1.1, Theorem 2.2] S_{B} is a spread (of lines) of P G(5, q). Moreover, for each pair of distinct elements ℓ_{1}, ℓ_{2} of S_{B} the 3−space determined by ℓ_{1} and ℓ_{2} contains exactly q^{2} + 1 elements of S_{B}; that is, S_{B} is a Bose spread of P G(5, q). We call the Bose spread S_{B} the coordinate Bose spread of P G(5, q). □

In this coordinate setting, Bose proved the representation of Baer subplanes of P G(2, q^{2}) in P G(5, q) as given in Theorem 3.2.7. For example, consider the Baer subplane P G(2, q) of P G(2, q^{2}), and let ρ_{i} = r_{i0} + αr_{i1}, i = 1, 2, …, q + 1 be q + 1 elements of GF (q^{2}), one from each coset of GF (q)\{0} in the multiplicative group GF (q^{2})\{0}. We may choose ρ_{1} = 1, the identity. Then each point (x, y, z) ∈ P G(2, q) corresponds to a line of S_{B} in P G(5, q) whose points are given by,

{(xr_{i0}, xr_{i1}, yr_{i0}, yr_{i1}, zr_{i0}, zr_{i1}) | i = 1, 2, …, q + 1}.

The collection of q^{2} + q + 1 lines of S_{B} obtained in this way are then the set of maximal spaces (transversals) of the Segre variety ρ_{1;2} in P G(5, q) defined by the following equations, for points with coordinates (x_{0}, x_{1}, y_{0}, y_{1}, z_{0}, z_{1}),

\begin{aligned}x_{0}y_{1} − x_{1}y_{0} & = & 0 \\ x_{0}z_{1} − x_{1}z_{0} & = & 0 \\ y_{0}z_{1} − y_{1}z_{0} & = & 0; \end{aligned}

(a Segre variety in P G(5, q) by Theorem 1.8.6 and the subsequent remarks.) The opposite system of maximal spaces of the Segre variety ρ_{1;2}, the q + 1 axis planes contained in this Segre variety, are each defined by a set of equations of the form,

\begin{aligned}r_{i1}x_{0} − r_{i0}x_{1} & = & 0 \\ r_{i1}y_{0} − r_{i0}y_{1} & = & 0 \\ r_{i1}z_{0} − r_{i0}z_{1} & = & 0, \end{aligned}

for a fixed i ∈ {1, 2, …, q + 1}. Also, for each non-zero element ρ ∈ GF (q^{2}), the transformation (x, y, z) ↦ (ρx, ρy, ρz) of P G(2, q^{2}) fixes every point in P G(2, q^{2}) and therefore fixes the canonical Bose spread S_{B} of P G(5, q). Finally, we examine the relationship between the Bose representation of P G(2, q^{2}) in P G(5, q) and the Bruck-Bose representation of P G(2, q^{2}) in P G(4, q) with hyperplane Σ_{∞} at infinity and S_{∞} a regular 1−spread of Σ_{∞}. Consider the Bose representation of P G(2, q^{2}) defined by a Bose spread S of P G(5, q) as presented in this section and with the same notation. Let P G(4, q) be a hyperplane of P G(5, q). By Theorem 3.2.3, the set of 3−spaces H_{3} associated to S is a dual spread of P G(5, q) and hence the hyperplane P G(4, q) contains a unique element of H_{3}; denote this element of H_{3} by Σ_{∞}. By Corollary 3.2.5, the 3−space Σ_{∞} contains exactly q^{2} + 1 elements of S and these q^{2} + 1 lines constitute a regular 1−spread of Σ_{∞}. Moreover, each element of S not in Σ_{∞} is skew to Σ_{∞} and therefore intersects P G(4, q) in a unique point of P G(4, q)\Σ_{∞}. By the Bose correspondence between the points of P G(2, q^{2}) and the elements of S, each point of P G(2, q^{2}) corresponds either to an element of S in Σ_{∞} or to a unique point of P G(4, q)\Σ_{∞}. Also, as the lines of P G(2, q^{2}) correspond to the elements of H_{3}, and any two elements of H_{3} intersect exactly in a line of S, each line of P G(2, q^{2}) corresponds to either Σ_{∞} or to a plane of P G(4, q) not contained in Σ_{∞} and which intersects Σ_{∞} in an element of S. We therefore have by Section 1.10,

Theorem 3.2.13 [19] Given a Bose representation of P G(2, q^{2}) in P G(5, q), a Bruck-Bose representation of P G(2, q^{2}) in P G(4, q) is obtained by considering any fixed hyperplane P G(4, q) of P G(5, q) and redefining each \underline{point} and \underline{line} of P G(2, q^{2}) to be the intersection of the corresponding subspace in the Bose representation with P G(4, q). □

Corollary 3.2.14 [19] In the Bruck-Bose representation of P G(2, q^{2}) in P G(4, q) defined by a regular spread S_{∞} in a hyperplane Σ_{∞}: each Baer subplane of P G(2, q^{2}) is either a plane of P G(4, q) not contained in Σ_{∞} and which intersects Σ_{∞} in line not in S_{∞}, or a ruled cubic surface V_{2}^{3} not contained in Σ_{∞} and which intersects Σ_{∞} in a line ℓ ∈ S_{∞} which is the line directrix of V_{2}^{3}.

Proof Consider a Bruck-Bose representation of P G(2, q^{2}) in P G(4, q) obtained from a Bose representation of P G(2, q^{2}) in P G(5, q) as in Theorem 3.2.13 and with the notation introduced there. By Theorem 3.2.7, in the Bose representation of P G(2, q^{2}) each Baer subplane of P G(2, q^{2}) is a Segre variety ρ_{1;2}, with transversal lines all in the Bose spread S. Each such variety has order 3 and dimension 3 and we shall denote it by R_{3}^{3}. A Segre variety R_{3}^{3} intersects the hyperplane P G(4, q) in a variety of order 3 and dimension 2; there are two cases to consider. Let R_{3}^{3} be the Segre variety in P G(5, q) which is the Bose representation of a Baer subplane B of P G(2, q^{2}). Suppose the Baer subplane B of P G(2, q^{2}) contains the line at infinity as a line, then in the Bose representation Σ_{∞} ∈ H_{3} intersects the Segre variety R_{3}^{3} in a 1−regulus, a hyperbolic quadric H_{2}^{2}. In this case, the intersection P G(4, q) ∩ R_{3}^{3} is the variety V_{2}^{3} = H_{2}^{2} ∪ S_{2}^{1}, the union of the hyperbolic quadric H_{2}^{2} and an axis plane S_{2}^{1} of the Segre variety R_{3}^{3}. Alternatively, suppose the Baer subplane B of P G(2, q^{2}) intersects the line at infinity in a unique point, then in the Bose representation Σ_{∞} ∈ H_{3} intersects the Segre variety R_{3}^{3} exactly in a single element ℓ of S. In this case, the intersection P G(4, q) ∩ R_{3}^{3} is a variety V_{2}^{3} containing ℓ ∈ S. Each element of H_{3} which contains ℓ and which represents a line of B in the Bose representation, intersects R_{3}^{3} in a 1−regulus of elements of S. These 1−reguli each intersect P G(4, q) in a degenerate conic, namely the line ℓ and a transversal to the 1−regulus. Hence the variety V_{2}^{3} = P G(4, q) ∩ R_{3}^{3} consists of the line ℓ and q + 1 lines of P G(4, q)\Σ_{∞} which meet ℓ; such a variety is a ruled cubic surface with line directrix ℓ. □

3.3 The Bose representation of Conics in PG(2, q^{2})

Consider the conic C in P G(2, q^{2}) with points (x, y, z), x, y, z ∈ GF (q^{2}) and not all zero, satisfying the equation y^{2} = xz. Let GF (q^{2}) = GF (q)(α) where α ∈ GF (q^{2})\GF (q) has minimal polynomial p_{α}(x) = x^{2} − x − γ as in the previous section. Moreover, if q is even, then γ has trace(γ) = 1 and for q odd, γ has the property that 1 + 4γ is a non-square, since x^{2} − x − γ is irreducible in GF (q) by [48, Section 1.4]. The element \overline{α} = α^{q} is the second root of p_{α} and therefore

α + \overline{α} = 1
and α\overline{α} = −γ.  (3.3)

Consider the conic C in the Bose representation of P G(2, q^{2}) in P G(5, q), defined by the coordinate Bose spread S_{B} of the previous section. The points (x, y, z) of C in P G(2, q^{2}) correspond to the points (x_{0}, x_{1}, y_{0}, y_{1}, z_{0}, z_{1}) of P G(5, q) which satisfy

(y_{0} + αy_{1})^{2} = (x_{0} + αx_{1})(z_{0} + αz_{1})  (3.4)

where x = x_{0}+αx_{1}, y = y_{0}+αy_{1}, z = z_{0}+αz_{1} with x_{i}, y_{i}, z_{i} ∈ GF (q). By expanding (3.4) and using the equations (3.3) and α^{2} = α + γ to simplify the expression, we obtain

y_{0}^{2} + γy_{1}^{2} + α(2y_{0}y_{1} + y_{1}^{2}) = x_{0}z_{0} + γx_{1}z_{1} + α(x_{1}z_{0} + x_{0}z_{1} + x_{1}z_{1}).

Thus the conic C in P G(2, q^{2}) in the Bose representation is the subset of q^{2} + 1 elements of the Bose spread S_{B}, no three contained in the same 3−space of H_{3}, with points (x_{0}, x_{1}, y_{0}, y_{1}, z_{0}, z_{1}) in P G(5, q) contained in the intersection of the two quadrics Q_{1} and Q_{2} with equations

y_{0}^{2} + γy_{1}^{2} − x_{0}z_{0} − γx_{1}z_{1} = 0,  (3.5)
and y_{1}^{2} + 2y_{0}y_{1} − x_{1}z_{0} − x_{0}z_{1} − x_{1}z_{1} = 0  (3.6)

respectively. We now determine if each quadric Q_{i} is non-singular, and if so, we determine the characteristic (hyperbolic or elliptic) of the quadric. Our method and notation are consistent with that used in Section 1.4 ([50, Section 22.2] and in particular [50, Theorem 22.2.1]). Let x = (x_{0}, x_{1}, y_{0}, y_{1}, z_{0}, z_{1}), then Q_{1} and Q_{2} are given by the quadratic forms defined by matrices

M_{1} = [ \begin{matrix}0 & 0 & 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 0 & 0 & −γ \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & γ & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} ] and M_{2} = [ \begin{matrix}0 & 0 & 0 & 0 & 0 & −1 \\ 0 & 0 & 0 & 0 & −1 & −1 \\ 0 & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} ]

respectively. For the quadric Q_{1}, let

A = [ \begin{matrix}0 & 0 & 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 0 & 0 & −γ \\ 0 & 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2γ & 0 & 0 \\ −1 & 0 & 0 & 0 & 0 & 0 \\ 0 & −γ & 0 & 0 & 0 & 0 \end{matrix} ] .

Therefore |A| = 4γ^{3} ≠ 0 for q odd and |A| = 4γ^{3} = 0 for q even. When q is odd, let a = −4γ^{3}. By Theorem 1.4.1 for q odd, the quadric Q_{1} is non-singular and Q_{1} is elliptic if and only if a is a non-square; for q even, the quadric Q_{1} is singular. For the quadric Q_{2}, let

A = [ \begin{matrix}0 & 0 & 0 & 0 & 0 & −1 \\ 0 & 0 & 0 & 0 & −1 & −1 \\ 0 & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 2 & 0 & 0 \\ 0 & −1 & 0 & 0 & 0 & 0 \\ −1 & −1 & 0 & 0 & 0 & 0 \end{matrix} ] .

Therefore |A| = 4 ≠ 0 for q odd and |A| = 4 = 0 for q even. When q is odd, let a = −4. By Theorem 1.4.1 for q odd, the quadric Q_{2} is non-singular and Q_{2} is elliptic if and only if a is a non-square; for q even, the quadric Q_{2} is singular. Thus for q odd, the quadrics Q_{1} and Q_{2} are both non-singular and for q even the quadrics Q_{1} and Q_{2} are both singular. Consider the case when q is even. In this case the conic C in P G(2, q^{2}) has nucleus N(0, 1, 0). In the coordinate Bose representation of P G(2, q^{2}) in P G(5, q), the nucleus is represented by the line joining points (0, 0, 1, 0, 0, 0) and (0, 0, 0, 1, 0, 0); denote this line by ℓ_{N}. The line ℓ_{N} intersects the quadric Q_{1} in the point P_{1}(0, 0, \sqrt{γ}, 1, 0, 0) (since q is even, γ is a square) and ℓ_{N} intersects Q_{2} in the point P_{2}(0, 0, 1, 0, 0, 0). Consider Π_{P_{1}} the tangent space to Q_{1} at the point P_{1}. Following Section 1.4, the partial derivatives of the quadratic form defining Q_{1} are,

\frac{∂Q_{1}}{∂x_{0}} = −z_{0} \frac{∂Q_{1}}{∂x_{1}} = −γz_{1} \frac{∂Q_{1}}{∂y_{0}} = 2y_{0} \frac{∂Q_{1}}{∂y_{1}} = 2γy_{1} \frac{∂Q_{1}}{∂z_{0}} = −x_{0} \frac{∂Q_{1}}{∂z_{1}} = −γx_{1},

which all equal zero at P_{1}(0, 0, \sqrt{γ}, 1, 0, 0). Thus Π_{P_{1}} is the entire space and therefore P_{1} is contained in the vertex of Q_{1}.

