Generalized Quadrangles and Associated Structures

Matthew Ross Brown, B.Sc (Hons) (Adelaide)

Thesis submitted for the degree of

Doctor of Philosophy

in Pure Mathematics

at

The University of Adelaide

(Faculty of Mathematical and Computer Sciences)

Department of Pure Mathematics

August 21, 1997

Contents

Abstract iii

Signed Statement iv

Preface v

Acknowledgements vii

1 Preliminaries 1 1.1 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Quadrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 An introduction to geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3.1 Examples of geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Generalized Quadrangles (GQs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4.1 Combinatorics of GQs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4.2 The classical GQs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4.3 The non-classical GQ of Tits . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4.4 Isomorphisms between GQ . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4.5 The q-clan GQs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4.6 Ovoids and rosettes of GQs . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Algebraic topology on a simplicial complex . . . . . . . . . . . . . . . . . . . . . 11 1.5.1 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5.2 Simplicial complexes, homology and cohomology . . . . . . . . . . . . . . 12 1.5.3 Cohomology groups of a simplicial complex . . . . . . . . . . . . . . . . . 14 1.5.4 The simplicial complex of a graph . . . . . . . . . . . . . . . . . . . . . . 15 1.5.5 Homology and cohomology over ℤ_{2} . . . . . . . . . . . . . . . . . . . . . . 15

1.6 Covers of a graph and covers of a geometry . . . . . . . . . . . . . . . . . . . . . 16

2 The Kantor ovoid K1(σ) 18 2.1 Flocks and Ovoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2 Symmetry properties of the K1(σ) ovoid . . . . . . . . . . . . . . . . . . . . . . . 20 2.3 Intersections of θ_{σ} and elliptic quadrics on Q(4, q), containing X_{σ} . . . . . . . . . 21

2.3.1 Case 1: ℓ_{σ} ⊂ Σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.2 Case 2: ℓ_{σ} ⊄ Σ but X_{σ} ∈ Σ . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4 More properties of the Σ_{t,t′} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.5 Characterisations of the K1(σ) ovoid . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.6 Rosettes containing K1(σ) ovoids . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.6.1 Elation rosettes of K1(σ) ovoids . . . . . . . . . . . . . . . . . . . . . . . 26 2.6.2 Rosettes, trades and towers . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.6.3 Rosettes of Q(4, 9) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3 SPGs and GQs of order (r, r^{2}) 40

3.1 SPGs from GQs of order (r, r^{2}) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.1.1 Algebraic 2-fold covers of SPGs and the GQ condition . . . . . . . . . . . 44 3.2 Isomorphisms of SPGs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.3 A GQ of Kantor, an ovoid of Kantor and a new SPG . . . . . . . . . . . . . . . . 47 3.4 SPG from q-clan GQs, q even . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.4.1 Algebraic conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4 Characterisations of GQs of order (r, r^{2}) 60

4.1 Characterisations of Q(5, q), q even . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.1.1 The intersection and subtending of ovoids of W (q), q = 2^{e}, e odd . . . . . 61 4.1.2 Applications to Q(5, q), q even . . . . . . . . . . . . . . . . . . . . . . . . 63

4.2 GQs of order (r, r^{2}) with a doubly subtended subquadrangle of order r . . . . . . 64 4.2.1 Introducing the cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.2.2 Calculating the homology . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.3 Application to Q(5, q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.3.1 Explicit homology calculation for Q(4, q), q odd . . . . . . . . . . . . . . . 70 4.3.2 The graph G_{{A,C(λ)}}, λ ≠ 0, −4η . . . . . . . . . . . . . . . . . . . . . . . 73

4.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5 Affine planes and GQs of order s 78 5.1 Covering affine planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.2 Algebraic covers and GQs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.2.1 A cover associated with the GQ W (q) . . . . . . . . . . . . . . . . . . . . 83 5.2.2 A cover associated with the GQ T_{2}(O), q even . . . . . . . . . . . . . . . 84 5.2.3 A “geometric” construction of a cover of T_{2}(O), with regular point (∞) . 85

5.3 Equivalence of covers of π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.3.1 Normalised algebraic covers . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.3.2 Equivalent algebraic covers of π . . . . . . . . . . . . . . . . . . . . . . . . 86

5.4 The automorphism group of \overline{π} and the group of S fixing (∞) . . . . . . . . . . . 91 5.4.1 Elations and symmetries about (∞) . . . . . . . . . . . . . . . . . . . . . 92 5.4.2 Example: The subgroup of AΓL(3, q) admitted by x_{1}y_{2} − x_{2}y_{1} . . . . . 93

5.4.3 Example: Covers of AG(2, q), q even, of the form (x_{1} + y_{1})F (\frac{x_{2} + y_{2}}{x_{1} + y_{1}} ) 93

5.5 Automorphisms of A and coboundaries associated with

collineations of π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.6 Covers associated with T_{2}(O)^{∧} . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.6.1 Case (a) X = (0, 0, 1, 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.6.2 Case (b) X = (0, 1, t, F (t)), t ∈ GF(q) . . . . . . . . . . . . . . . . . . . . 99

5.7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Bibliography 101

Abstract

Our aim in this thesis has been to consider questions concerning the relationship between a Generalized Quadrangle (GQ) and various substructures, with a view to proving characterisation and classification results. We also lay the groundwork for new GQ construction methods, although no new GQs are constructed here.

In Chapter 1 we introduce preliminary concepts and results required for the rest of the thesis, involving graphs, quadrics, geometries, GQs, algebraic topology on a simplicial complex and covers of geometries.

Chapter 2 contains a detailed investigation of the ovoid K1(σ) of Q(4, q) constructed by Kantor in [30], including construction of non-elation rosettes of Q(4, q) containing only K1(σ) ovoids and rosettes containing both K1(σ) ovoids and elliptic quadric ovoids.

In Chapter 3 we show that if S is a GQ of order (s, s^{2}) and S′ is a subquadrangle of order s doubly subtended in S, then the subtended ovoid/rosette structure is a Semi-Partial Geometry (SPG). A new SPG is constructed from a GQ of Kantor ([31]) and a Q(4, q) subquadrangle. For a q-clan GQ S, q even, Payne constructed a family of subquadrangles S_{α} of order q ([45]). We derive the algebraic conditions under which S_{α} is doubly subtended in S, and hence gives an SPG.

In Chapter 4 it is shown that if q is even a non-classical GQ of order (q, q^{2}) containing a subquadrangle isomorphic to Q(4, q) implies the existence of a new ovoid of PG(3, q). Also, by a homology calculation, it is shown that if S is a GQ of order (q, q^{2}), q odd, such that S contains a Q(4, q) subquadrangle, with each ovoid of Q(4, q) subtended by S an elliptic quadric ovoid, then S is isomorphic to Q(5, q).