Consider Π_{P_{2}} the tangent space to Q_{2} at the point P_{2}. The partial derivatives of the quadratic form defining Q_{2} are given by,

\frac{∂Q_{2}}{∂x_{0}} = −z_{1} \frac{∂Q_{2}}{∂x_{1}} = −z_{1} − z_{0} \frac{∂Q_{2}}{∂y_{0}} = 2y_{1} \frac{∂Q_{2}}{∂y_{1}} = 2y_{1} + 2y_{0} \frac{∂Q_{2}}{∂z_{0}} = −x_{1} \frac{∂Q_{2}}{∂z_{1}} = −x_{1} − x_{0},

which all equal zero at P_{2}(0, 0, 1, 0, 0, 0). Thus Π_{P_{2}} is the entire space and therefore P_{2} is contained in the vertex of Q_{2}. Moreover, for i = 1, 2 respectively, P_{i} is the only point of Q_{i} such that Π_{P_{i}} = P G(5, q) and therefore each of the quadrics Q_{1}, Q_{2} has a point vertex for q even and the vertex is incident with ℓ_{N}. In the notation of [50], Q_{i} = Π_{0}P_{4}, that is Q_{i} has a point vertex and base a parabolic quadric in a hyperplane disjoint from the vertex. A further verification that the vertex of Q_{i} is a point vertex is to check that a hyperplane of P G(5, q) not incident with P_{i} intersects Q_{i} in a non-singular quadric. Consider the hyperplane Σ_{0} of P G(5, q) defined by the equation y_{0} = 0 and which contains neither P_{1} nor P_{2}. The intersection Σ_{0} ∩ ℓ_{N} is the point with coordinates (0, 0, 0, 1, 0, 0). For i = 1, 2, let Q_{i,0} be the quadric in Σ_{0} such that Q_{i,0} = Q_{i} ∩ Σ_{0}. Then each quadric Q_{i,0} has points with coordinates (x_{0}, x_{1}, 0, y_{1}, z_{0}, z_{1}) satisfying the equations

\begin{aligned}Q_{1,0} : & y_{1}^{2} − x_{1}z_{0} − x_{0}z_{1} − x_{1}z_{1} & = & 0 \\ Q_{2,0} : & γy_{1}^{2} − x_{0}z_{0} − γx_{1}z_{1} & = & 0 \end{aligned}

respectively. For each of Q_{1,0} and Q_{2,0}, \frac{1}{2}|A| ≠ 0 and therefore the quadrics Q_{1,0} and Q_{2,0} are non-singular parabolic quadrics by Theorem 1.4.1. Furthermore, for i = 1, 2, the point (0, 0, 0, 1, 0, 0) is the unique point in Σ_{0} at which \frac{∂Q_{i,0}}{∂x_{j}} = \frac{∂Q_{i,0}}{∂y_{j}} = \frac{∂Q_{i,0}}{∂z_{j}} = 0, and therefore (0, 0, 0, 1, 0, 0) is the nucleus of Q_{i,0}. Hence for each i = 1, 2, the quadric Q_{i} in P G(5, q) in non-singular with point vertex and a parabolic quadric base. Moreover, since the partial derivatives of both Q_{1} and Q_{2} all equal zero when evaluated at any point of the line ℓ_{N}, if Σ is a hyperplane of P G(5, q) which intersects ℓ_{N} in a unique point distinct from P_{1} and P_{2}, then the intersections Σ ∩ Q_{1} and Σ ∩ Q_{2} are both parabolic quadrics in Σ with common nucleus Σ ∩ ℓ_{N}. It also follows that if Σ_{k} is a hyperplane of P G(5, q) such that the intersection Σ_{k} ∩ ℓ_{N} is exactly the point P_{1}, then the intersection Σ_{k} ∩ Q_{1} is a singular quadric with point vertex P_{1} and the intersection Σ_{k} ∩ Q_{2} is a non-singular quadric with nucleus P_{1}. Similarly, if Σ_{k} is a hyperplane of P G(5, q) such that the intersection Σ_{k} ∩ ℓ_{N} is exactly the point P_{2}, then the intersection Σ_{k} ∩ Q_{2} is a singular quadric with point vertex P_{2} and the intersection Σ_{k} ∩ Q_{1} is a non-singular quadric with nucleus P_{2}.

For S a Bose spread of P G(5, q) and π_{q^{2}}(S) the Bose representation of P G(2, q^{2}) in P G(5, q) defined by S and H_{3} the dual spread associated with S, we have therefore shown,

Theorem 3.3.1 If C is a non-degenerate conic in P G(2, q^{2}), then in the Bose representation of P G(2, q^{2}) in P G(5, q), the conic is a collection C_{B} of q^{2} + 1 elements of S, no three contained in the same 3−space of H_{3}. Furthermore, for q odd, the points of C_{B} lie in the intersection of two non-singular quadrics in P G(5, q); for q even, the points of C_{B} lie in the intersection of two singular quadrics Q_{1} and Q_{2}, each of which has a point vertex and a parabolic quadric base. In the q even case, the nucleus N of C is a line ℓ_{N} in the Bose representation and the point vertices of Q_{1} and Q_{2} are distinct points of ℓ_{N}. □

Consider a non-degenerate conic C in P G(2, q^{2}) and let ℓ_{∞} denote an external line of C; call this line the line at infinity of P G(2, q^{2}). If q is even, then the nucleus N of C is not incident with ℓ_{∞}. In the Bose representation of P G(2, q^{2}) in P G(5, q), the line at infinity is a 3−space element Σ_{∞} of H_{3} which is disjoint to the set of q^{2} +1 elements of S in the Bose representation C_{B} of the conic C. Moreover Σ_{∞} is disjoint from the element ℓ_{N} of S which corresponds to the nucleus N in the q even case. In Theorem 3.2.13 we presented the relationship between a Bose representation of P G(2, q^{2}) in P G(5, q) and a Bruck-Bose representation of P G(2, q^{2}) in P G(4, q). Let Π_{4} be a hyperplane of P G(5, q) which contains the 3−space Σ_{∞}; note that for the q even case, the intersection Π_{4} ∩ ℓ_{N} is a unique point. By Theorems 3.2.13, 3.3.1 and the remarks preceding Theorem 3.3.1 we have,

Theorem 3.3.2 If C is a non-degenerate conic in P G(2, q^{2}) disjoint from the line at infinity ℓ_{∞}, then in the Bruck-Bose representation of P G(2, q^{2}) in P G(4, q) the conic C is a collection of q^{2} +1 affine points contained in the intersection of two quadrics Q_{1} and Q_{2}. For q even the two quadrics are either both non-singular with a common nucleus or exactly one of the quadrics is singular with a point vertex which is then the nucleus of the second quadric. □

From our work in the earlier chapters, a Baer subline of a line of P G(2, q^{4}) which is disjoint from the line at infinity, is represented in 4−dimensional Bruck-Bose by a nondegenerate conic in a plane about a spread element and such that the conic is disjoint from the hyperplane at infinity (see Section 2.2). If we then represent P G(2, q^{4}) in 8−dimensional Bruck-Bose we obtain a representation of the non-degenerate conic, which in turn represents a Baer subline of a line of P G(2, q^{4}). Theorem 3.3.2 provides additional information about this representation which was first discussed in Theorem 2.7.2 in a slightly different setting.

3.4 The Bruck-Bose representation of PG(2, q^{4}) in

PG(8, q) revisited

The Bruck-Bose representation of P G(2, q^{4}) in P G(8, q) is determined by a regular 3−spread S_{3} in a fixed hyperplane Σ_{7,q} of P G(8, q). We denote this representation by Π_{8,q} and we denote by ℓ_{∞} the line of P G(2, q^{4}) which corresponds to the spread S_{3} in Σ_{7,q}; we call ℓ_{∞} the line at infinity. In the first section of this chapter we investigated the representation of the affine Baer subplanes of P G(2, q^{4}) in Π_{8,q}. In Corollary 3.1.5 we characterised the affine Baer subplanes of P G(2, q^{4}) in terms of this representation. Moreover we determined how Baer sublines b_{ℓ} of lines of P G(2, q^{4}), and such that b_{ℓ} is incident with ℓ_{∞}, are represented in Π_{8,q}. We now consider the non-affine Baer subplanes of P G(2, q^{4}) and the Baer sublines which are disjoint from ℓ_{∞}. Let ℓ be a line in P G(2, q^{4}) distinct from ℓ_{∞} and let P be the unique point of intersection of ℓ and ℓ_{∞}. Let b_{ℓ} be a Baer subline of ℓ such that b_{ℓ} is disjoint from ℓ_{∞}, so that P is not incident with b_{ℓ}. Also let Π_{4,q^{2}} be the Bruck-Bose representation of P G(2, q^{4}) in P G(4, q^{2}) defined by a regular 1−spread S_{1} of a hyperplane Σ_{3,q^{2}} of P G(4, q^{2}). In Π_{4,q^{2}}, ℓ is represented by a plane ℓ^{∗} of P G(4, q^{2})\Σ_{3,q^{2}} and the intersection ℓ^{∗} ∩ Σ_{3,q^{2}} is a line P^{∗} which is an element of the spread S_{1}. By Theorem 2.2.3, the Baer subline b_{ℓ} is represented by a non-degenerate conic b_{ℓ}^{∗} in the plane ℓ^{∗} such that b_{ℓ}^{∗} is disjoint from P^{∗}; in Section 2.2 we called such a conic a Baer conic. Note that the plane ℓ^{∗}\{P^{∗}}, namely ℓ^{∗} with the line P^{∗} and all its points removed, is isomorphic to the affine plane AG(2, q^{2}). By the results on internal structures of a Miquelian inversive planes discussed in Section 1.14 we have: the points of ℓ^{∗}\{P^{∗}} correspond to the points of ℓ distinct from P; the lines of ℓ^{∗}\{P^{∗}} correspond to the Baer sublines of ℓ which contain P; incidence is containment.

In Π_{8,q}, the line ℓ is represented by a 4−space ℓ^{∗∗} of P G(8, q)\Σ_{7,q} and the intersection ℓ^{∗∗} ∩Σ_{7,q} is a 3−space element P^{∗∗} of the regular 3−spread S_{3} of Σ_{7,q}. By Theorem 3.1.2, there exists a fixed induced regular 1−spread S_{ℓ}^{1} in P^{∗∗}. By Corollary 3.1.5 the planes of ℓ^{∗∗}\P^{∗∗} which intersect P^{∗∗} in a unique line of S_{ℓ}^{1} represent the Baer sublines of ℓ which contain the point P. Hence this regular 1−spread S_{ℓ}^{1} in P^{∗∗} defines a 4−dimensional Bruck-Bose representation of P G(2, q^{2}) in the 4−space ℓ^{∗∗}; this is the 4−dimensional Bruck-Bose representation of the plane ℓ^{∗}. Let b_{ℓ}^{∗∗} denote the representation in Π_{8,q} of the Baer subline b_{ℓ} of ℓ. In Π_{4,q^{2}}, since b_{ℓ}^{∗} is a non-degenerate conic in ℓ^{∗}, disjoint from the line P^{∗} in ℓ^{∗}, it follows from the preceding paragraph that b_{ℓ}^{∗∗} ⊆ ℓ^{∗∗} is precisely a representation in 4−dimensional Bruck-Bose of a non-degenerate conic in the plane ℓ^{∗} = P G(2, q^{2}) and disjoint from the line at infinity, P^{∗}. This representation was explicitly determined in Section 3.3, in particular in Theorem 3.3.2. By Theorem 2.2.9, a Baer subplane B of P G(2, q^{4}) which intersects ℓ_{∞} in a unique point R is represented in Π_{4,q^{2}} by a ruled cubic surface B^{∗} with line directrix R^{∗} where R^{∗} is an element of S_{1}. Moreover, the intersection B^{∗} ∩Σ_{3,q^{2}} in Π_{4,q^{2}} is exactly the line R^{∗} and the points of B^{∗} lie on q^{2} + 1 distinct lines of Π_{4,q^{2}}\Σ_{3,q^{2}}, one through each point of R^{∗}. These lines represent the Baer sublines in B which are incident with R. The remaining Baer sublines in B are represented in Π_{4,q^{2}} by q^{2} Baer conics on the ruled cubic surface B^{∗}. In Π_{8,q}, the Baer subplane B is represented by a structure B^{∗∗} in P G(8, q). The point R in P G(2, q^{4}) is represented by an element R^{∗∗} of the spread S_{3} of Σ_{7,q}. By Theorem 3.1.2 and Corollary 3.1.5 and by considering the situation in Π_{4,q^{2}} above, the Baer sublines in B incident with R are represented in Π_{8,q} by q^{2} + 1 distinct planes in P G(8, q)\Σ_{7,q} each of which intersect Σ_{7,q} in a distinct line of the induced 1−spread S_{j}^{1} in R^{∗∗}. Moreover, as B^{∗} contains q^{2} Baer conics in Π_{4,q^{2}}, the structure B^{∗∗} contains q^{2} representations of Baer conics in Π_{8,q}, where each has the structure determined in Theorem 3.3.2 for a given 4−space of Π_{8,q}, which corresponds to a line of P G(2, q^{4}). It is difficult to determine in more helpful detail the Bruck-Bose representation of the non-affine Baer subplanes of P G(2, q^{4}). To conclude the chapter we present one more geometric construction which may help to clarify some of the geometric properties of these representations. For the plane P G(2, q^{4}) we have been moving back and forth between the 4−dimensional Bruck-Bose representation in P G(4, q^{2}) and the 8−dimensional Bruck-Bose representation in P G(8, q); the following construction provides a concrete link between the two representations. Let S be a Bose spread of lines of P G(5, q^{2}), so that by the results of Section 3.2, S defines a Bose representation of P G(2, q^{4}). Let H_{3} denote the dual spread of P G(5, q^{2}) associated with S in the usual way. Let P G(4, q^{2}) denote a fixed hyperplane of P G(5, q^{2}). By Theorem 3.2.13, P G(4, q^{2}) determines a fixed 4−dimensional Bruck-Bose representation Π_{4,q^{2}} of P G(2, q^{4}). Denote by Σ_{3,q^{2}} the unique 3−space in H_{3} which is contained in P G(4, q^{2}). By Theorem 3.2.5, the set S_{1} of q^{4} + 1 elements of S contained in Σ_{3,q^{2}} constitutes a regular 1−spread of Σ_{3,q^{2}}. Embed P G(5, q^{2}) as a subspace in P G(11, q^{2}) in such a way that P G(5, q^{2}) is skew to P G(11, q); this is possible by Construction 3.1.1 with h = 3. The 3−space Σ_{ℓ} spanned by an element ℓ of the Bose spread S and its conjugate \overline{ℓ}, with respect to the extension GF (q^{2}) of GF (q), intersects P G(11, q) in a 3−space; the join of each point P ∈ ℓ to its conjugate \overline{P} yields a regular 1−spread of the 3−space Σ_{ℓ} ∩ P G(11, q). The collection of such 3−spaces in P G(11, q) constitutes a 3−spread of P G(11, q). The 7−space of P G(11, q^{2}) spanned by Σ_{3,q^{2}} and its conjugate space \overline{Σ}_{3,q^{2}} intersects P G(11, q) in a 7−space which we shall denote by Σ_{7,q}. Note that since Σ_{3,q^{2}} contains q^{4} + 1 distinct elements of S, Σ_{7,q} contains exactly q^{4} + 1 elements of the 3−spread of P G(11, q) which therefore constitute a 3−spread of Σ_{7,q}; denote this 3−spread of Σ_{7,q} by S_{3}. The hyperplane P G(4, q^{2}) of P G(5, q^{2}) together with its conjugate \overline{P G(4, q^{2})} spans a 9−space of P G(11, q^{2}) which intersects P G(11, q) in a 9−space which we shall denote by P G(9, q). Note that Σ_{7,q} is a subspace of P G(9, q). Each point P ∈ P G(4, q^{2})\Σ_{3,q^{2}} is incident with a line of P G(9, q), namely the line P \overline{P}. If we let P G(8, q) be a hyperplane of P G(9, q) which contains Σ_{7,q}, then P G(8, q) intersects each line P \overline{P} in a unique point, where P ∈ P G(4, q^{2})\Σ_{3,q^{2}}. By Theorem 3.1.1 the 3−spread S_{3} in Σ_{7,q} is regular since the 1−spread S_{1} in Σ_{3,q^{2}} is regular. Therefore S_{3} defines an 8−dimensional Bruck-Bose representation Π_{8,q} of P G(2, q^{4}) in P G(8, q). Moreover, in this construction the correspondence between Π_{4,q^{2}} and Π_{8,q} arises in a natural way.

Baer subplanes and

Buekenhout-Metz Unitals

In this chapter we investigate the relationship between Baer subplanes and unitals in planes where both of these objects are defined. In particular, in a finite projective plane π_{q^{2}} of order q^{2}, we consider the problem of classifying the subsets of points of the plane, which are the set of points in the intersection of a Baer subplane and a unital. The earliest work on this problem is due to Seib [70] and the relevant paper is written in German; the following statement of Seib’s result is taken from [16, Lemma 2.1, (1) (2)] [17, Result 2.4].

Theorem 4.0.1 [70] Let σ be a Baer involution which leaves invariant a unital \overline{U} of a finite projective plane π_{q^{2}}, of square order q^{2}. Then B, the Baer subplane fixed pointwise by σ, contains exactly q + 1 points of \overline{U}, and exactly q + 1 tangents of \overline{U} are lines of B. If q is even, then the q + 1 points in B ∩ \overline{U} are collinear in B. If q is odd, then the q + 1 points in B ∩ \overline{U} form a (q + 1)-arc in B. □

In [51] Hölz discussed classical unitals and Baer subplanes in P G(2, q^{2}) and used his results to define two new designs. The results Hölz obtained on the intersection of a classical unital and a Baer subplane in P G(2, q^{2}) are as follows. In the paper [51], Hölz defines, Property (T): For each point P in P G(2, q^{2}), which lies in both a Baer subplane B and a classical unital \overline{U}, the tangent line t_{P} to \overline{U} at P is a line of B.

Theorem 4.0.2 [51, Lemma 2.2] Let B be a Baer subplane of P G(2, q^{2}) satisfying property (T), which contains at least three distinct points of a classical unital \overline{U}. Then B has exactly q + 1 points in common with \overline{U}. If q is even, these points are collinear. If q is odd, these points are either collinear or they form an oval in B. □

In [23] Bruen and Hirschfeld gave many combinatorial results for the intersection of a set of type (m, n) and a set of type (m′, n′) in a plane of order q, including specific results when the two sets concerned are a Baer subplane and a unital in a plane π_{q^{2}} of order q^{2}. In [44] Grüning gave similar results including the following result which has proved to be very useful in characterising unitals of P G(2, q^{2}) (see the next chapter).

Theorem 4.0.3 [44] [23] Let B be a Baer subplane and let \overline{U} be a unital in a projective plane π_{q^{2}} of order q^{2}. Denote by b_{1} the number of lines of B which when extended are tangent lines of \overline{U} and let |B ∩ \overline{U}| denote the number of points in the intersection of B and \overline{U}. Then,

|B ∩ \overline{U}| + b_{1} = 2(q + 1)
□

For a projective plane π_{q^{2}} of order q^{2} and a unital \overline{U} in π_{q^{2}}, the set of tangent lines to \overline{U} constitutes the set of points of a unital in the dual plane π_{q^{2}}^{d} of π_{q^{2}} (see Section 1.13.2); this unital is the dual unital of \overline{U} and is denoted by \overline{U}^{d}. Recall also, that for any Baer subplane B of π_{q^{2}} the set of lines of B constitutes a set of points of a Baer subplane B^{d} in the dual plane π_{q^{2}}^{d} and B^{d} is the dual Baer subplane of B in π_{q^{2}}^{d}. By Theorem 4.0.3, we have

Corollary 4.0.4 Let B be a Baer subplane and let \overline{U} be a unital in a projective plane π_{q^{2}} of order q^{2}. If we let |B ∩ \overline{U}| denote the number of points in the intersection of B and \overline{U}, then in the dual plane π_{q^{2}}^{d} of π_{q^{2}} we have,

|B^{d} ∩ \overline{U}^{d}| = 2(q + 1) − |B ∩ \overline{U}|

where B^{d} and \overline{U}^{d} are the dual structures of B and \overline{U} respectively. □

Theorem 4.0.3 also provides a bound on the maximum possible number of points in the intersection of a unital \overline{U} and Baer subplane B in a projective plane π_{q^{2}}, namely,

0 ≤ |B ∩ \overline{U}| ≤ 2(q + 1).

The exact values of |B ∩\overline{U}| for which there exists a set of intersection of a Baer subplane and a unital in π_{q^{2}} of cardinality |B ∩ \overline{U}|, has been determined in [23] for \overline{U} a classical unital \overline{U} and a Baer subplane B in P G(2, q^{2}). In [23] Bruen and Hirschfeld used the canonical equation of a classical unital in P G(2, q^{2}) and an algebraic proof to obtain the following result.

Theorem 4.0.5 [23] In P G(2, q^{2}), for \overline{U} a classical unital and B a Baer subplane we have

|B ∩ \overline{U}| = 1, q + 1 or 2q + 1

where the intersection sets are a unique point, q + 1 points of a line of B or a conic in B, or a line pair in B respectively. □

We extend this work by giving a geometric proof of the above result which we obtain as a corollary to our results concerning the Buekenhout-Metz unitals in P G(2, q^{2}).

4.1 The intersection of a Baer subplane and a

Buekenhout-Metz unital in PG(2, q^{2})

We begin with a theorem which generalises Theorem 4.0.5 in certain cases. We acknowledge that recently in the literature some of the results in Theorem 4.1.1 have been proved independently in papers discussing derivation of Buekenhout-Metz unitals. See for example [11], [12], [31].

Theorem 4.1.1 Let \overline{U} be a Buekenhout-Metz unital re (T, ℓ_{∞}) in P G(2, q^{2}) and let B be a Baer subplane in P G(2, q^{2}). Then,

(i) if |B ∩ ℓ_{∞}| = q + 1 then |B ∩ \overline{U}| = 1, q + 1 or 2q + 1 where the intersection sets are a unique point, q + 1 points of a line of B or an oval in B, or a line pair in B respectively.

(ii) if B ∩ ℓ_{∞} = \{T \} then |B ∩ \overline{U}| = q + 1 or 2q + 1 where the intersection sets are a union of q points of B distinct from T and incident with distinct lines of B through T either the unique point T or q + 1 points of a Baer subline in B containing T respectively.

(Note: The remaining case B ∩ ℓ_{∞} = {P } with P ≠ T is considered later in the chapter.)

Proof: As the setting for our proof we use the 4−dimensional Bruck-Bose representation Π_{4} of P G(2, q^{2}), defined by a regular 1−spread S of a hyperplane Σ_{∞} of P G(4, q). As \overline{U} is a Buekenhout-Metz unital in P G(2, q^{2}), in Bruck-Bose \overline{U} is an ovoidal cone \overline{U}^{∗} with base ovoid O and vertex V, where V is incident with an element t of the spread S, and t represents the unique point T of \overline{U} at infinity in P G(2, q^{2}). The line at infinity ℓ_{∞} of P G(2, q^{2}) corresponds to the hyperplane Σ_{∞} and in Bruck-Bose, \overline{U}^{∗} ∩ Σ_{∞} = \{t\}.

(i) In this case, the line at infinity is a line of the Baer subplane B and therefore B is represented in Bruck-Bose by a plane B of P G(4, q)\Σ_{∞} which intersects Σ_{∞} in a line which is not an element of the spread S. Suppose that V ∈ B, then, since \overline{U}^{∗} is an ovoidal cone with vertex V, the intersection B ∩ \overline{U}^{∗} is the unique point V, a generator line of \overline{U}^{∗} or a pair of distinct generator lines of \overline{U}^{∗}. In these cases the number |B ∩ \overline{U}| of points in the intersection equals 1, q + 1 or 2q + 1 respectively. Alternatively, suppose that V ∉ B. Note that the hyperplane Σ_{∞} is the unique hyperplane of P G(4, q) which intersects the ovoidal cone \overline{U}^{∗} in exactly the line t; since in the quotient 3−space determined by V, Σ_{∞} corresponds to the unique tangent plane to the ovoid determined by \overline{U}^{∗} at the ovoid point corresponding to t. Therefore in P G(4, q) and since B is not contained in Σ_{∞}, the hyperplane ⟨V, B⟩, spanned by the plane B and the vertex V of \overline{U}^{∗} intersects \overline{U}^{∗} in either an oval cone or in a unique line on \overline{U}^{∗} distinct from t. In the latter case, B ∩ \overline{U}^{∗} is a unique point of P G(4, q)\Σ_{∞} and hence the Baer subplane B intersects \overline{U} in a unique point. Consider the case where the hyperplane ⟨V, B⟩ intersects \overline{U}^{∗} in an oval cone. Since V ∉ B, the transversal plane B is either tangent to this oval cone or intersects the oval cone in an oval of q + 1 points of \overline{U}^{∗}. In these two cases the number |B ∩ \overline{U}| of points in the intersection equals 1 or q + 1 respectively.

(ii) In this case, the line at infinity intersects B in the unique point T and therefore in Bruck-Bose, B is a Baer ruled cubic surface B with line directrix t. Furthermore, since T is the unique point at infinity of \overline{U}, in Bruck-Bose we have the intersection B ∩ Σ_{∞} = \overline{U}^{∗} ∩ Σ_{∞} = \{t\}. Denote the generators of B by g_{1}^{∗}, …, g_{q+1}^{∗}, where g_{1}^{∗} denotes the unique generator line of B incident with the point V of t. Recall that each generator line of \overline{U}^{∗} passes through V and each plane of P G(4, q)\Σ_{∞} about t contains a unique generator line of \overline{U}^{∗}. Thus each plane ⟨g_{i}^{∗}, t⟩ i = 1, …, q + 1 contains a generator line, l_{i}^{∗} say, of \overline{U}^{∗}. Note that as g_{1}^{∗} passes through V, g_{1}^{∗} is either the generator line l_{1}^{∗} of \overline{U}^{∗} or intersects l_{1}^{∗} in the unique point V. Each line g_{i}^{∗} (i ≠ 1) does not pass through V and therefore in the plane ⟨g_{i}^{∗}, t⟩, the line g_{i}^{∗} intersects the generator line l_{i}^{∗} of \overline{U}^{∗} in a unique point of P G(4, q)\Σ_{∞}. Therefore, for such a Baer subplane B, and for these two cases the number |B ∩ \overline{U}| of points in the intersection of the Baer subplane and the Buekenhout-Metz unital equals 2q + 1 or q + 1 respectively. □ Note that by the proof of Theorem 4.1.1, if a Baer subplane contains an oval of points of a Buekenhout-Metz unital \overline{U} in P G(2, q^{2}) as in case (i), then the oval B ∩ \overline{U} is related to an oval plane section of the 3−dimensional base ovoid of \overline{U} in the following way.

Corollary 4.1.2 Let \overline{U} be a Buekenhout-Metz unital re (T, ℓ_{∞}) in P G(2, q^{2}) and let B be a Baer subplane in P G(2, q^{2}) such that ℓ_{∞} is a line of B. Let O denote the base ovoid of \overline{U}. If the intersection B ∩ \overline{U} is an oval O in B, then the oval O is projectively equivalent to an oval contained in a 3−dimensional oval cone with base oval a plane section of the ovoid O. □

We now obtain the Bruen and Hirschfeld result (Theorem 4.0.5) as a corollary to Theorem 4.1.1 as follows.

Corollary 4.1.3 [23] In P G(2, q^{2}), for \overline{U} a classical unital and B a Baer subplane we have

|B ∩ \overline{U}| = 1, q + 1 or 2q + 1

where the intersection sets are a unique point, q + 1 points of a line of B or q + 1 points of a conic in B and a line pair in B respectively.

Proof: By Section 1.13.3 the classical unital \overline{U} is Buekenhout-Metz re (T, l_{T}) for all points T ∈ \overline{U} and the corresponding tangent line l_{T} to \overline{U} at T. We begin by showing that for every Baer subplane B of P G(2, q^{2}) there exists at least one line of B which when extended is a tangent line of \overline{U}. Suppose B is a Baer subplane of P G(2, q^{2}) such that B contains no line tangent to \overline{U}, then by Theorem 4.0.3, |B ∩ \overline{U}| = 2q + 2. As \overline{U} is classical, by Theorem 1.13.1.2 every Baer subline of P G(2, q^{2}) intersects \overline{U} in 0, 1, 2 or q + 1 points; in particular, every Baer subline in B intersects B ∩ \overline{U} in 0, 1, 2 or q + 1 points. The set B ∩ \overline{U} has too many points to be an oval in B. If B ∩ \overline{U} is the disjoint union of a Baer subline in B and an oval in B, then a secant (Baer sub)line of the oval intersects B ∩ \overline{U} in three points, contradicting Theorem 1.13.1.2. If B ∩ \overline{U} is the union of two lines in B plus a further point Q say, then there exist q Baer sublines in B through Q which intersect B ∩ \overline{U} in three points, contradicting Theorem 1.13.1.2. Alternatively, one could argue that by Theorem 1.13.1.2 B ∩ \overline{U} is a Tallini set in B and no Tallini set in a plane of order q has cardinality 2q + 2 (see [55].) Therefore, the number of points |B ∩\overline{U}| in the intersection of B and \overline{U} is necessarily less than 2q + 2 and by Theorem 4.0.3 this implies that B contains at least one line which when extended is a tangent line of \overline{U}; denote this line by ℓ_{∞}. Since \overline{U} is Buekenhout-Metz with respect to the line ℓ_{∞} and since B contains ℓ_{∞} as a line, by Theorem 4.1.1, B intersects \overline{U} in 1, q + 1 or 2q + 1 points. Moreover since a classical unital has as base ovoid an elliptic quadric, by Corollary 4.1.2 if a Baer subplane B of P G(2, q^{2}) intersects \overline{U} in an oval, then the oval is a non-degenerate conic in B. □

Theorem 4.1.1 does not exhaust the possible intersections of a Buekenhout-Metz (B-M) unital \overline{U} and a Baer subplane B of P G(2, q^{2}). It remains to consider the case when \overline{U} is B-M re (T, ℓ_{∞}) and B is a Baer subplane of P G(2, q^{2}) such that B ∩ ℓ_{∞} is a unique point P on ℓ_{∞} distinct from T. We partially solve the problem in this case, by improving the restriction on the number of points |B ∩ \overline{U}| in the intersection of B and \overline{U} which was given in Theorem 4.0.3

Theorem 4.1.4 Let B be a Baer subplane in P G(2, q^{2}). Let \overline{U} be a Buekenhout-Metz unital re (T, ℓ_{∞}) in P G(2, q^{2}). If the base ovoid of \overline{U} is an elliptic quadric, then for q > 13,

1 ≤ |B ∩ \overline{U}| ≤ 2q + 1.