In Chapter 5 we show a GQ S of order s with a regular point (∞) gives rise to a cover of the affine plane constructed from S and (∞), as in [49, 1.3.1]. Given an affine plane π of order s and an s-fold cover of π satisfying special conditions we construct a GQ of order s with a regular point. If the cover of π is algebraic the condition on the cover is interpreted in cohomological terms; we investigate these for the remainder of the Chapter 5.

Signed 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 consent to this copy of my thesis, when deposited in the University Library, being available for loan and photocopying.

SIGNED: ....................... DATE: .......................

Preface

Generalized Quadrangles (GQs) were introduced by J. Tits in 1959 ([66]) in his work on triality. Tits was also responsible for the construction of the first non-classical GQ of order (s, s^{2}), the GQ T_{3}(θ) of order (q, q^{2}) where q = 2^{e}, e odd, and θ is the Suzuki-Tits ovoid of PG(3, q), which first appeared in [19]. After the classical GQ Q(5, q) of order (q, q^{2}) and T_{3}(θ), the next GQ of order (s, s^{2}) was constructed by Kantor via the group coset geometry method ([29]). A number of new GQs of order (s, s^{2}) followed using the group coset geometry method of Kantor, all of which, except the Roman GQ of Thas and Payne [65], arise from q-clans (see [43, 45] and [31]). In all cases the parameter s is a prime power.

Relatively little is known about subquadrangles of the known GQs of order (s, s^{2}) although there are a number of examples of subquadrangles of order s. The GQ Q(5, q) has subquadrangles isomorphic to Q(4, q); the GQ of Kantor in [31] and the Roman GQ both have subquadrangles isomorphic to Q(4, q), while the GQs derived from q-clans for q even have subquadrangles isomorphic to the GQ T_{2}(O), for O some oval of PG(2, q). This thesis is largely devoted to investigating the combinatorics of a subquadrangle of order s of a GQ of order (s, s^{2}).

A result of Payne and Thas ([49]) shows that if S is a GQ of order (s, s^{2}) and S′ is a subquadrangle of order s, then each point of S \ S′ is collinear to the s^{2} + 1 points of an ovoid of S′. A line of S \ S′ contains s such points of S \ S′ and the corresponding set of q ovoids of S′ has the property that any two elements intersect in a fixed point (the point where the line meets S′). A suggestion of Christine O’Keefe and Tim Penttila was to study these “rosettes” of ovoids of a GQ S′ of order s (defined combinatorially as a set of s ovoids meeting pairwise in a fixed point) to see what they revealed about GQs of order (s, s^{2}) containing S′ as a subquadrangle. A result in this direction had already been proved by Payne and Thas ([65, VII.1.]) showing that the GQ Q(5, q) is characterised by the existence of a Q(4, q) subquadrangle with each point of Q(5, q) \ Q(4, q) being collinear to an elliptic quadric ovoid of Q(4, q).

An obvious first step in considering ovoids and rosettes of GQs of order s is to consider ovoids and rosettes of Q(4, q). For q even the only known non-classical ovoid of Q(4, q) is for the case q = 2^{e}, e odd, and is isomorphic to the Tits ovoid of W (q). In Chapter 4 rosettes of these “Tits” ovoids of Q(4, q) are used to show that the existence of a non-classical GQ of order (q, q^{2}), q even, containing a subquadrangle isomorphic to Q(4, q) implies the existence of a new ovoid of PG(3, q). (Note that the latter is fairly unlikely!).

For q odd there are a number of non-classical ovoids of Q(4, q) known, the most “structured” of these being the K1(σ) ovoid constructed by Kantor in [30]. Chapter 2 is devoted to investigating the properties of this ovoid and the rosettes of Q(4, q) containing it.

The other obvious problem for q odd it to show that Q(5, q) is characterised by the existence of a Q(4, q) subquadrangle with each point of Q(5, q) \ Q(4, q) being collinear to an elliptic quadric ovoid of Q(4, q). Consideration of this problem led to considering the general combinatorial setting: that is, a GQ S of order (s, s^{2}) with a subquadrangle S′ of order s such that if the points of an ovoid of S′ are collinear with a point of S \ S′, then they are collinear with exactly two points of S \ S′. A consequence of a result of Bose and Shrikhande ([8], see [49, 1.2.4]) is that any set of points of an ovoid of S′ may be collinear with at most two points of S \ S′ and so S′ is a combinatorially “extreme” subquadrangle. It was observed by Metz (see [18]) and Hirschfeld and Thas (see [26]) that in the case where S = Q(5, q) and S′ = Q(4, q) that the ovoids and rosettes of Q(4, q) associated with the points and lines of S \ S′ form a geometry which is a Semi-Partial Geometry (SPG). In Chapter 3 it is shown that this is also true in general for any such S and S′ described above. A new SPG is constructed from a GQ of Kantor in [31] and a Q(4, q) subquadrangle.

In this “extreme” case the relationship between the SPG and the geometry of S \ S′ is a cover, and more specifically an algebraic cover. The question of how many “different” GQs of order (s, s^{2}) give rise to a given SPG in this manner becomes a problem in cohomology. In Chapter 4 this approach allows us to prove [65, VII.1] in the case where q is odd.

This idea of the geometry of a “substructure” of a GQ being related to the geometry of the GQ “outside” the substructure is explored again in Chapter 5. In this case the GQ is of order s with a regular point (∞), and the substructure is an affine plane of order s, as defined in [49, 1.3.1.]. It is shown that such a GQ always gives rise to a cover of the associated affine plane (and also a distance-regular cover of the complete graph on s^{2} vertices). The rest of Chapter 5 is spent exploring the idea of algebraic covers of an affine plane and their relation to GQs of order s with a regular point. This seems a promising approach to constructing new GQs of order s, although none, as yet, have been found.

Acknowledgements

I would like to thank my supervisor Christine O’Keefe for her steadfast and unwavering support, on both academic and personal grounds, during the course of my PhD. Without her patience, dedication to the ideals of research and standing in the academic community my PhD. would have been much the poorer.

Thanks also to the geometry seminar at the University of Adelaide, especially including Wen-Ai Jackson, Catherine Quinn and Sue Barwick.

Particular thanks also to the now scattered Combinatorial Computing Research Group of the University of Western Australia: Tim Penttila, Gordon Royle, Ivano Pinneri and Nicholas Hamilton (in no particular order). All extremely gifted mathematicians and more importantly all extremely generous, friendly and cultured people.

During the course of my PhD. many geometers made their way to Adelaide and I was grateful for the opportunity to meet and interact with them. In particular I would like to thank Frank De Clerck, Jef Thas, Stan Payne and Bill Cherowitzo for their visits to Adelaide.

I am greatly in debt to my family, parents Barbara and Graham and siblings Anna and David, for their love and unqualified support at all times.

For the (almost) countless, privileged hours spent in his company during the four years of my PhD. I would like to thank Paul McCann. I am inestimably grateful to Shane Richards who despite putting up with my domestic habits for nearly three years still remains a true friend. Special thanks to Catherine Quinn, Deb Brown (my big sister), Dave Standingford, David Beard, Keith Martin, Steve Cox, Wen-Ai Jackson and Sue Barwick for their friendship during my PhD.

Finally, I would like to acknowledge the financial support of an Australian Postgraduate Research Award.

Chapter 1

Preliminaries

In this chapter we give definitions and results that will be used throughout the thesis. We will assume that reader is familiar with the theories of finite fields and projective spaces over finite fields; an excellent and thorough introduction to projective spaces over finite fields is given in [24]. For notational convenience we will associate a point of the projective space PG(n, q) with the homogeneous coordinate vector that represents it.

1.1 Graphs

A graph G is an ordered pair (V, E) where V is a non-empty set of elements called vertices and E is a set of unordered pairs of distinct vertices, called edges. Two vertices of G, u and v, that are contained in E are said to be adjacent and we write u ∼ v.

A path of G is a finite, ordered set of vertices v_{0}, v_{1}, …, v_{n} such that v_{i} ∼ v_{i+1} for 0 ≤ i ≤ n − 1. The path is said to have length n. Two vertices u and v are said to be connected if there exists a path v_{0}, v_{1}, …, v_{n} such that v_{0} = u and v_{n} = v. A graph G is connected if each pair of vertices of G is connected.

A path v_{0}, v_{1}, …, v_{n} is said to be closed if v_{0} = v_{n}. Alternatively, we may think of a closed path as a set of edges of G such that each vertex of G appears an even number of times in edges in the path. This set of edges is called the edge set of the path. A circuit of G is a closed path such that no edge is repeated. We will represent the circuit v_{0}, v_{1}, …, v_{n} with v_{0} = v_{n} by (v_{0}, v_{1}, …, v_{n−1}). For convenience we will often consider two circuits that differ only by a cyclic permutation of their vertices to be the same circuit. An elementary circuit is a circuit v_{0}, v_{1}, …, v_{n−1}, v_{0} such that v_{i} ≠ v_{j} for 0 ≤ i, j ≤ n − 1, i ≠ j. The elementary circuit C is said to be induced if each vertex of C is adjacent to exactly two others in C.

Let G = (V, E) and G′ = (V′, E′) be graphs. The graph G′ is a subgraph of G if V′ ⊆ V and E′ = {(u, v) : u, v ∈ V′ and (u, v) ∈ E}. If v is a vertex of G, then G_{v} is the subgraph of G with vertex set {u : u ∼ v}. In general, if A ⊆ V with A = {v_{1}, v_{2}, …, v_{n}}, then G_{A} is the subgraph of G with vertex set {u : u ∼ v_{i} for i = 1, …, n}. A clique of G is defined to be a complete subgraph of G. A connected component of G is a maximal connected subgraph.

A connected graph G = (V, E) is regular if for every vertex u ∈ V, |{v ∈ V : u ∼ v}| = k, for some fixed k. The number k is called the valence of the graph G.

Let u and v be two vertices of the connected graph G = (V, E) and let u = u_{0}, u_{1}, …, u_{n−1}, u_{n} = v be a path of shortest length from u to v. The distance between u and v is defined to be n and we denote this by d(u, v) = n or d(v, u) = n. The diameter of G is defined to be the number d = max{d(u, v) : u, v ∈ V }. If u is a vertex of G, then let G_{i}(u) be the set {v : d(u, v) = i}. Thus G_{0}(u) = {u} and G_{1}(u) = G_{u}.

A connected graph G = (V, E) is called distance-regular if there exists integers b_{i}, c_{i} (i ≥ 0) such that for any two vertices u, v ∈ V, with i = d(u, v), there are precisely c_{i} vertices contained in G_{i−1}(u) that are adjacent to v and b_{i} vertices contained in G_{i+1}(u) that are adjacent to v. The sequence {b_{0}, b_{1}, …, b_{d−1}; c_{1}, c_{2}, …, c_{d}}, where d is the diameter of G, is called the intersection array of G. Note that it follows that G is regular with valence k = b_{0} and that b_{d} = c_{0} = 0 and c_{1} = 1. For more information on distance-regular graphs see [9] or [5]. A distance-regular graph of diameter 2 is called a strongly regular graph. If G = (V, E) is a strongly regular graph, then it has parameters v, k, μ, λ, where v = |V |, k is the valence of G, μ = c_{2} and λ = k − b_{1} − 1.

For any connected graph G = (V, E) of diameter d > 1 we denote by G_{i} (for i = 1, 2, …, d) the graph with vertex set V and with vertices u and v adjacent if d(u, v) = i in G. The graph G is antipodal if G_{d} is a disjoint union of cliques.

1.2 Quadrics

In this section we introduce quadrics of PG(3, q); for a more detailed treatment see [24, Chapters 5, 6 and 7], [25, Chapters 15, 16, 17 and 20] and [27, Chapter 22].

A quadric Q_{n} of PG(n, q) is the set of points of PG(n, q) whose homogeneous coordinates satisfy a quadratic form F (x_{0}, x_{1}, …, x_{n}) = ∑_{i=0}^{n} a_{i}x_{i}^{2} + ∑_{i<j} a_{ij}x_{i}x_{j}. If F cannot be reduced to a form in fewer than n + 1 variables by a linear transformation, then F is non-degenerate and Q_{n} is non-singular. If n is even, then every non-singular quadric in PG(n, q) is projectively equivalent to the parabolic quadric, denoted by P_{n}. If n is odd, then PG(n, q) has exactly two projectively distinct non-singular quadrics, the elliptic quadric, denoted by E_{n} and the hyperbolic quadric, denoted by H_{n}. Canonical forms for the non-singular quadrics are as follows:

P_{n} = V (x_{0}^{2} + x_{1}x_{2} + … + x_{n−1}x_{n}),
E_{n} = V (f(x_{0}, x_{1}) + x_{2}x_{3} + … + x_{n−1}x_{n}),
H_{n} = V (x_{0}x_{1} + x_{2}x_{3} + … + x_{n−1}x_{n}),

where f is an irreducible binary quadratic form.

If we allow F to be degenerate, then the quadric defined by F may be singular, that is, a cone Π_{k}Q_{s}. This cone is the join of the vertex Π_{k}, a subspace of PG(n, q) of dimension k and a non-singular quadric Q_{s} in the subspace Π_{s} of dimension s, with Π_{k} and Π_{s} skew and k + s = n − 1.