Proof: Consider a unital \overline{U} in P G(2, q^{2}) which is Buekenhout-Metz re (T, ℓ_{∞}) and which has an elliptic quadric as base. By Theorem 4.0.3 and for B any Baer subplane of P G(2, q^{2}),

0 ≤ |B ∩ \overline{U}| ≤ 2q + 2.

If \overline{U} is a classical unital then by Theorem 4.0.5, |B ∩ \overline{U}| = 1, q + 1 or 2q + 1 for all q as required. If \overline{U} is a non-classical Buekenhout-Metz unital, then by Theorem 4.1.1 if B is a Baer subplane of P G(2, q^{2}) which contains ℓ_{∞} as a line or if B intersects ℓ_{∞} in the unique point T, then |B ∩ \overline{U}| = 1, q + 1 or 2q + 1 as required. The remaining case to consider is the case where \overline{U} is a non-classical Buekenhout-Metz unital re (T, ℓ_{∞}) with elliptic quadric as base and B is a Baer subplane of P G(2, q^{2}) such that B intersects the line at infinity in a unique point P distinct from T. It remains to prove for this case that 1 ≤ |B ∩ \overline{U}| ≤ 2q + 1 when q > 13. Our proof is by contradiction making use of several preliminary results. Suppose in this case the intersection B ∩ \overline{U} contains 2q+2 distinct points, then by Theorem 4.0.3 B∩\overline{U} contains exactly 2q+2 points. By Theorem 1.13.3.1 and since P ∈ ℓ_{∞} is not a point of the unital, each Baer subline in B which contains the point P intersects \overline{U} in at most two points; as |B ∩ \overline{U}| = 2q + 2, each Baer subline in B which contains P contains exactly two distinct points of \overline{U}. The unital \overline{U} is a set of points \overline{U}^{∗} in Bruck-Bose, where \overline{U}^{∗} is an elliptic quadric cone in P G(4, q). In the Bruck-Bose setting, the Baer subplane B is a Baer ruled cubic surface B with line directrix p in P G(4, q). The line at infinity of P G(2, q^{2}) corresponds to a hyperplane Σ_{∞} of P G(4, q) and the line {p} = B ∩ Σ_{∞} is an element of the regular spread S of Σ_{∞} which defines the Bruck-Bose representation. In Bruck-Bose, the element p ∈ S represents the unique point P of B on the line at infinity. The Baer sublines in B which contain P are, in Bruck-Bose, the generator lines of the ruled cubic surface B; from above, each such generator line contains exactly two distinct points of \overline{U}^{∗} in P G(4, q)\Σ_{∞}. In particular the line directrix p of B contains no point of \overline{U}^{∗} in P G(4, q). Denote these q + 1 generator lines of B by g_{1}^{∗}, g_{2}^{∗}, …, g_{q+1}^{∗}. The q^{2} Baer sublines in B which do not contain P are represented in Bruck-Bose by q^{2} distinct Baer conics on B. For each secant line ℓ of \overline{U} not incident with T the intersection ℓ ∩ \overline{U} in Bruck-Bose is a non-degenerate conic, namely the plane section of the quadric \overline{U}^{∗} by the plane ℓ^{∗} of P G(4, q), which corresponds to ℓ via Bruck-Bose. By Theorem 1.13.3.2 and since \overline{U} is non-classical with elliptic quadric as base, no such intersection ℓ ∩ \overline{U} is a Baer subline. It follows that no Baer subline in B is contained in \overline{U} and therefore, in Bruck-Bose, no Baer conic in B coincides with a conic ℓ^{∗} ∩ \overline{U}^{∗} (a plane section of the quadric \overline{U}^{∗} in P G(4, q)). Moreover, since two distinct non-degenerate conics in P G(2, q) intersect in at most four points, every Baer subline in B contains at most four points of B ∩ \overline{U}. Hence the 2q + 2 points in the intersection B ∩ \overline{U} constitute a \{2q + 2; 4\}−arc in the Baer subplane B. Note that by Theorem 1.11.2 there exists a Baer subline in B which contains exactly four distinct points of \overline{U}. The quadric \overline{U}^{∗} intersects the hyperplane Σ_{∞} in the spread line t. Since the line directrix p of B is distinct from t and B contains no further point in Σ_{∞}, the line directrix p is disjoint from the quadric \overline{U}^{∗} in P G(4, q). In particular the point vertex V of \overline{U}^{∗} is not a point of B. Let γ denote the variety which is the intersection B ∩ \overline{U}^{∗} in P G(4, q). Note that the 2q + 2 points of γ in P G(4, q) are disjoint from Σ_{∞}. Since the 2q + 2 points of γ lie two each on each generator of B and since B is not contained in any hyperplane of P G(4, q), the variety γ is not contained in any hyperplane of P G(4, q). Also from our above remarks and since γ does not contain p, any generator line of B or any Baer conic in B in P G(4, q), we have that γ contains no lines or conics in P G(4, q). For the remainder of the proof, the points of P G(4, q) will be called rational points; hence the variety γ has 2q + 2 rational points. Since γ is the intersection of a ruled cubic surface B, of order 3 and dimension 2, and a quadric \overline{U}^{∗}, of order 2 and dimension 3, the variety γ has order 6 and dimension 1; γ is therefore a curve of order 6 in P G(4, q). If γ is reducible, then the order of γ may be partitioned in the following ways: 6 = 1 + 5 = 1 + 1 + 4 = 1 + 1 + 1 + 3 = 1 + 1 + 1 + 1 + 2 = 1 + 1 + 1 + 1 + 1 + 1 = 2 + 4 = 2 + 3 + 1 = 2 + 2 + 2 = 3 + 3 = 1 + 1 + 1 + 3 = 1 + 1 + 2 + 2. Hence if γ is reducible then one of the following holds:

(a) γ contains two twisted cubic curves.
(b) γ contains an irreducible conic and an irreducible quartic curve.
(c) γ contains three irreducible conics.
(d) γ is a curve with line components.

Note that the components of γ may coincide or may belong to some field extension of GF (q). We denote by K̂ the algebraic closure of K = GF (q). To show that γ is absolutely irreducible, by the above remarks it suffices to show that over K̂, γ contains no lines, conics or twisted cubic curves. By [15, Note 1.] each variety B = V_{2}^{3} of P G(4, q) is the set of rational points of a variety V̂_{2}^{3} of P G(4, K̂) obtained by a projectivity ϕ̂ between a line p̂ and a conic Ĉ of P G(4, K̂) where ϕ̂|_{p} is a projectivity ϕ between the line p and a (Baer) conic C contained in B of P G(4, q), that is ϕ ∈ P GL(2, q) (see also Section 2.4.1). So in P G(4, K̂) the points of the ruled cubic surface V̂_{2}^{3} are partitioned by the generators of V̂_{2}^{3} and distinct generators of V̂_{2}^{3} intersect p̂ in distinct points. The quadric \overline{U}^{∗} in P G(4, q) extends to a quadric Q̂_{U} in P G(4, K̂), by considering the equation which defines \overline{U}^{∗} in P G(4, q) over the field K̂. Each generator line of V̂_{2}^{3} intersects the quadric Q̂_{U} in 1 or 2 points unless the generator is contained in Q̂_{U}. Consider the line directrix p of B in P G(4, q). The line p is disjoint to \overline{U}^{∗} in P G(4, q), and therefore in the quadratic extension, the intersection p ∩ \overline{U}^{∗} is a pair of points A, A^{q} of p, conjugate with respect to the extension GF (q^{2}) of GF (q). Thus for the line p̂ in P G(4, K̂), the two points A, A^{q} are the only points of the quadric Q̂_{U} incident with p̂. In P G(4, K̂) the sextic curve γ is the intersection of the ruled cubic surface V̂_{2}^{3} and the quadric Q̂_{U}. Suppose the sextic curve γ contains a line component g. The only lines of V̂_{2}^{3} are the generators and the line directrix p̂. Since no generator of B in P G(4, q) is contained in \overline{U}^{∗} and since p̂ is not contained in γ, the line component g of γ is then a generator of V̂_{2}^{3} and is such that the points of g belong to some field extension of GF (q). The lines g and p̂ of V̂_{2}^{3} intersect in a unique point and since g ⊆ γ ⊆ Q̂_{U} the point g ∩ p̂ is then a point of the quadric Q̂_{U}. Hence g ∩ p̂ is either the point A or A^{q}. Suppose, without loss of generality, that g ∩p̂ is the point A; hence the generator g of V̂_{2}^{3} has one and therefore all of its points in the quadratic extension P G(4, q^{2}) of P G(4, q) (see Section 2.4.1). Every generator of V̂_{2}^{3} contains a unique point of the base conic Ĉ of V̂_{2}^{3}, hence we denote by X the unique point of Ĉ incident with g. By definition of the ruled cubic V̂_{2}^{3}, the points A ∈ p̂ and X ∈ Ĉ of g are related by the projectivity ϕ̂. Since XX^{q} is a line of P G(4, q) in the plane containing the base conic C of B, the point X^{q}, conjugate to X with respect to the quadratic extension, is therefore a point of the conic Ĉ. The points A^{q} ∈ p̂ and X^{q} ∈ Ĉ are therefore related by the projectivity ϕ ∈ P GL(2, q) and thus the line g^{q} = X^{q}A^{q} is a generator line of the ruled cubic surface V̂_{2}^{3}. Since g is a line component of γ, g is contained in the quadric Q̂_{U} which is defined by a quadratic form with coefficients in GF (q), namely the quadratic form which defines \overline{U}^{∗} in P G(4, q). It follows that the line g^{q}, conjugate to g with respect to the extension GF (q^{2}) of GF (q), is contained in Q̂_{U} and hence is a line component of γ. We have that the sextic curve γ contains two line components g, g^{q}, such that g and g^{q} contain no rational points, and therefore γ\{g, g^{q}} is residually a curve C_{1}^{4} of order 4 which contains 2q + 2 distinct points in P G(4, q). If C_{1}^{4} is absolutely irreducible, then since the 2q + 2 rational points of C_{1}^{4} are not contained in any hyperplane of P G(4, q), we have by Theorem 1.6.8 that C_{1}^{4} has genus g = 0 and by Theorem 1.6.10 the curve C_{1}^{4} has exactly q + 1 rational points, a contradiction. Thus the curve C_{1}^{4} must be reducible over some field extension of GF (q). The curve C_{1}^{4} has no line components, since by the above above arguments the lines g, g^{q} are the unique lines in V̂_{2}^{3} incident with p̂ in the points {A, A^{q}}, the only two points of p̂ in γ = Q̂_{U} ∩ V̂_{2}^{3}. The only possibility is that C_{1}^{4} has a pair of conic components and therefore the rational points of C_{1}^{4} are contained in conics of P G(4, q). But the only conics in P G(4, q) contained in B are Baer conics and since γ contains no Baer conic in B, from the earlier comments in the proof, we have a contradiction. We have established that the sextic curve γ has no line components. Suppose the sextic curve γ contains an irreducible conic component C_{1}^{2}. Since γ contains no conics in P G(4, q), the conic C_{1}^{2} in γ is a conic on the surface V̂_{2}^{3} in P G(4, K̂) and therefore contains at most one rational point. The remaining 2q + 1 rational points of γ are then contained in a curve C_{1}^{4} = γ\{C_{1}^{2}} of order 4. By the argument presented above, the curve C_{1}^{4} cannot be an absolutely irreducible component of γ, and since γ contains no line components, the curve C_{1}^{4} must be the union of two conic components of γ. We obtain a contradiction as γ contains no conics in P G(4, q) and yet γ has 2q + 2 rational points. Hence the sextic curve γ contains no conic components. Suppose the sextic curve γ contains a twisted cubic component C_{1}^{3}. Since γ contains no line or conic components, γ must be the union of two irreducible cubic curve components. Suppose that C_{1}^{3} is contained in P G(4, q). By Theorem 2.3.1 the irreducible cubic curves on B contain a rational point of p. Since γ contains no rational point of p, an irreducible cubic component C_{1}^{3} of γ is not contained in P G(4, q). Thus if γ is the union of two irreducible cubic curve components and neither is contained in P G(4, q), then we have a contradiction since γ has 2q + 2 rational points. The sextic curve γ therefore contains no components of lower order over any field extension, hence γ is an absolutely irreducible sextic curve with 2q + 2 rational points. By Theorem 1.6.10 for γ absolutely irreducible and with 2q + 2 points in P G(4, q),

|2q + 2 − (q + 1)| ≤ 2g\sqrt{q}.