Suppose that Q_{n} is a non-singular quadric of PG(n, q) defined by the quadratic form F (x_{0}, x_{1}, …, x_{n}). Then the bilinear form β associated with F is given by β(X, Y) = F (X +Y)−F (X)−F (Y). The bilinear form gives the geometry of the quadric, as the following lemma shows.

Lemma 1.2.1 [27, Lemma 22.3.1] Let Y ∈ Q_{n}. Then

(i) If X ∉ Q_{n}, then β(X, Y) = 0 ⟺ ⟨X, Y ⟩ is tangent to Q_{n}.

(ii) If X ∈ Q_{n}, then β(X, Y) = 0 ⟺ ⟨X, Y ⟩ ⊂ Q_{n}.

(iii) β(X, Y) ≠ 0 ⟺ |⟨X, Y ⟩ ∩ Q_{n}| = 2.

Lemma 1.2.2 [27, Lemma 22.3.3] If q and n are not both even, the correspondence Y ⟷ V (β(Y, X)) is a polarity. For q odd, the set of self-polar points is Q_{n}. For q even, the polarity is null and every point in PG(n, q) is self-polar.

We will denote the action of the polarity by ⊥. If Y is a point of PG(3, q), then the polar hyperplane V (β(Y, X)) is called the tangent space or tangent hyperplane of Y. For details of the action of the polarity on subspaces of PG(n, q), see [27, Theorem 22.7.2].

Now we consider the symmetries of Q_{n}, where q and n are not both even. Let Q be a point of PG(n, q) \ Q_{n} and let μ_{Q} : Q_{n} → Q_{n} be defined as follows. For P in Q_{n},

μ_{Q}(P) = \{ \begin{matrix}P & if ⟨P, Q⟩ is a tangent to Q_{n} \\ P′ & if ⟨P, Q⟩ meets Q_{n} at P′ ≠ P. \end{matrix}

The map μ_{Q} may be extended to an automorphism of PG(n, q) (fixing Q_{n}) given by

X ↦ X − \frac{β(X, Q)}{F (Q)}Q

([27, Lemma 22.6.3]).

1.3 An introduction to geometries

An (incidence) geometry 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 ⊆ (P × B) ∪ (B × P) is a symmetric point-line incidence relation. In addition, we require that two distinct points of S are incident with at most one line and two distinct lines of S are incident with at most one point. Two points of S that are incident with a common line are said to be collinear and two lines of S that are incident with a common point are concurrent.

The dual of a geometry S = (P, B, I) is the geometry S^{∧} = (B, P, I). So, the points of S^{∧} are the lines of S, the lines of S^{∧} are the points of S and the incidence relation between P and B is the same in both S and S^{∧}.

An automorphism of a geometry S = (P, B, I) is a bijection ϕ : P ∪ B → P ∪ B, such that ϕ maps points to points, lines to lines and preserves incidence. By preserving incidence we mean that if P ∈ P and ℓ ∈ B, then P I ℓ if and only if ϕ(P) I ϕ(ℓ). The point graph of S is the graph with vertex set P and edge-set the set of collinear pairs of points of S.

The geometry S′ = (P′, B′, I′) is a subgeometry of S if P′ ⊆ P, B′ ⊆ B and I′ is the restriction of I to (P′ × B′) ∪ (B′ × P′). If S′ is a subgeometry of S, then we write S′ ⊆ S. If S ≠ S′, then we say that S′ is a proper subgeometry of S and write S′ ⊂ S. If S′ ⊂ S, then it follows that P′ ≠ P or B′ ≠ B.

1.3.1 Examples of geometries

A (finite) partial geometry (PG) is an incidence structure W = (P, B, I), satisfying the following axioms:

(i) Each point is incident with 1 + t lines (t ≥ 1) and two distinct points are incident with at most one line;

(ii) Each line is incident with 1 + s points (s ≥ 1) and two distinct lines are incident with at most one point;

(iii) If X is a point and ℓ is a line not incident with X, then the number of pairs (Y, m) ∈ P ×B for which X I m I Y I ℓ is a constant α (α ≥ 0).

The integers s, t and α are called the parameters of W. It follows that |P| = (s+1)(st+α)/α, |B| = (t + 1)(st + α)/α and that the point graph of W is strongly regular. Partial geometries were introduced by Bose in [7] to investigate strongly regular graphs.

A (finite) semi-partial geometry (SPG) is an incidence structure T = (P, B, I), satisfying the following axioms:

(i) Each point is incident with 1 + t lines (t ≥ 1) and two distinct points are incident with at most one line;

(ii) Each line is incident with 1 + s points (s ≥ 1) and two distinct lines are incident with at most one point;

(iii) If X is a point and ℓ is a line not incident with X, then the number of pairs (Y, m) ∈ P ×B for which X I m I Y I ℓ is either 0 or a constant α (α ≥ 0); (iv) For any pair of non-collinear points (X, Y) there are μ (μ ≥ 0) points Z such that Z is

collinear with both X and Y.

The integers s, t, α, μ are the parameters of T . It follows that |P| = 1 + (t + 1)s(μ + t(s − α + 1))/μ and |B| = μ(t + 1)/(s + 1). Note that axiom (iii) of a semi-partial geometry is a generalisation of axiom (iii) for a partial geometry, while axiom (iv) forces the point graph of a semi-partial geometry to be strongly regular. Thus, partial geometries are a subclass of semi-partial geometries. The point graph of T has parameters

v = 1 + \frac{(t + s)s(μ + t(s − α + 1))}{μ}, k = (t + 1)s, , μ, λ = s − 1 + t(α − 1).

Semi-partial geometries were first introduced by Debroey and Thas in [18]. For more information on partial geometries and semi-partial geometries see [17].

An important class of partial geometries (and hence semi-partial geometries) is the case α = 1, the generalized quadrangles. A (finite) generalized quadrangle (GQ) is an incidence structure S = (P, B, I), satisfying the following axioms:

(i) Each point is incident with 1 + t lines (t ≥ 1) and two distinct points are incident with at most one line;

(ii) Each line is incident with 1 + s points (s ≥ 1) and two distinct lines are incident with at most one point;

(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.

It follows that |P| = (s + 1)(st + 1) and |B| = (t + 1)(st + 1). We say that S has order (s, t) while if s = t, then we say that S has order s. A subgeometry of a GQ S that is also a GQ is called a subquadrangle or subGQ. Generalized quadrangles were introduced as part of a larger class of geometries called generalized n-gons, by Tits in [66]. For a complete list of the known GQs see [64].

1.4 Generalized Quadrangles (GQs)

1.4.1 Combinatorics of GQs

In this section we give some combinatorial definitions and properties of GQs. For more details see [49].

Let S be a GQ of order (s, t). Given two points X, Y of S, we write X ∼ Y if X and Y are collinear, or if X = Y, otherwise we write X ≁ Y. Dually, for two lines ℓ, m of S, we write ℓ ∼ m if ℓ and m are concurrent, or if ℓ = m, otherwise we write ℓ ≁ m.

For X ∈ P, define X^{⊥} = {Y ∈ P : Y ∼ X}. The set X^{⊥} is called the trace of X. The trace of a pair of distinct points X and Y, is the set X^{⊥} ∩ Y^{⊥} and is denoted {X, Y }^{⊥}. The size of {X, Y }^{⊥} is s+1 or t+1, according as X ∼ Y or X ≁ Y. For X ≠ Y, the span of the pair (X, Y) is {X, Y }^{⊥⊥} = {U ∈ P : U ∈ Z^{⊥} for all Z ∈ {X, Y }^{⊥}}, that is, {X, Y }^{⊥⊥} = ({X, Y }^{⊥})^{⊥}. In general, if P′ ⊂ P, then P′^{⊥} = ∩_{X∈P′}X^{⊥} and P′^{⊥⊥} = (P′^{⊥})^{⊥}.

If X ∼ Y, X ≠ Y, or if X ≁ Y and |{X, Y }^{⊥⊥}| = t + 1, we say that the pair (X, Y) is regular. The point X is regular if (X, Y) is regular for each Y ∈ P such that Y ≠ X.

A triad (of points) of S is a triple of distinct, pairwise non-collinear points.

An ovoid of a GQ S of order (s, t) is a set θ of points such that each line of S is incident with precisely one point of θ. It follows that θ has st + 1 points. Dually, a spread of S is a set Ω of st + 1 lines of S, such that each point of S is incident with exactly one line of Ω.

A rosette based at a point X of S is a set R of ovoids with pairwise intersection {X} and such that {θ \ {X} : θ ∈ R} is a partition of the points of S not collinear with X. The point X is called the base point of R. It follows that a rosette R has s ovoids. A rosette is homogeneous if all of its ovoids are of the same isomorphism class in S. A rosette that isn’t homogeneous is inhomogeneous.

Let the GQ S′ = (P′, B′, I′) of order (s′, t′) be a subGQ of the GQ S = (P, B, I). If S′ is a proper subGQ of S, that is S′ ≠ S, and ℓ ∈ B then by [49, 2.1], exactly one of the following occurs: (i) ℓ ∈ B′; (ii) ℓ ∉ B′ and ℓ is incident with a unique point X ∈ P′, in which case we say ℓ is tangent to S′ at X; or (iii) ℓ ∉ B′ and ℓ is incident with no point of P′, in which case we say ℓ is external to S′. Dually, we may define external points and tangent points.

Lemma 1.4.1 ([61], [41], see [49, 2.2.1]) Let S = (P, B, I) be a GQ of order (s, t) with a subquadrangle S′ = (P′, B′, I′) of order (s, t′). Each point of S, external to S′, is collinear with the 1 + st′ points of an ovoid of S′.

An ovoid arising as in Lemma 1.4.1 is said to be subtended by P, or just subtended. The ovoids of S′ subtended by the points in P \ P′ are said to be the ovoids subtended by S or just the subtended ovoids.

If ℓ is a line of S that is tangent to S′ then each of the s points incident with ℓ and not contained in P′, is external to S′. Furthermore, the set of s ovoids subtended by these points is a rosette of S′. To see this, first observe that if X, Y I ℓ and X, Y ∈ P \ P′, then the ovoids θ_{X} and θ_{Y}, subtended by X and Y respectively, are both incident with the point ℓ ∩ S′. Also, if θ_{X} and θ_{Y} are both incident with a further point Z of S′, then X, Y, Z form a triangle. We say that the rosette constructed as above is subtended by the line ℓ, or just subtended.

We define subtended spreads and subtended rosettes of S′ dually.

1.4.2 The classical GQs

The classical GQs are those GQs corresponding to the classical groups and are as follows (see [49, Section 3.1]).

1(a) The GQ Q(3, q) of order (q, 1) arises as the points and lines of the non-singular hyperbolic quadric in PG(3, q), given by the equation x_{0}x_{1} + x_{2}x_{3} = 0. Since t = 1, the structure of Q(3, q) is trivial.

1(b) The GQ Q(4, q) of order q arises as the points and lines of the non-singular (parabolic) quadric in PG(4, q), given by the equation x_{0}^{2} + x_{1}x_{2} + x_{3}x_{4} = 0.

1(c) The GQ Q(5, q) of order (q, q^{2}) arises as the points and lines of the non-singular elliptic quadric in PG(5, q) given by the equation x_{0}^{2} + ax_{0}x_{5} + bx_{5}^{2} + x_{1}x_{2} + x_{3}x_{4} = 0, where

x_{0}^{2} + ax_{0}x_{5} + bx_{5}^{2} is an irreducible binary quadratic form.

Note that Q(5, q) contains subquadrangles isomorphic to Q(4, q) (see [49, 3.5(a)]).

2(a) The GQ H(3, q^{2}) of order (q^{2}, q) arises as the points and lines of the non-singular hermitian variety of PG(3, q^{2}), given by the equation x_{0}^{q+1} + x_{1}^{q+1} + x_{2}^{q+1} + x_{3}^{q+1} = 0.

2(b) The GQ H(4, q^{2}) of order (q^{2}, q^{3}) arises as the points and lines of the non-singular hermitian variety of PG(4, q^{2}), given by the equation x_{0}^{q+1} + x_{1}^{q+1} + x_{2}^{q+1} + x_{3}^{q+1} + x_{4}^{q+1} = 0.

3(a) The GQ W (q) of order q arises as the absolute points and absolute lines of a symplectic polarity in PG(3, q), with canonical bilinear form x_{0}y_{1} − x_{1}y_{0} + x_{2}y_{3} − x_{3}y_{2} = 0.

It has been shown, independently by Buekenhout and Lefèvre in [11] and Olanda in [39, 40], that the classical GQs are precisely the GQs S = (P, B, I) such that P is a subset of the pointset of some projective space PG(n, q) and B is a set of lines of PG(n, q).

1.4.3 The non-classical GQ of Tits

The following constructions of the non-classical GQs T_{2}(O) and T_{3}(θ) are due to Tits and first appeared in [19].