Since by Theorem 1.6.8, the sextic curve γ has genus g at most 2, we consider the possibilities g = 0, 1, 2. Both g = 0, 1 give rise to a contradiction. For g = 2, on rearranging, we have q − 4\sqrt{q} + 1 ≤ 0 and since q is a positive prime power, q must satisfy 2 − \sqrt{3} ≤ \sqrt{q} ≤ 2 + \sqrt{3}; a contradiction if q > 13. We now give a “dual” argument to show that if |B ∩ \overline{U}| ≠ 2q + 2 for any B-M unital and any Baer subplane of P G(2, q^{2}), then |B ∩ \overline{U}| ≠ 0 for any B-M unital and any Baer subplane of P G(2, q^{2}). The same argument can be used to show that if there is no B-M unital and Baer subplane of P G(2, q^{2}) with exactly 2q + 2 − m points in common for some fixed m satisfying 0 ≤ m ≤ 2q + 2 then there is no B-M unital and Baer subplane of P G(2, q^{2}) with exactly m points in common. Above we have shown for B a Baer subplane and \overline{U} a B-M unital, with base ovoid an elliptic quadric, in P G(2, q^{2}), with q > 13, that |B ∩ \overline{U}| ≠ 2q + 2. Recall that the plane P G(2, q^{2}) is isomorphic to the dual plane of P G(2, q^{2}). The dual B^{d} of a Baer subplane B of P G(2, q^{2}) is a Baer subplane of the dual plane. The dual of a B-M unital \overline{U}, with base ovoid an elliptic quadric, is a B-M unital \overline{U}^{d}, with base ovoid an elliptic quadric, in the dual plane (see [6], [26]). Using the definition of dual structures and Theorem 4.0.4,

|B^{d} ∩ \overline{U}^{d}| = (The number of lines of B which when extended are tangent lines of \overline{U})
= 2q + 2 − |B ∩ \overline{U}|

and since |B ∩ \overline{U}| < 2q + 2, we have |B^{d} ∩ \overline{U}^{d}| > 0. This concludes our proof. □

So by Theorem 4.1.4, for a Baer subplane of P G(2, q^{2}) and a B-M unital \overline{U} with elliptic quadric as base, there are restrictions on the values |B ∩ \overline{U}| can take, for q > 13. In fact the arguments used in the proof of this theorem can be used to show that as q increases the bounds on |B ∩ \overline{U}| further restrict the possible values, for a Baer subplane B and unital \overline{U} as in the statement of the theorem, as follows. Let B and \overline{U} be as in the statement of Theorem 4.1.4. Suppose that B is tangent to the line at infinity at a point P, where P is distinct from the unique point of \overline{U} on the line at infinity. In Bruck-Bose B is represented by a ruled cubic surface B and \overline{U} is a set of points \overline{U}^{∗} in P G(4, q). Let γ denote the intersection B ∩ \overline{U}^{∗}. If γ is an absolutely irreducible sextic curve then, by the arguments used in the proof of Theorem 4.1.4, the number of points R of γ in P G(4, q) is restricted by q, according to the following table.

Restriction on R = |γ| = |B ∩ \overline{U}|
q > 13  0 < R < 2q + 2
q > 16  1 < R < 2q + 1
q > 17  2 < R < 2q
q > 19  3 < R < 2q − 1
q > 21  4 < R < 2q − 2
⋮ ⋮
etc.  etc.

Corollary 4.1.5 Let \overline{U} be a unital in P G(2, q^{2}), q > 13, and q odd. If there exists a Baer subplane B of P G(2, q^{2}) with no point in common with \overline{U}, then \overline{U} is not a Buekenhout-Metz unital.

Proof: Suppose \overline{U} is a Buekenhout-Metz unital in P G(2, q^{2}), then in Bruck-Bose \overline{U} is an ovoidal cone \overline{U}^{∗} with elliptic quadric as base since q is odd. Then by Theorem 4.1.4 for any Baer subplane B of P G(2, q^{2}) we have |B ∩ \overline{U}| ≥ 1 and so no Baer subplane of P G(2, q^{2}) is disjoint from \overline{U}. The result now follows. □

Note that at present all known unitals in P G(2, q^{2}) are Buekenhout-Metz unitals (see [26]). Finally, we include the statement of the very recent result of Barwick, O’Keefe and Storme [14] which characterises Buekenhout-Metz unitals in translation planes π_{q^{2}} of order q^{2} which can be represented in 4−dimensional Bruck-Bose. Note, in the following theorem, a parabolic unital in π_{q^{2}} is a unital for which the translation line ℓ_{∞} of the plane is a tangent line of \overline{U}; also, linear Baer subplanes of π_{q^{2}} are those Baer subplanes B of π_{q^{2}} such that B is represented by a (transversal) plane in the 4−dimensional Bruck-Bose representation of π_{q^{2}}.

Theorem 4.1.6 [14] Let \overline{U} be a parabolic unital in a translation plane π_{q^{2}} of order q^{2} with kernel containing GF (q). Then \overline{U} is a Buekenhout-Metz unital if and only if every linear Baer subplane of π_{q^{2}} meets \overline{U} in 1 modulo q points. □

4.2 Examples and quartic curves

Let \overline{U} be a Buekenhout-Metz unital re (T, ℓ_{∞}) and with an elliptic quadric as base in P G(2, q^{2}). Let B be a Baer subplane of P G(2, q^{2}) such that |B ∩ \overline{U}| = 2q + 2, the maximum possible number of points by Theorem 4.0.3. By the results of the previous section, the unital \overline{U} is necessarily non-classical and the Baer subplane B is necessarily tangent to the line at infinity ℓ_{∞} at a point P distinct from the unique point T of \overline{U} on ℓ_{∞}. In this section we shall give some specific examples in P G(2, q^{2}), q < 13, of this situation; further, we show that in this case the points B ∩ \overline{U} are points of a quartic curve in B with P a double point of the curve. In Bruck-Bose, B is a Baer ruled cubic surface B with line directrix p a line of the regular 1−spread S in a hyperplane Σ_{∞} of P G(4, q), in the usual notation. The element p of S is the Bruck-Bose representation of the point P = B ∩ ℓ_{∞} in P G(2, q^{2}). The unital \overline{U} in Bruck-Bose is a quadric cone \overline{U}^{∗} in P G(4, q) and so has an associated quadratic form with coefficients in GF (q); a point in P G(4, q) has coordinates given by (x_{0}, x_{1}, x_{2}, x_{3}, x_{4}) for some x_{i} ∈ GF (q) and not all zero. By Theorem 1.7.3, the ruled cubic surfaces in P G(4, q) are projectively equivalent and hence we can choose a coordinate representation of P G(4, q) such that the ruled cubic surface B is the ruled cubic surface R_{2}^{3} whose points are given by,

{(x^{2}, xy, y^{2}, zx, zy); x, y ∈ GF (q), (x, y) ≠ (0, 0), z ∈ GF (q) ∪ {∞} }.

By the results of Section 2.6, the Baer subplane B may be identified with P G(2, q) with point coordinates

{(x, y, z); x, y, z ∈ GF (q) x, y, z not all zero }

and such that the line directrix of the ruled cubic surface corresponds to the point P (0, 0, 1) in B. In an abuse of notation, we have chosen coordinates conveniently to represent B as the Baer subplane P G(2, q). The unital \overline{U} and the line at infinity ℓ_{∞} will therefore be given by new coordinates but we retain the same notation; thus P (0, 0, 1) is the unique point of B on ℓ_{∞}. In Bruck-Bose, \overline{U} is still a quadric cone \overline{U}^{∗}, the image of the original quadric cone under a projectivity of P G(4, q), and so denote its quadratic form by

Q(x_{0}, x_{1}, x_{2}, x_{3}, x_{4}).

The points of P G(2, q^{2}) in the intersection B ∩ \overline{U} are therefore the points (x, y, z) in P G(2, q) satisfying,

Q(x^{2}, xy, y^{2}, zx, zy) = 0,

this is a polynomial of degree 4, homogeneous in x, y, z and so represents a quartic curve C_{1}^{4} in B = P G(2, q). Moreover the highest degree of z in the polynomial is 2 and so the quartic curve has a double point at P (0, 0, 1) by Section 1.6. This is consistent with the fact that in Bruck-Bose, the spread element p which represents P (0, 0, 1), is a line and so intersects the quadric \overline{U}^{∗} in two points; these two points lie in a quadratic extension P G(4, q^{2}) of P G(4, q), since in P G(4, q), \overline{U}^{∗} and p are disjoint. Thus in P G(2, q^{2}), every line of B = P G(2, q) through P intersects the quartic curve C_{1}^{4} twice at P. We note also that for small values of q computer searches have verified that there exist examples of Buekenhout-Metz unitals with elliptic quadric as base and Baer subplanes of P G(2, q^{2}) such that the unital and Baer subplane are disjoint; hence there are also examples of the dual case where such a unital and Baer subplane intersect in 2q + 2 points. We include one such example. An example in PG(2, 9): Consider the primitive polynomial x^{2} − x − 1 with root (primitive element) ω. The elements of the fields GF (3) and GF (9) can be represented as follows:

\begin{aligned}GF (3) & = & \{0, 1, 2\} \\ GF (9) & = & \{0, 1, ω, ω^{2}, ω^{3}, ω^{4} ≡ 2, ω^{5}, ω^{6}, ω^{7}\} \end{aligned}

Multiplication in GF (9) is the usual operation with ω^{8} ≡ 1. Field addition is given in the following table.

\begin{matrix}0 & 1 & ω & ω^{2} & ω^{3} & ω^{4} & ω^{5} & ω^{6} & ω^{7} \\ 1 & ω^{4} & ω^{2} & ω^{7} & ω^{6} & 0 & ω^{3} & ω^{5} & ω \\ ω & & ω^{5} & ω^{3} & 1 & ω^{7} & 0 & ω^{4} & ω^{6} \\ ω^{2} & & & ω^{6} & ω^{4} & ω & 1 & 0 & ω^{5} \\ ω^{3} & & & & ω^{7} & ω^{5} & ω^{2} & ω & 0 \\ ω^{4} & & & & & 1 & ω^{6} & ω^{3} & ω^{2} \\ ω^{5} & & & & & & ω & ω^{7} & ω^{4} \\ ω^{6} & & & & & & & ω^{2} & 1 \\ ω^{7} & & & & & & & & ω^{3} \end{matrix} .

Consider the B-M unital \overline{U}_{ω0} with elliptic quadric as base with pointset given by,

\overline{U}_{ω0} = \{(x, ωx^{2} + r, 1); r ∈ GF (3), x ∈ GF (9) \} ∪ \{(0, 1, 0)\}.

The line at infinity is the line with equation z = 0, which is tangent to the unital at the point (0, 1, 0) which we shall call the vertex of the unital. This form of a B-M unital with elliptic quadric as base in P G(2, q^{2}), q odd, is given by Baker and Ebert in [6]. Let P G(2, 3) denote the Baer subplane of P G(2, 9) with points given by the coordinates {(x, y, z); x, y, z ∈ GF (3), x, y, z not all zero }. Consider the following matrix with columns the coordinates of the 13 distinct points of P G(2, 3).

[ \begin{matrix}0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 2 & 2 & 2 \\ 1 & 0 & 1 & 2 & 0 & 1 & 2 & 0 & 1 & 2 & 0 & 1 & 2 \end{matrix} ] .

Consider the projectivity ϕ of P G(2, 9) associated with the matrix,

H_{ϕ} = [ \begin{matrix}ω^{6} & ω^{4} & 0 \\ 1 & ω^{5} & 0 \\ ω^{6} & ω^{2} & 1 \end{matrix} ]

that is

\begin{aligned}ϕ & : & P G(2, 9) & ⟶ & P G(2, 9) \\ & & ( \begin{matrix}x \\ y \\ z \end{matrix} ) & ↦ & H_{ϕ} ( \begin{matrix}x \\ y \\ z \end{matrix} ) . \end{aligned}

Let B be the Baer subplane which is the image of P G(2, 3) under the projectivity ϕ. The homogeneous coordinates of the 13 distinct points of B are given as columns on the following matrix.

[ \begin{matrix}0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 0 & ω & ω & ω & ω^{2} & ω^{2} & ω^{2} & 1 & 1 & 1 & ω^{5} & ω^{5} & ω^{5} \\ 1 & ω^{6} & ω^{3} & ω^{5} & 1 & ω^{7} & ω^{5} & 0 & ω^{5} & ω & ω^{5} & ω^{2} & ω^{4} \end{matrix} ] .

Of the above points of B, the following eight are points of \overline{U}_{ω0}.

\begin{aligned}(0, 0, 1) & ≡ & (0, ω0^{2} + 0, 1) \\ (1, ω, ω^{6}) & ≡ & (ω^{2}, ωω^{4} + 1, 1)ω^{6} \\ (1, ω, ω^{3}) & ≡ & (ω^{5}, ωω^{2} + 1, 1)ω^{3} \\ (1, ω^{2}, 1) & ≡ & (1, ω + 1, 1) \\ (1, ω^{2}, ω^{7}) & ≡ & (ω, ωω^{2} + 0, 1)ω^{7} \\ (1, 1, ω) & ≡ & (ω^{7}, ωω^{6} + 0, 1)ω \\ (1, ω^{5}, ω^{2}) & ≡ & (ω^{6}, ωω^{4} + 1, 1)ω^{2} \\ (1, ω^{5}, ω^{4}) & ≡ & (ω^{4}, ω + 0, 1)ω^{4}. \end{aligned}

We now analyse some properties of this set B ∩ \overline{U}_{ω0} in the Baer subplane B of P G(2, 9). The unique point of B on the line at infinity is the point P (1, 1, 0). Each line of B on P contains exactly two points of the set B ∩ \overline{U}_{ω0}, as expected, since we aim to prove that B ∩ \overline{U}_{ω0} is the set of points in B of a quartic curve with double point P. The line ωx + z = 0 in B is an external line of the set. The remaining eight lines of B are 3-secants of the set B ∩ \overline{U}_{ω0}. Note that each line of B contains four points and therefore if B ∩ \overline{U}_{ω0} had a 4-secant in B, disjoint from the point P, the unital \overline{U}_{ω0} would be classical (by Lefèvre-Percsy Theorem 1.13.3.2), a contradiction to Theorem 4.0.5 since B contains 2q + 2 = 8 points of the unital. We want to verify in this case that B ∩ \overline{U}_{ω0} is a quartic curve in B with double point P. Let θ be the projectivity associated with the matrix,

H_{θ} = [ \begin{matrix}0 & 0 & 1 \\ 1 & 2 & 0 \\ 1 & 0 & 0 \end{matrix} ]

that is

\begin{aligned}θ & : & P G(2, 9) & ⟶ & P G(2, 9) \\ & & ( \begin{matrix}x \\ y \\ z \end{matrix} ) & ↦ & H_{θ} ( \begin{matrix}x \\ y \\ z \end{matrix} ) . \end{aligned}

Now consider the map σ = θϕ^{−1} which maps B to the real Baer subplane P G(2, 3) and maps P to the point (0, 0, 1). The image of the pointset B ∩ \overline{U}_{ω0} under σ is the pointset in P G(2, 3) whose coordinates are given by the columns in the following matrix.

[ \begin{matrix}1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 1 & 1 & 0 & 2 & 1 \\ 0 & 0 & 0 & 1 & 1 & 2 & 1 & 2 \end{matrix} ] .

(Note that by [48, Section 2.6(iii)] a projectivity of the plane does not change the order (degree) of a curve in the plane.) It is now a brief exercise to verify that the above set of points in P G(2, 3) lies on the quartic curve with equation

z^{2}(x^{2} + xy + 2y^{2}) + z(x^{3} + xy^{2} + y^{3}) + x^{2}y^{2} + xy^{3} = 0

which has the point (0, 0, 1) as a double point (see Section 1.6).