Let Σ = PG(3, q) and H = PG(2, q) be a hyperplane of Σ. Further, let O be an oval of H. Define T_{2}(O) to be the following incidence structure:

Points:  (i)  The points of Σ \ H
(ii)  The planes of Σ, not H, meeting H in a tangent to O
(iii)  The symbol (∞)
Lines: (a) Lines of Σ, not contained in H, that meet O
(b)  Points of O
Incidence: (i)(a) Incidence in Σ
(ii)(a) Containment in Σ
(iii)(a) None
(i)(b) None
(ii)(b) Incidence in Σ
(iii)(b) All

Let Σ′ = PG(4, q) and H′ = PG(3, q) be a hyperplane of Σ′. Further, let θ be an ovoid of H′. Define T_{3}(θ) to be the following incidence structure:

Points:  (i)  The points of Σ′ \ H′
(ii)  The hyperplanes of Σ′, not H′, meeting θ in a unique point
(iii)  The symbol (∞)
Lines: (a) Lines of Σ′, not contained in H′, that meet θ
(b)  Points of θ
Incidence: (i)(a) Incidence in Σ′
(ii)(a) Containment in Σ′
(iii)(a) None
(i)(b) None
(ii)(b) Incidence in Σ′
(iii)(b) All

1.4.4 Isomorphisms between GQ

In this section we give some isomorphisms between the GQs defined in Section 1.4.2 and Section 1.4.3. For proofs of the isomorphisms, see the cited result in [49].

Theorem 1.4.2 [49, 3.2.1] The GQ Q(4, q) is isomorphic to the dual of W (q). Moreover, Q(4, q) (or W (q)) is self-dual if and only if q is even.

Theorem 1.4.3 [49, 3.2.2] The GQ T_{2}(O) is isomorphic to Q(4, q) if and only if O is an irreducible conic; it is isomorphic to W (q) if and only if q is even and O is a conic.

Theorem 1.4.4 [49, 3.2.3] The GQ Q(5, q) is isomorphic to the dual of H(3, q^{2}).

Theorem 1.4.5 [49, 3.2.4] The GQ T_{3}(θ) is isomorphic to Q(5, q) if and only if θ is an elliptic quadric of PG(3, q).

1.4.5 The q-clan GQs

In this section we give a summary of results on the q-clan construction of GQs of order (q, q^{2}). First consider the group coset construction of a GQ as introduced by Kantor [29]. Let G be a finite group of order s^{2}t, s ≥ 2, t ≥ 2 and suppose that G admits two families of subgroups F = {S_{0}, …, S_{t}}, F^{⋆} = {S_{0}^{⋆}, …, S_{t}^{⋆}} such that |S_{i}| = s, |S_{i}^{⋆}| = st and S_{i} ⊆ S_{i}^{⋆}. If F and F^{⋆} satisfy:

K1 S_{i}S_{j} ∩ S_{k} = 1 for k ≠ i, j and i ≠ j and
K2 S_{i}^{⋆} ∩ S_{j} = 1 for i ≠ j,

then F is a 4-gonal family for G. The following point-line geometry S(G, F) is a GQ of order (s, t).

Points : (i) Elements of G
(ii) Right cosets, S_{i}^{⋆}g, i = 0, …, t, g ∈ G
(iii) (∞)
Lines : (a) Right cosets S_{i}g, i = 0, …, t, g ∈ G
(b) Symbols [S_{i}], i = 0, …, t.

A point g of type (i) is incident with each line S_{i}g. A point S_{i}^{⋆}g of type (ii) is incident with [S_{i}] and with each line S_{i}h ⊂ S_{i}^{⋆}g. The point (∞) is incident with each line [S_{i}] of type (b). These are all of the incidence relations.

We now consider q-clans and give the construction of a 4-gonal family from a q-clan, the development of which is due to Payne [43, 45] and Kantor [31]. For q a prime power a q-clan is a set C = {A_{t} : t ∈ GF(q)} of 2 × 2 matrices over GF(q) such that for distinct s, t ∈ GF(q), (a, b)(A_{s} − A_{t})(a, b)^{T} = 0 has only the trivial solution a = b = 0. We can normalise a q-clan

such that for q odd A_{t} = ( \begin{matrix}x_{t} & y_{t} \\ y_{t} & z_{t} \end{matrix} ) and for q even A_{t} = ( \begin{matrix}x_{t} & y_{t} \\ 0 & z_{t} \end{matrix} ).

Now consider the particular group G = {(α, c, β) : α, β ∈ GF(q)^{2}, c ∈ GF(q)} and define a binary operation on G by

(α, c, β) · (α′, c′, β′) = (α + α′, c + c′ + β(α′)^{T} , β + β′).

This binary operation makes G into a group where (α, c, β)^{−1} = (−α, αβ^{T} − c, −β). Let C be a normalised q-clan and let K_{t} = A_{t} + A_{t}^{T} for t ∈ GF(q). Define the following subgroups of G:

A(∞) = \{(\overline{0}, 0, β) : β ∈ GF(q)^{2}\}
A^{⋆}(∞) = \{(\overline{0}, c, β) : c ∈ GF(q), α ∈ GF(q)^{2}\}
A(t) = {(α, αA_{t}α^{T} , αK_{t}) : α ∈ GF(q)^{2}} for t ∈ GF(q)
A^{⋆}(t) = {(α, c, αK_{t}) : c ∈ GF(q), α ∈ GF(q)^{2}} for t ∈ GF(q),

where \overline{0} is the identity element of GF(q)^{2}. Then F(C) = \{A(t) : t ∈ GF(q) ∪ \{∞\}\} is a 4-gonal family for G which gives rise to a GQ S(G, F(C)) of order (q^{2}, q). We will denote S(G, F(C)) by S(C).

The following three theorems are results on the automorphism group of a q-clan GQ.

Theorem 1.4.6 [48, III.(1)] Suppose G_{1}, G_{2} are groups and F is a 4-gonal family for G_{1}. If θ : G_{1} → G_{2} is a group isomorphism or a group anti-isomorphism, then S(G_{1}, F) and S(G_{2}, θ(F)) are isomorphic GQs.

(A group anti-isomorphism is a one-to-one and onto map θ : G_{1} → G_{2} such that θ(gg′) = θ(g′)θ(g) for all g, g′ ∈ G_{1}.)

In particular, if we have a group automorphism of G then it induces an automorphism of S(G, F).

Theorem 1.4.7 [48, IV.1.] Let C be a normalised q-clan, q odd, with A_{0} = K_{0} = 0. Let S(C) = (P, B, I) be the associated GQ of order (q^{2}, q). Let Θ be a collineation of S(C) which fixes (∞), [A(∞)], and (0, 0, 0). Then there is

a permutation t ↦ \overline{t} of the elements of GF(q), a nonzero element λ of GF(q), an automorphism σ of GF(q) and , a 2 × 2 matrix D ∈ GL(2, q)

for which the following holds

(D^{−1})^{T} (A\overline{t} − A\overline{0}) D^{−1} − λA_{t}^{σ} is a skew symmetric matrix (with 0 diagonal)
for all t ∈ GF(q).  (1.4.1)

Moreover, Θ is induced by an automorphism of G, and is given by

Θ : (α, c, β) ↦ (α^{σ}λ^{−1}D^{−T} , λ^{−1}c^{σ} + λ^{−2}α^{σ}D^{−T} A\overline{0}D^{−1}(α^{σ})^{T} , β^{σ}D + α^{σ}λ^{−1}D^{−T} K\overline{0}).