4.3 Concerning Classical Unitals in PG(2, q^{2})

In [24] Buekenhout showed that a classical unital \overline{U} in P G(2, q^{2}) is Buekenhout-Metz with respect to any tangent line of \overline{U}; denote by ℓ_{∞} a tangent line of \overline{U}. Moreover, in the 4−dimensional Bruck-Bose representation of P G(2, q^{2}) with respect to ℓ_{∞}, the unital \overline{U} is represented by an ovoidal cone \overline{U}^{∗} in P G(4, q) with an elliptic quadric as base. Since the plane is the Desarguesian plane P G(2, q^{2}), the spread S in the Bruck-Bose representation is a regular 1−spread of a fixed hyperplane Σ_{∞} of P G(4, q), in the usual notation. In [60], Metz showed that for a given regular spread in Σ_{∞} there exist non-Baer conics in P G(4, q) in planes about the spread elements. Metz used this fact to construct Buekenhout-Metz unitals with elliptic quadric as base and which are non-classical unitals in P G(2, q^{2}). Let \overline{U} be a non-classical unital with the above Metz construction in P G(2, q^{2}). So in Bruck-Bose with respect to a given fixed spread S, the unital is an ovoidal cone \overline{U}^{∗} with elliptic quadric as base and such that there exists a plane ℓ^{∗} of P G(4, q) about an element p of the spread S, for which ℓ^{∗} ∩ \overline{U}^{∗} is a non-degenerate non-Baer conic C^{∗}. In P G(2, q^{2}), ℓ^{∗} corresponds to a secant line ℓ of \overline{U} for which ℓ ∩ \overline{U} is not a Baer subline. Denote by t the spread element of S in the ovoidal cone \overline{U}^{∗}; note that t is distinct from p. By setting up a projectivity ϕ between t and C^{∗} in P G(4, q) we obtain a set V_{2}^{3} of q + 1 lines XX^{ϕ} where X ranges over the q + 1 points of t. The set so obtained is a ruled cubic surface V_{2}^{3} with line directrix t and a conic directrix C^{∗} by Section 1.7. Bernasconi and Vincenti [15, Section 2] proved that there exists a regular spread S′ of Σ_{∞} for which V_{2}^{3} is a Baer ruled cubic for the Desarguesian plane π = π(S′) (see Theorem 2.4.1). Note that the line p is a spread element of S′ and C^{∗} is then a Baer conic representing a Baer subline C of π(S′). Consider \overline{U}^{∗} in P G(4, q) with respect to this new spread S′; \overline{U}^{∗} corresponds to a set of points \overline{U}′ in the Desarguesian plane π(S′), which is by definition a Buekenhout-Metz unital (with elliptic quadric as base) in π(S′). Moreover, there exists a secant line of \overline{U}′, not on the vertex point of \overline{U}′ and which intersects \overline{U}′ in a Baer subline C. Hence by Theorem 1.13.3.2, \overline{U}′ is a classical unital in the Desarguesian plane π(S′). Hence for any non-classical B-M unital \overline{U}, with base ovoid an elliptic quadric, in P G(2, q^{2}), in Bruck-Bose it is easy to construct, by the above procedure, a new regular spread S′ in Σ_{∞}, for which \overline{U} is a classical unital in the Desarguesian plane π(S′). We have shown,

Theorem 4.3.1 [27] In P G(2, q^{2}), every non-classical Buekenhout-Metz unital \overline{U} with elliptic quadric as base is inherited from a classical unital \overline{U}_{C} in P G(2, q^{2}), by a procedure of switching regular 1−spreads of Σ_{∞} = P G(3, q) in the 4−dimensional Bruck-Bose representation of P G(2, q^{2}) in P G(4, q). □

A characterisation of

Buekenhout-Metz unitals

The known unitals in P G(2, q^{2}) are to within collineation Buekenhout-Metz unitals, namely a unital whose representation in Bruck-Bose is isomorphic to that given in Section 1.13.3. In [24] the classical unitals were shown to be B-M and it was proved that for certain q even that there exist non-classical (B-M) unitals in translation planes of dimension 2 over their kernel. Metz showed in [60] this class of unitals contained non-classical unitals for all q > 2. The Buekenhout unitals, that is the unitals with construction given in [24, Section 3., Theorem 4] were shown in [10] to be classical and therefore B-M. In [26], see also [6], the dual of a Buekenhout-Metz unital in P G(2, q^{2}) was shown to be B-M in P G(2, q^{2}) and thus all known unitals in P G(2, q^{2}) are B-M unitals. There exist many characterisations of B-M unitals and classical unitals, see for example Theorems 1.13.3.1, 1.13.3.2. The B-M unitals in P G(2, q^{2}) were characterised by Lefèvre-Percsy as follows (a variant of Theorem 1.13.3.1):

Theorem 5.0.1 [56] Let \overline{U} be a unital in P G(2, q^{2}) where q > 2 and let ℓ_{∞} be some tangent line to \overline{U}. If all Baer sublines having a point on ℓ_{∞} intersect \overline{U} in 0, 1, 2 or q + 1 points, then \overline{U} is a B-M unital re (T, ℓ_{∞}) for T the unique point of \overline{U} on ℓ_{∞}. □

In [26] the Lefèvre-Percsy (Theorem 5.0.1) characterisation in P G(2, q^{2}) was improved in the cases q even and q = 3 by weakening the hypotheses to give the result:

Theorem 5.0.2 [26, Theorem 1.3] Let \overline{U} be a unital in P G(2, q^{2}), where q > 2 is even or q = 3. Then \overline{U} is a B-M unital if and only if there exists a point T of \overline{U} such that the points of \overline{U} on each of the q^{2} secant lines to \overline{U} through T form a Baer subline. □

In this chapter, we extend the result of Theorem 5.0.2 by proving it for q > 3; we therefore obtain

Theorem 5.0.3 [68] Let \overline{U} be a unital in P G(2, q^{2}), q > 2. Then \overline{U} is a B-M unital if and only if there exists a point T of \overline{U} such that the points of \overline{U} on each of the q^{2} secant lines to \overline{U} through T form a Baer subline.

5.1 Proof of theorem

Let \overline{U} be a unital in P G(2, q^{2}), where q > 3, with the line at infinity ℓ_{∞} a tangent line of \overline{U}. Let T = ℓ_{∞} ∩ \overline{U} and suppose that the points of \overline{U} on each line of P G(2, q^{2}) through T, and distinct from ℓ_{∞}, form a Baer subline. Represent P G(2, q^{2}) in P G(4, q) as in Section 2.1 with the notation introduced there. The unital \overline{U} corresponds to a set of points \overline{U}^{∗} in P G(4, q). As observed in [26], the above hypothesis is equivalent to the hypothesis that \overline{U}^{∗} consists of a spread element t together with a union of q^{2} lines l_{1}^{∗}, l_{2}^{∗}, …, l_{q^{2}}^{∗} of P G(4, q)\Σ_{∞}, each meeting t but pairwise having no common point in P G(4, q)\Σ_{∞} . We call l_{i}^{∗} (i = 1, …, q^{2}) a generator line of \overline{U}^{∗}. In [26], with a sequence of lemmata, the following result is obtained:

Lemma 5.1.1 [26] \overline{U} is either a B-M unital or \overline{U}^{∗} has the following structure: The generator lines of \overline{U}^{∗} fall into q oval cones C_{1}, …, C_{q}, with distinct vertices V_{1}, …, V_{q} respectively. Each cone has q+1 generators, namely the line t and q generator lines of \overline{U}^{∗}. The cones pairwise intersect in t and have a common tangent plane π (about t) which is contained in Σ_{∞} . Cone C_{i} lies in a 3-dimensional space Σ_{i} (i = 1, …, q) and the spaces Σ_{∞}, Σ_{1}, …, Σ_{q} have the plane π as common intersection. We call each Σ_{i} (i = 1, …, q) a conespace. □

We now prove Theorem 5.0.3 Theorem 5.1.2 Let \overline{U} be a unital in P G(2, q^{2}), where q > 3. Then \overline{U} is a B-M unital if and only if there exists a point T of \overline{U} such that the points of \overline{U} on each of the q^{2} secant lines to \overline{U} through T form a Baer subline.

Proof: The “necessary” result is well known; it follows from the construction of a B-M unital, see Theorem 1.13.3.1. We now prove “sufficiency”. In P G(2, q^{2}), q > 3, let T be a point of a unital \overline{U} such that the points of \overline{U} on each of the q^{2} secant lines to \overline{U} through T form a Baer subline. We may assume the tangent line to \overline{U} at T is ℓ_{∞}. Suppose \overline{U} is not a B-M unital. By Lemma 5.1.1, \overline{U}^{∗} has the q-cone structure defined there. Let α be a plane of P G(4, q)\Σ_{∞} representing a secant line of \overline{U}, not through T, in P G(2, q^{2}). Then α is a plane about a line m of the spread S, m being distinct from t. Choose two generator lines l_{1}^{∗}, l_{2}^{∗} from distinct cones of \overline{U}^{∗} and incident with α. Let

L_{1}^{∗} = l_{1}^{∗} ∩ α
L_{2}^{∗} = l_{2}^{∗} ∩ α.

By Lemma 2.2.6, there exist q Baer conics in α\m containing the points L_{1}^{∗} and L_{2}^{∗}. By Lemma 2.2.11, for each such Baer conic there exist q +1 Baer ruled cubics containing the Baer conic and t; each Baer ruled cubic is determined completely by joining L_{1}^{∗} to a point of t. Thus, there exist q Baer ruled cubics containing the line l_{1}^{∗} and the point L_{2}^{∗}. By Theorem 1.2.1, no two Baer ruled cubics containing l_{1}^{∗} and L_{2}^{∗} contain the same generator line through L_{2}^{∗} and therefore there exists a unique Baer ruled cubic B containing the generator lines l_{1}^{∗} and l_{2}^{∗} of \overline{U}^{∗}. Let C^{∗} be the Baer conic B ∩ α and let B be the Baer subplane of P G(2, q^{2}) represented in P G(4, q) by B. Let g_{3}^{∗}, …, g_{q+1}^{∗} together with l_{1}^{∗} and l_{2}^{∗} be the generators of B; let l_{1}^{∗}, l_{2}^{∗}, g_{3}^{∗}, …, g_{q}^{∗} pass through vertices V_{1}, V_{2}, V_{3}, …, V_{q} respectively on t and g_{q+1}^{∗} through the unique nonvertex point of t (see Lemma 5.1.1). Clearly g_{q+1}^{∗} is not a generator line of \overline{U}^{∗}. None of g_{3}^{∗}, …, g_{q}^{∗} are generator lines of \overline{U}^{∗} as by Lemma 4.0.3, B can intersect \overline{U} in at most 2q + 2 points. The plane ⟨g_{q+1}^{∗}, t⟩ is a plane of P G(4, q)\Σ_{∞} about t and therefore contains exactly one generator line l^{∗} of \overline{U}^{∗}. As l^{∗} ∩g_{q+1}^{∗} is necessarily an affine point, by counting the number of affine points in B ∩ \overline{U}^{∗} together with t we obtain,

|B ∩ \overline{U}| ≥ 2q + 2.

By Lemma 4.0.3 we have,

|B ∩ \overline{U}| = 2q + 2.

Hence no line g_{i}^{∗} (i = 3, …, q) can intersect \overline{U}^{∗} in an affine point. The plane ⟨g_{i}^{∗}, t⟩ (i = 3, …, q) is a plane of P G(4, q)\Σ_{∞} about t and therefore contains a line l_{i}^{∗} of \overline{U}^{∗} which, from above, intersects t in the vertex V_{i}. Since the generators of cone C_{i} are contained in the 3-dimensional space Σ_{i} we have that B has a generator in each conespace Σ_{i} (i = 1, …, q). Let Q be the unique point of intersection of the plane α and the plane π, the common tangent plane to the cones C_{i} (i = 1, …, q). Note that Q = α ∩ π = m ∩ π, and m is not a tangent to the Baer conic C^{∗} in α. Hence, if q is even Q is not the nucleus of the C^{∗}. Since q > 3, there exists a secant line of the Baer conic C^{∗} through Q, incident with C^{∗} in two (distinct) points M_{i}^{∗}, M_{j}^{∗} say, such that neither point is on the line g_{q+1}^{∗}. But then M_{i}^{∗}, M_{j}^{∗} belong to two distinct generators of B belonging to distinct conespaces Σ_{i}, Σ_{j} say. Hence both M_{i}^{∗} and M_{j}^{∗} belong to π, a contradiction. Hence, by Lemma 5.1.1, \overline{U} is a B-M unital. □ Note that in the case q = 3, the possibility exists that there is no secant line of C^{∗} through Q which does not intersect g_{q+1}^{∗} and in that case the secant line may lie in a unique conespace. For this reason the above proof is not valid when q = 3.

Maximal Arcs, Inversive planes and

T_{3}(O)

In this chapter we investigate the relationship between Thas maximal arcs and egglike inversive planes. We show that a Thas maximal arc has an associated egglike inversive plane isomorphic in a natural way to an inversive plane obtained from the generalized quadrangle T_{3}(O), by the method given in Theorem 1.16.3. We also show the inversive plane obtained from a Thas maximal arc is isomorphic in a natural way to an inversive plane obtained from a certain Buekenhout-Metz unital. The relationship between inversive planes and Buekenhout-Metz unitals was recently explored by Barwick and O’Keefe in [13]; see also [6, Section 5.] and [92].