Conversely, given D, σ, λ and t ↦ \overline{t} as just described, the Θ as above is a collineation of S(C). When q is odd, and if each A_{t} ∈ C is symmetric, 1.4.1 may be replaced by

A\overline{t} = λD^{T} A_{t}^{σ}D + A\overline{0} for all t ∈ GF(q).

The automorphism Θ is denoted by Θ(π, λ, σ, D).

The automorphisms in Theorem 1.4.7 all fix the line [A(∞)], in the following theorem we consider automorphisms not fixing [A(∞)].

Theorem 1.4.8 [48, III.5.] If C = \{A_{t} : t ∈ GF(q)\} is a q-clan, q odd, with A_{0} = ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} ),

then C′ = {A_{t}^{−1} : 0 ≠ t ∈ GF(q)} ∪ {A_{0}} is a q-clan with S(C) ≅ S(C′). In fact, the switch from C to C′ just amounts to interchanging the roles of A(∞) and A(0) in the coordinatisation of S.

Note that from the proof of Theorem 1.4.8 in [48] the isomorphism in Theorem 1.4.8 is given by the group anti-automorphism θ : G → G : (α, c, β) ↦ (β, 2c, 2α). Thus the subgroup Q_{1} = (GF(q) × 0) × GF(q) × (GF(q) × 0) is fixed under the isomorphism.

1.4.6 Ovoids and rosettes of GQs

Ovoids of Q(4, q)

Let Q(4, q) ⊂ PG(4, q), for q odd or even, and let Σ be a hyperplane of PG(4, q) such that Σ ∩ Q(4, q) = θ is a non-singular elliptic quadric of Σ. Then θ is an ovoid of Q(4, q) and an ovoid that arises in such a way is called classical.

If q is even, then Q(4, q) is isomorphic to W (q) and so an ovoid of Q(4, q) is equivalent to an ovoid of W (q). If W (q) ⊂ PG(3, q), then an ovoid of W (q) is also an ovoid of PG(3, q), that is a set of q^{2} + 1 points of PG(3, q), no three collinear. Further, any ovoid of PG(3, q) is an ovoid of a GQ W (q)′, where W (q)′ ⊂ PG(3, q) and W (q)′ is isomorphic to W (q) ([55]). Two ovoids of W (q) are isomorphic under an automorphism of W (q) if and only if they are isomorphic under an automorphism of PG(3, q) ([2, Prop. 1]). Thus, the distinct classes of ovoids of W (q) are exactly the projectively distinct ovoids of PG(3, q).

If q = 2^{e}, e even, then every known ovoid of PG(3, q) is classical. For q = 2, it is straightforward to show that the only ovoids of PG(3, q) (and hence of W (q)) are the classical ovoids. Further, PG(3, q) has only the classical ovoid when q = 4, [4] (see also [25, 16.1.7]) and q = 16, [36]. If q = 2^{e}, e odd, e = 2h + 1 say, and e ≥ 3, then in addition to the classical ovoid PG(3, q) also contains the Tits ovoid. The canonical form of the Tits ovoid is the following set of points of PG(3, q).

{(1, s^{σ} + st + t^{σ+2}, s, t) : s, t ∈ GF(q)} ∪ {(0, 1, 0, 0)}

where σ is the automorphism of GF(q) such that σ : x ↦ x^{2^{h+1}}. For q = 8, PG(3, q) contains only the classical ovoid and the Tits ovoid, which was shown with the assistance of a computer by Fellegara in [20] and without the use of a computer by Penttila and Praeger in [52]. For q = 32, PG(3, q) contains only the classical ovoid and the Tits ovoid [37]. For more information on ovoids of PG(3, q) see [35, 25] and for a discussion of the connection between ovoids of Q(4, q) and ovoids (and spreads) of PG(3, q) see [49, 3.2.1, 3.4.1].

We now consider ovoids of Q(4, q) when q is odd. As we have seen, Q(4, q) contains the classical ovoids and further there are three non-classical ovoids of Q(4, q). The first two were constructed by Kantor in [30]. The ovoid K1 exists for the case q = p^{h}, with p an odd prime and h ≥ 2. In [30] K1 is defined to be the set of points

{(1, y, z, −my^{σ}, −z + my^{σ+1}) : y, z ∈ GF(q)} ∪ {(0, 0, 0, 0, 1)}

of Q(4, q) with the form x_{0}x_{5} + x_{1}x_{4} + x_{2}x_{3}, where m is a non-square element of GF(q) and σ is a non-trivial automorphism of GF(q). In this thesis we shall indicate that a K1 ovoid has associated field automorphism σ by calling it a K1(σ) ovoid. The ovoid K2 of Q(4, q) exists for the case q = 3^{2r+1} with r ≥ 1. The third type of non-classical ovoid of Q(4, q) was constructed by Thas and Payne ([65]) for q = 3^{h} with h ≥ 3. The ovoid of Thas and Payne arises as a subtended ovoid of the embedding of Q(4, q) as a subquadrangle in the dual of the Roman GQ ([65]). For q = 3, 5 or 7, every ovoid of Q(4, q) is classical [10, 34, 16, 14, 15, 33]. For more information on ovoids of Q(4, q) and ovoids Q(2n, q) in general, see [38].

Elation rosettes of Q(4, q)

We saw in Section 1.4.4 that the GQ Q(4, q) is dual to W (q) and that for q even, Q(4, q) is also isomorphic to W (q). We shall use this and elations of the ambient PG(3, q) of W (q) to construct rosettes of Q(4, q).

An elation of P G(n, q) is an automorphism of P G(n, q) that fixes a given hyperplane pointwise and a given point on the hyperplane linewise. The hyperplane is the axis of the elation and the point is the centre of the elation. For more details on elations see [1].

Lemma 1.4.9 [1, VIII.5 lemma 2] Let P be a point and π a plane of P G(3, q) with P ∈ π and let ℓ be a line of P G(3, q) such that P ∈ ℓ and ℓ ⊈ π. The group consisting of the elations with centre P and axis π, acts regularly on the set of points l \ P.

Let ψ be an ovoid of Q(4, q) and suppose that q is even. Since Q(4, q) and W (q) are isomorphic GQs for q even, there is an ovoid θ of W (q) that is the image of ψ under the isomorphism from Q(4, q) to W (q). We will use specific elations of PG(3, q) to construct rosettes containing θ.

Lemma 1.4.10 [59, p. 71] Let σ be an elation of the ambient P G(3, q) of W (q) with centre P and axis P^{⊥}, then σ can be naturally restricted to an automorphism of W (q).