6.1 Maximal arcs and Thas maximal arcs

Let K be a Thas maximal arc in a translation plane π_{q^{2}} of order q^{2} with associated Bruck-Bose construction as given in Section 1.15 and the notation introduced there. Thus π_{q^{2}} has a Bruck-Bose representation Π_{4} in P G(4, q), where Σ_{∞} denotes a fixed hyperplane of P G(4, q) and S is a fixed 1−spread of Σ_{∞}. The translation line of π_{q^{2}}, which we denote by ℓ_{∞} and call the line at infinity, is represented in Bruck-Bose by the hyperplane Σ_{∞}; the points of ℓ_{∞} corresponding to the elements of the spread S of Σ_{∞}. The Thas maximal arc K is defined by a 3−dimensional ovoid O in Σ_{∞} with the property that each element of S contains exactly one point of O. The points of K in Bruck-Bose, are the points of P G(4, q)\Σ_{∞} contained in an ovoidal cone with base ovoid O and vertex a point X in P G(4, q)\Σ_{∞}. Note that by definition ℓ_{∞} is an external line of the Thas Maximal arc K. In an abuse of notation we shall use X to denote the base point of K in π_{q^{2}} and also to denote the image of this point in the Bruck-Bose representation. Denote by o_{1}, …, o_{q^{2}+1} the points of the ovoid O in Σ_{∞} which defines the Thas maximal arc K. Call the lines Xo_{i}, i = 1, …, q^{2} + 1, in P G(4, q), generator lines of K. Let π_{o_{i}} denote the unique tangent plane to O in Σ_{∞} at the point o_{i}, i = 1, …, q^{2} + 1. Recall that the unique spread line through a point o_{i} of O is contained in the tangent plane π_{o_{i}} at o_{i}, since the plane π_{o_{i}} contains a spread line and each spread line contains a unique (and therefore at least one) point of O. Denote by s_{i} the spread line incident with the point o_{i} of O. There exist q + 1 hyperplanes of P G(4, q) which contain the plane ⟨X, s_{i}⟩, for a fixed point o_{i} of the ovoid O. The hyperplane ⟨X, π_{o_{i}}⟩ contains the unique generator line Xo_{i} of K and therefore the q−1 planes in ⟨X, π_{o_{i}}⟩ about the spread line s_{i}, besides π_{o_{i}} and the plane ⟨X, s_{i}⟩, represent the q − 1 external lines of K on the point at infinity represented by s_{i}. These q − 1 external lines together with ℓ_{∞} are all the external lines to K on the point at infinity of π_{q^{2}} represented by s_{i}. The remaining q hyperplanes on ⟨X, s_{i}⟩ each intersect the ovoidal cone in an oval cone. Let Σ be such a hyperplane, so that Σ contains q generator lines of K besides Xo_{i}. Planes about s_{i} in Σ, besides Σ ∩ Σ_{∞}, intersect the oval cone in q points of K and of these planes all, except ⟨X, s_{i}⟩, intersect the same q generator lines of K. We have the following well known result:

Result 6.1.1 Let K be a Thas maximal arc with base point X and axis line ℓ_{∞} in a translation plane π_{q^{2}} of order q^{2}, where ℓ_{∞} is the translation line of π_{q^{2}}. Let P be a point of ℓ_{∞}, then the secant lines of K incident with P besides XP are partitioned into q classes of q − 1 lines such that the lines in a class intersect the same generator lines of K.

6.2 Excursion into T_{3}(O)

In this section we recall the definition of the generalized quadrangle T_{3}(O) and some of the well known properties of this generalized quadrangle.

Consider the generalized quadrangle T_{3}(O), defined with the ovoid O in P G(3, q) as per Result 1.16.4 with the notation introduced there. Let X be a point of type (i) in T_{3}(O) and let Y be the point (∞) and so X ≁ Y. Now

{X, Y }^{⊥} = {⟨X, π_{o_{1}}⟩, …, ⟨X, π_{o_{q^{2}+1}}⟩}

where o_{1}, …, o_{q^{2}+1} are the points of the ovoid O and π_{o_{i}} is the tangent plane to O in P G(3, q) at the point o_{i}, i = 1, …, q^{2} = 1. The points in {X, Y }^{⊥} are points of type (ii) in T_{3}(O). Let Z_{1}, Z_{2}, Z_{3} be three distinct points in {X, Y }^{⊥}, then {Z_{1}, Z_{2}, Z_{3}} is necessarily a triad by GQ axiom (iii). Considering Z_{1}, Z_{2}, Z_{3} as hyperplanes of P G(4, q), Z_{1} ∩ Z_{2} ∩ Z_{3} is a line XQ on X which intersects Σ_{∞} in a unique point Q and Q ∉ O. Thus in T_{3}(O),

{Z_{1}, Z_{2}, Z_{3}}^{⊥} = {points of the line XQ\{Q}} ∪ {Y }.

A point of {Z_{1}, Z_{2}, Z_{3}}^{⊥⊥} is therefore a point of type (ii), since such a point is collinear with Y, contains X and contains all points o_{i} of O such that Q is incident with the plane π_{o_{i}}. For any ovoid O in P G(3, q), a point Q ∉ O lies in the tangent planes of exactly q + 1 points of O and the corresponding q + 1 points of O constitute an oval in O [69]. We have therefore that |{Z_{1}, Z_{2}, Z_{3}}^{⊥⊥}| = q + 1 and since Z_{1}, Z_{2}, Z_{3} are three arbitrary and distinct points in {X, Y, }^{⊥} we have that every triad in {X, Y }^{⊥} is 3-regular. Alternatively one could argue that since Y ∈ {Z_{1}, Z_{2}, Z_{3}}^{⊥} and since Y is 3-regular, the flag (Y, ℓ) has property (G) , for all lines ℓ of type (b) and therefore every triad {Z_{1}, Z_{2}, Z_{3}} is 3-regular. If we let X be a point of type (i) in T_{3}(O) and let Y be the point (∞), then from the above discussion we can apply Theorem 1.16.3 and construct an inversive plane I_{X3}(O) from T_{3}(O) as follows (see also the proof of [67, Theorem 5.3.1]).

Result 6.2.1 For the generalized quadrangle T_{3}(O) let X be a point of type (i) and let Y be the point (∞). The associated inversive plane I_{X3}(O) is defined as follows:

Points: Hyperplanes ⟨X, π_{o_{i}}⟩, i = 1, …, q^{2} + 1.

Circles: All sets {⟨X, π_{o′_{1}}⟩, …, ⟨X, π_{o′_{q+1}}⟩} where {o′_{1}, …, o′_{q+1}} are the points of an oval (plane section) of O.

6.3 Thas maximal arcs and Inversive planes

Motivated by [92] we have the following definition.

Definition 6.3.1 An O’Nan configuration is a set of six distinct points with the following properties. The set contains four distinct points A, B, C, D of which no three are collinear and the remaining two points E, F are such that {E} = AC ∩ BD and {F } = AB ∩ CD. The six points A, B, C, D, E, F are called the vertices of the configuration.

Let K be a maximal arc in a projective plane π_{q} of order q. Let X be a point of K. We say K satisfies property

I_{X}: If K contains no O’Nan configurations with X a vertex.

II_{X}: If l is a secant line of K not through X, m a secant line of K through X meeting l in a point of K and Y (≠ X) a point of K on m, then there exists a line l′ ≠ m incident with Y and meeting every line through X that meets l and such that l′ intersects each such line in a point of K.)

We now show that a Thas maximal arc K with base point X satisfies I_{X} and II_{X} and these properties lead to defining an inversive plane associated to the Thas maximal arc. Let K be a Thas maximal arc with base point X in a translation plane π_{q^{2}} of order q^{2} with translation line ℓ_{∞}. Note that π_{q^{2}} has a Bruck-Bose representation in P G(4, q) with the usual notation.

Lemma 6.3.1 K satisfies I_{X}.

Proof: Suppose there exists an O’Nan configuration in K with X a vertex. Let m_{1} and m_{2} be the two secant lines of K not incident with X in the configuration. Let P_{i} be the point of intersection of m_{i} and ℓ_{∞}, i = 1, 2. The three points of K on m_{1} in the O’Nan configuration correspond to three generator lines l_{1}, l_{2}, l_{3} of K and m_{2} intersects these same generator lines of K in the O’Nan configuration. By Result 6.1.1 and the comments preceding it, in the Bruck-Bose representation of π_{q^{2}}, l_{1}, l_{2}, l_{3} generate a hyperplane of P G(4, q)\Σ_{∞} which contains the spread lines corresponding to P_{1}, P_{2} ∈ ℓ_{∞}, a contradiction since Σ_{∞} is the only hyperplane of P G(4, q) which contains two distinct elements of the spread S. Therefore there exist no O’Nan configurations in K with X a vertex. □

Lemma 6.3.2 K satisfies II_{X}

Proof: Let l be a secant line of K not on X and let l ∩ ℓ_{∞} = {P }. The result now follows from Result 6.1.1. □. Consider the incidence structure IK^{′} defined by:

Points: generator lines of K

Blocks: secant lines of K not incident with X; identifying blocks with their

points and using the property II_{X} to eliminate repeated blocks

Incidence: is inherited from the translation plane

Lemma 6.3.3 IK^{′} is a 2-(q^{2} + 1, q, q − 1) design.

Proof: There are q^{2} + 1 generator lines of K, corresponding to the points of the ovoid in the construction of K, therefore the number v′ of points of IK^{′} is q^{2} + 1. A secant line of K which is not incident with X intersects q generator lines of K, hence the number k′ of points in a block is q. By Result 6.1.1, each point of ℓ_{∞} corresponds q distinct blocks of IK^{′} and since each secant line of K intersects ℓ_{∞} in a unique point, blocks corresponding to distinct points of ℓ_{∞} are distinct. Therefore the number b′ of blocks of IK^{′} is therefore q(q^{2} + 1) = q^{3} + q. By Result 6.1.1 there exist q − 1 secants on a point P ∈ ℓ_{∞} which define the same block of IK^{′}. A generator line of K has q − 1 points of K besides X and there exist q^{2} secant lines not containing X through each such point. Therefore in IK^{′}, the number r′ of blocks containing a point is q^{2}(q − 1)/(q − 1) = q^{2}. Consider two generator lines of K; they each have q − 1 points besides X. From above a block is defined by q − 1 distinct secant lines of K and therefore the number λ2^{′} of blocks containing two fixed points is (q − 1)^{2}/(q − 1) = q − 1. It follows that IK^{′} is a 2-(q^{2} + 1, q, q − 1) design. □ We have that a block, B_{P} say, of IK^{′} is determined by q − 1 distinct secant lines of K each incident with a common point P ∈ ℓ_{∞}. Thus to each block B_{P} in IK^{′} is associated a unique point not incident with the block, namely, the generator line of K on the line XP. We use this fact to define a new incidence structure as follows.

Definition 6.3.4 Let I_{K} be the incidence structure defined by:

Points: generator lines of K

Circles: {{Block B_{P} of IK^{′}} ∪ { the generator line of K in XP } ; for all blocks B_{P}

in IK^{′}}

Incidence: containment

Lemma 6.3.5 The incidence structure I_{K} is a 3-(q^{2} + 1, q + 1, 1) design, namely a finite inversive plane of order q.

Proof: I_{K} has the same number of points and blocks as IK^{′} therefore v = v′ = q^{2} + 1 and b = b′ = q^{3} + q. The number k of points in a block of I_{K} is b = b′ + 1 = q + 1. The number r of blocks on a fixed point of I_{K} is given by

r = r′+{ the number of blocks of IK^{′} determined by secant lines on a fixed point of ℓ_{∞}}

Using the definition of blocks of IK^{′} and Result 6.1.1 we have r = r′ + q = q^{2} + q. It remains to show that for any three distinct points of I_{K} there exists a unique block containing them. Let l_{1}, l_{2}, l_{3} be three distinct points of I_{K}, that is, l_{1}, l_{2}, l_{3} are three generator lines of K in the Bruck-Bose representation of the translation plane. The three lines span a hyperplane Σ in P G(4, q) which intersects Σ_{∞} in a plane containing a unique spread element; denote this spread element by P. Since the hyperplane Σ intersects the ovoidal cone of the Thas maximal arc in three generator lines, Σ contains an oval cone of generator lines. Thus the planes in Σ about P represent secant lines of K and define a unique block of I_{K} containing the points l_{1}, l_{2}, l_{3}. We have shown therefore that I_{K} is an inversive plane. □

Theorem 6.3.6 The inversive plane I_{K} associated to a Thas maximal arc K with base point X in a translation plane π_{q^{2}} is isomorphic to the inversive plane I_{X3}(O) obtained from the generalized quadrangle T_{3}(O) (defined in the P G(4, q) with ovoid O of the construction of K.) The inversive planes are egglike.

Proof: The result follows from the above discussion of the construction in P G(4, q) of I_{K} and Result 6.2.1. □

Remark: The inversive plane associated to a Buekenhout-Metz unital (see Barwick and O’Keefe [13]) is isomorphic in a natural way to the inversive planes of Theorem 6.3.6 defined with the same ovoid O of P G(3, q), since both Thas maximal arcs and Buekenhout-Metz unitals are defined using a 4−dimensional ovoidal cone with base ovoid a 3−dimensional ovoid O.

6.4 A characterisation of Thas maximal arcs

In this section we endeavour to find a converse to the main result of Section 6.3. We attempt to characterise Thas Maximal Arcs with the configurational properties I_{X} and II_{X}. We weaken our hypothesis and obtain a partial converse.

6.4.1 A sequence of lemmata

Let K be a (maximal) {q^{3} − q^{2} + q; q}-arc in a translation plane π_{q^{2}} of order q^{2} with kernel GF (q), so that π_{q^{2}} has a Bruck and Bose representation in P G(4, q) defined by a spread in the hyperplane Σ_{∞} of P G(4, q). Denote by ℓ_{∞} the translation line of π_{q^{2}} and suppose ℓ_{∞} is an external line of K. Let X be a fixed point of K. We say K satisfies:

I_{X}: (As in Section 6.3.)

II_{X}^{∗}: If l is a secant line of K not through X and P is the point of intersection of lines l and ℓ_{∞}, then there exist q − 2 further secant lines of K incident with P and which intersect every line through X that meets l (these intersections are all in K).

Suppose K satisfies properties I_{X} and II_{X}^{∗}. We proceed with a sequence of lemmata determining some properties of K, but first we introduce some terminology. Each line on X contains q − 1 points of K besides X; call such a set of q − 1 points of K on a line through X a variety. For a variety V (on a line l through X), label the point at infinity of l, namely l ∩ ℓ_{∞}, by P_{V}. We shall sometimes refer to P_{V} as the point at infinity of the variety V.

Let l be a secant line of K not on X. Then l is incident with q varieties and by II_{X}^{∗} there exist q − 2 further secants of K incident with these same q varieties and concurrent with l in a point P on ℓ_{∞}. Call such a collection of q varieties a block b and call the associated point P on ℓ_{∞} the point at infinity of the block b and say b is a block of P.

Lemma 6.4.1.1 For a point P ∈ ℓ_{∞},

(i) Distinct blocks of P are disjoint (they have no varieties in common.)
(ii) P is the point at infinity of exactly q blocks.

Proof: Let P be a point on ℓ_{∞}.

(i) Let b_{1} and b_{2} be two blocks of P. Suppose b_{1} and b_{2} intersect in a variety V_{1}. Let l_{1} be a secant line of K on P incident with b_{1} (and therefore incident with every variety in b_{1}). Since l_{1} is incident with the variety V_{1} of block b_{2} and l_{1} passes through P, then l_{1} must be one of the q − 1 secant lines of K on P incident with every variety in b_{2} by II_{X}^{∗}. Since l_{1} intersects K in exactly q points, blocks b_{1} and b_{2} must coincide. We have shown therefore that distinct blocks of P are disjoint. (ii) There exist q^{2} − q secant lines of K on P besides the line XP. For each block of P there exist q − 1 secant lines of K on P which determine that block and since by (i) distinct blocks of P are disjoint, there are exactly q blocks of P. □

Lemma 6.4.1.2 Let P and Q be two points on ℓ_{∞} and let b_{P}, b_{Q} be a block of P,Q respectively. Then the blocks b_{P} and b_{Q} intersect in exactly 0, 1, 2 or q varieties.

Proof: If P = Q then by Lemma 6.4.1.1 b_{P} intersects b_{Q} in 0 or q varieties. If P ≠ Q, suppose b_{P} and b_{Q} have three varieties V_{1}, V_{2}, V_{3} in common; V_{i} contained in line l_{i}, i = 1, 2, 3, incident with X. Let R be a point of K in V_{1}. By II_{X}^{∗}, the line RP is a secant line of K incident with P and incident with the varieties in b_{P}; also the line RQ is a secant line of K on Q incident with b_{Q}. The lines RP, RQ, l_{2} and l_{3} are four lines of an O’Nan configuration in K with X as a vertex; a contradiction, as K satisfies I_{X}, thus in this case b_{P} and b_{Q} have at most 2 varieties in common. □

Lemma 6.4.1.3 There are exactly q^{3} + q blocks in K.

Proof: By Lemma 6.4.1.1 there are q blocks corresponding to each of the q^{2} + 1 points of ℓ_{∞} and by definition (or the proof of Lemma 6.4.1.2) a block corresponds to a unique point at infinity. The result follows. □

Lemma 6.4.1.4 Let V_{1} and V_{2} be two distinct varieties. There exist exactly q − 1 blocks containing both V_{1} and V_{2}.

Proof: Let V_{1} be on line l_{1} through X and let V_{2} be on line l_{2} through X. Let R be a point (of K) in V_{1}. The join of R to each point of V_{2} defines q − 1 secant lines m_{i} (i = 1, …, q − 1) of K, not on X and with distinct points P_{1}, … P_{q−1} on the line at infinity. The line m_{i} defines block B_{i}, containing both varieties V_{1} and V_{2}, and with point at infinity P_{i} (for i = 1, …, q − 1). Thus there exist at least q − 1 blocks containing both V_{1} and V_{2}. By II_{X}^{∗}, for each block B_{i} there exist q − 2 further lines through P_{i} incident with both V_{1} and V_{2}, thus giving all the possible lines joining a point of V_{1} and a point of V_{2}. Thus there exist exactly q − 1 blocks containing both V_{1} and V_{2}. □

Lemma 6.4.1.5 There are exactly q^{2} blocks containing a given variety V.

Proof: Let P_{V} be the point at infinity of a fixed variety V. For each point P on the line at infinity besides P_{V}, V lies in a block of P, since there exist secant lines of K on P incident with points in V. Therefore by Lemmata 6.4.1.1 and 6.4.1.2, V lies in exactly one block of P (P ∈ ℓ_{∞}\{P_{V} }), with no two distinct points at infinity determining the same block containing V. Since there are q^{2} points on ℓ_{∞} besides P_{V}, there exist exactly q^{2} blocks containing the variety V. □ Let V be the set of varieties and B be the set of blocks and with incidence I the natural containment relation. We define an incidence structure I′ = (V, B, I).

Lemma 6.4.1.6 The incidence structure I′ = (V, B, I) is a 2-(q^{2} + 1, q, q − 1) design with parameters:

\begin{aligned}v′ & = & q^{2} + 1 \\ k′ & = & q \\ b′ & = & q^{3} + q \\ r′ & = & q^{2} \\ λ2^{′} & = & q − 1 \end{aligned}

Proof: Lemmata 6.4.1.1, 6.4.1.2, 6.4.1.3, 6.4.1.3, 6.4.1.4 and 6.4.1.5 determine the parameters of I′. □ Next we define a new incidence structure I = (V, C, I) based on I′. Let the set of varieties V of I′ be the points P of I and let

C = {{varieties in a block B_{P} of a point P } ∪ {the variety contained in the line XP }

: for all blocks B_{P} of a point P, for all points P on ℓ_{∞}}.

Call the elements of C circles and call C the set of circles in I. There is a natural one-to-one correspondence between blocks of I′ and circles of I since each block of I′ is contained in a unique circle and conversely each circle of I contains a unique block of I′.

Lemma 6.4.1.7 The incidence structure I = (V, C, I) is a 2-(q^{2} + 1, q + 1, q + 1) design with parameters

\begin{aligned}v & = & q^{2} + 1 \\ k & = & q + 1 \\ b & = & q^{3} + q \\ r & = & q^{2} + q \\ λ_{2} & = & q + 1 \end{aligned}

Proof: Now v = v′ = q^{2} + 1 and b = b′ = q^{3} + q using the definition of I and the natural one-to-one correspondence between circles and blocks. The number k of varieties in a circle is one more than the number k′ of varieties in a block, therefore k = k′ + 1 = q + 1. For a variety V with point at infinity P, the number of circles containing V equals the number of blocks containing V plus the number of blocks of P, therefore r = r′ + q = q^{2} + q. Lastly, consider two varieties V_{1} and V_{2} with points at infinity P_{1} and P_{2} respectively. Variety V_{1} lies in a unique block of P_{2} and similarly variety V_{2} lies in a unique block of P_{1} and there are q − 1 blocks containing both V_{1} and V_{2}. Therefore the number λ_{2} of circles containing both V_{1} and V_{2} is q + 1. □

Corollary 6.4.1.8 The following four statements are equivalent for the incidence structure I.

(i) three distinct varieties are contained in at least one circle
(ii) three distinct varieties are contained in at most one circle
(iii) the design I has parameter λ_{3} = 1
(iv) the design I is a finite inversive plane

Proof: If three distinct varieties are contained in a unique circle, for any choice of three distinct varieties, then I is a 3-(q^{2} + 1, q + 1, 1) design with the parameters given in Lemma 6.4.1.7 together with λ_{3} = 1, that is, I is a finite inversive plane. Let λ_{3_{i}}, i = 1, …, \binom{v}{3}, be the number of circles containing three given (distinct) varieties V_{1}, V_{2}, V_{3}, for all \binom{v}{3} possible choices of V_{1}, V_{2}, V_{3}. We now count in two ways the number of 3-flags of I

\begin{matrix}( & {v \atop 3} & ) \\ ∑ \\ i=1 \end{matrix} λ_{3_{i}} = b\binom{k}{3}.

Thus the average number λ_{3_{i}ave} of circles on three varieties is given by

\begin{aligned}λ_{3_{i}ave} & = & b\binom{k}{3}/\binom{v}{3} \\ & = & 1. \end{aligned}

Therefore if λ_{3_{i}} ≥ 1 for all i then λ_{3_{i}} = 1 for all i. Similarly if λ_{3_{i}} ≤ 1 for all i then λ_{3_{i}} = 1 for all i. □

Lemma 6.4.1.9 Let V_{1} and V_{2} be two distinct varieties in a block b_{P} of a point P (P ∈ ℓ_{∞}). Let l_{i} be the lines on X containing V_{i}, with the point at infinity of l_{i} denoted by Q_{i}, i = 1, 2. If a Baer subplane B of π_{q^{2}} contains P, Q_{1}, Q_{2} and X then

either B contains no points of V_{1} or V_{2}
or  B contains the same number of points of V_{1} as of V_{2}.

Proof: Let R be a point of V_{1} in B. Since P R and l_{2} are lines of B, the point P R ∩ l_{2} is a point of B. Since l_{1} and l_{2} lie in the block b_{P} of P, by II_{X}^{∗}, the point P R ∩ l_{2} of B is a point on l_{2} of the maximal arc K, that is P R ∩ l_{2} is a point of V_{2}. The same argument holds if we suppose R is a point of V_{2} in B. It follows that either B contains no points of V_{1} and V_{2} or B contains the same number of points of V_{1} as of V_{2}. □

We now use the results of Section 2.1 and Section 2.2 concerning the Bruck-Bose representation in P G(4, q) of some of the Baer subplanes and Baer sublines of π_{q^{2}}. In the following lemmata, a linear Baer subplane of π_{q^{2}} is a Baer subplane of π_{q^{2}} which is represented in Bruck-Bose by a (transversal) plane of P G(4, q)\Σ_{∞} which intersects Σ_{∞} in a line which is not a line of the spread S of Σ_{∞}; a linear Baer subline is a Baer subline of a line of π_{q^{2}} which is represented in Bruck-Bose by a line of P G(4, q)\Σ_{∞}.

Lemma 6.4.1.10 There exists a linear Baer subline in π_{q^{2}} containing X and which contains at least one further point of K. If a linear Baer subline which contains X contains exactly n further points of K, then every linear Baer subline which contains X and which contains further points of K contains exactly n points of K besides X.

Proof: Let l_{1} be a line on X containing a linear Baer subline l_{B1}, where l_{B1} contains X and contains n points of K besides X. Let l_{2}(≠ l_{1}) be any other line containing a linear Baer subline l_{B2}, with X ∈ l_{B2}, and such that l_{B2} contains further points of K. There exists a linear Baer subplane B of π_{q^{2}} containing l_{B1} and l_{B2} and since l_{B1} and l_{B2} both have points at infinity, the line at infinity is a line of B. Let l be a line not through X and such that l contains a point of K in l_{B1} and a point of K in l_{B2}, then l is a line of B and intersects ℓ_{∞} in a point P of B. Thus, as l is a secant line of K on P and hence the varieties in l_{1} and l_{2} lie together in a block of P. Now by Lemma 6.4.1.9, Baer sublines l_{B1} and l_{B2} contain the same number of points of K besides X. The result now follows. □ By Lemma 6.4.1.10, the linear Baer sublines of π_{q^{2}} which contain X contain either 0 or n further points of K, where 1 ≤ n ≤ q −1 is a fixed integer. Moreover, since each secant line of K incident with X contains exactly q − 1 points of K distinct from X, the integer n divides q − 1.

Lemma 6.4.1.11 If π_{q^{2}} is the Desarguesian plane P G(2, q^{2}), then each linear Baer subline of π_{q^{2}} which contains X contains either 0 or n further points of K, where 1 < n ≤ q − 1 is a fixed integer such that n divides q − 1.

Proof: If π_{q^{2}} is the Desarguesian plane P G(2, q^{2}), then by 1.15.2 and since π_{q^{2}} contains a maximal arc K we have that q is even. Moreover in the Bruck-Bose representation of π_{q^{2}} in P G(4, q) the 1−spread S of Σ_{∞} = P G(3, q) is then a regular spread. By the remarks preceding this lemma we have that each linear Baer subline of π_{q^{2}} which contains X contains exactly 0 or n further points of K, where 1 ≤ n ≤ q − 1 is a fixed integer and n divides q − 1. Suppose that n = 1. Firstly if q = 2, then K is necessarily a Thas maximal arc in π_{q^{2}}, so we consider the case q ≥ 4. Consider two distinct varieties V_{1} and V_{2} of I′ contained in lines ℓ_{1}, ℓ_{2} of π_{q^{2}} respectively. By definition ℓ_{1} and ℓ_{2} intersect in the point X of K. Denote by P_{1} and P_{2} the points at infinity of ℓ_{1} and ℓ_{2} respectively. In Bruck-Bose, the points P_{1}, P_{2} on ℓ_{∞} correspond to distinct elements P_{1}^{∗}, P_{2}^{∗} of the regular spread S of Σ_{∞}. In I′, there exist q − 1 distinct blocks which contain the varieties V_{1} and V_{2}; denote the points at infinity of these blocks by Q_{1}, Q_{2}, …, Q_{q−1}. In Bruck-Bose the points Q_{i} correspond to q − 1 distinct elements of the spread S; denote these spread elements by Q_{i}^{∗}, i = 1, …, q − 1. There exist q + 1 reguli in S containing P_{1}^{∗} and P_{2}^{∗}, therefore there exists at least one regulus R of lines of S which contains P_{1}^{∗} and P_{2}^{∗} but which contains no spread element Q_{i}^{∗}. Let R′ denote the opposite regulus of R in Σ_{∞}. In Bruck-Bose, the lines ℓ_{1} and ℓ_{2} correspond to planes ℓ_{1}^{∗} and ℓ_{2}^{∗} in P G(4, q) respectively; both planes contain X and a line P_{1}^{∗}, P_{2}^{∗} respectively of S. Since n = 1, the q points of K in ℓ_{1} are represented in Bruck-Bose by the point X and q −1 further points of ℓ_{1}^{∗}\{P_{1}^{∗}} on distinct lines of ℓ_{1}^{∗} through X. Similarly for the points of K incident with ℓ_{2}. In Bruck-Bose, since q ≥ 4 there exists a line m in the opposite regulus of R such that the plane B = ⟨m, X⟩ contains a point of K in ℓ_{1}^{∗} besides X and a point of K in ℓ_{2}^{∗} besides X; denote these two points of K in B, which are distinct from X, by Y_{1}^{∗}, Y_{2}^{∗} respectively. Each point Y_{i}^{∗} corresponds to a point Y_{i} in π_{q^{2}} incident with the variety V_{i} for i = 1, 2. The line Y_{1}Y_{2} is distinct from ℓ_{∞} and intersects ℓ_{∞} in a point Q which is necessarily the point at infinity of a block containing both varieties V_{1} and V_{2}. In Bruck-Bose Q corresponds to a spread element Q^{∗} contained in the regulus R of S; a contradiction, since the regulus R contains no element which is the Bruck-Bose representation of a point of infinity of a block containing the varieties V_{1} and V_{2}. Hence n ≠ 1 and therefore n > 1 as required. □ Note that a Mersenne prime is a prime number which can be written in the form 2^{p} − 1 for some positive integer p which is necessarily prime (see [47, Theorem 18]). There are 31 known Mersenne primes and it is conjectured that there exist an infinite number of Mersenne primes.

Corollary 6.4.1.12 Suppose K is a maximal {q^{3} − q^{2} + q; q}−arc in the Desarguesian plane P G(2, q^{2}) satisfying properties I_{X} and II_{X}^{∗} for some point X in K. If q − 1 is a Mersenne prime, then K is a Thas maximal arc with base point X and axis line ℓ_{∞}. □

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