Given an ovoid θ of W (q) containing a point P we will now consider a description of θ that will be useful in the work that follows. The points of W (q) collinear with P are exactly the points of the plane P^{⊥} and so the the q^{2} points of θ \ {P } are in the set P G(3, q) \ P^{⊥}. Also, no two of these points span a line of PG(3, q) that is incident with P, since in that case θ contains three collinear points, contradicting the definition of an ovoid in P G(3, q). Thus each of the q^{2} lines of P G(3, q) \ P^{⊥} incident with P is incident with exactly one further point of θ.

Lemma 1.4.11 Let P be a point of W (q) and θ an ovoid of W (q) such that P ∈ θ. An elation σ which has centre P and axis P^{⊥} maps θ to an ovoid of W (q) containing P. Furthermore the group consisting of the set of such elations, acting on the set of ovoids of W (q), has a rosette based at P and containing θ as the orbit containing θ.

Proof: By Lemma 1.4.10, the elation σ induces an automorphism of W (q) and fixes P, hence θ^{σ} is an ovoid of W (q) containing P. Recall that θ has one point per line through P not in P^{⊥} and the elation group acts regularly on the points of such lines, not P. So the q ovoids which are images of θ under the action of the group of such elations have P as their pairwise intersection and partition the points of P G(3, q) \ P^{⊥}. Thus the orbit of θ under this group is a rosette of W (q) based at P.

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We will call the rosette based at P, containing θ and generated as in Lemma 1.4.11 an elation rosette with respect to P and θ.

For q even the only known ovoids of W (q) are the elliptic quadric ovoids and the Tits ovoids. In Chapter 4 we shall see that if θ is either an elliptic quadric ovoid or a Tits ovoid of W (q) and P is a point of θ, then the elation rosette R constructed in Lemma 1.4.11 is the unique homogeneous rosette containing θ and with base point P.

Now, let θ be an ovoid of Q(4, q) and let q be odd or even. Since Q(4, q) and W (q) are dual GQs, there exists a spread S_{θ} of W (q) such that S_{θ} is the image of θ under the duality from Q(4, q) to W (q). Let ℓ be a fixed line of S_{θ} and P a point incident with ℓ. The plane P^{⊥} contains ℓ and consequently, every other line of S_{θ} meets P^{⊥} in a point.

We introduce the term rosette of spreads of a GQ to mean the dual of a rosette of ovoids.

Lemma 1.4.12 Let ℓ be a line of W (q) and S a spread of W (q) such that ℓ ∈ S. Let P be a point of W (q) incident with ℓ. An elation σ which has centre P and axis P^{⊥} maps S to a spread of W (q) containing ℓ. Furthermore, the group consisting of the set of such elations, acting on the set of spreads of W (q), has a rosette of spreads based at ℓ and containing S as the orbit containing S.

Proof: We consider the action of the elation σ on the lines of S. Since ℓ is contained in the axis of σ it follows that ℓ is fixed by σ. If m ∈ S with m ≠ ℓ, then m ∩ P^{⊥} is a point {Q}, say and < m, P > is a plane π_{m}, say. Since π_{m} contains P, it is fixed by σ and so σ(m) is a line contained in π_{m}, containing Q, but not P. By Lemma 1.4.9 the group of q elations with centre P and axis P^{⊥} map m onto the q distinct lines of π_{m} which contain Q, but not P. Thus if σ and σ′ are two elations with centre P and axis P^{⊥}, then σ(m) = σ′(m) if and only if σ = σ′. If m′ is a line of S such that m′ ≠ ℓ, m, meeting P^{⊥} in the point Q′, then Q ≠ Q′ and so no elation with centre P and axis P^{⊥} maps m onto m′. Thus, the orbit of S under the group of elations with centre P and axis P^{⊥} has size q and has the property that any two spreads in the orbit intersect in exactly ℓ.

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If θ is an ovoid of Q(4, q) and P is a fixed point of θ, then by dualising Lemma 1.4.12 we construct q + 1 homogeneous rosettes containing θ and based at P. We will call these rosettes, elation rosettes. Thus, if q is even, there are q + 1 elation rosettes containing θ and based at P, from Lemma 1.4.12, and 1 elation rosette containing θ and based at P from Lemma 1.4.11. However, for both known examples of ovoids of Q(4, q), where q is even, the elliptic quadric ovoid and the ovoid isomorphic to the Tits ovoid of W (q), there is a unique homogeneous rosette containing a given ovoid of that type and based at a given point. (Suppose that θ is an elliptic quadric ovoid of Q(4, q) with ambient hyperplane Σ and that θ′ is a second elliptic quadric ovoid with ambient hyperplane Σ′. If θ ∩ θ′ = {P }, then π = Σ ∩ Σ′ is the unique tangent plane to θ at P in Σ. Thus the only elliptic quadric ovoids of Q(4, q) that intersect θ in exactly P are the q elliptic quadric ovoids whose ambient hyperplane contains π. This is the unique rosette containing θ with base point P. In the case where θ is isomorphic to a Tits ovoid of W (q) see Lemma 4.1.7.) Thus the q + 2 elation rosettes are identical.

In the case where q is odd, there are q + 1 elation rosettes containing θ and based at P. For θ an elliptic quadric ovoid, the q + 1 elation rosettes are identical (by the same argument presented for the q even case in the preceding paragraph). In Chapter 2 we shall show that if θ is a K1 ovoid, then there exist a point P such that the q + 1 elation rosettes based at P are distinct. We shall also construct homogeneous rosettes of K1 ovoids, that are not elation rosettes.

1.5 Algebraic topology on a simplicial complex

In this section we introduce the concept of a simplicial complex and the homology and cohomology groups of the simplicial complex over an abelian group. Useful introductory texts to algebraic topology are [23], [58], [57] and [32]. We will not be concerned with the more general topological aspects of simplicial complexes only the calculation of their homology and cohomology, for which the results that follow will suffice.

1.5.1 Modules

Let R be a ring with identity 1 ≠ 0. A left R-module A is an additive abelian group together with a function p : R × A → A, written p(r, a) = ra, such that

(r + r′)a = ra + r′a,
(rr′)a = r(r′a),
r(a + a′) = ra + ra′,
1a = a

where r, r′ ∈ R and a, a′ ∈ A. It follows that 0a = 0 and (−1)a = −a.

We can define right R-modules in the obvious manner. Since we will not be using right Rmodules in what follows, we will refer to a left R-module as an R-module, or simply a module if the ring R is understood from context. If 𝔽 is a field, then an 𝔽-module is a vector space over 𝔽.

Note that for any abelian group A we may find a ring R such that A is an R-module. As an example, if A is any abelian group, then we may turn it into a ℤ-module. If m is any positive integer, and a any element of A, then define ma = a + a + … + a (m times). If we also define (−m)a = −(ma) then this makes A a ℤ-module.

A subset S of an R-module A is a submodule, if S is closed under addition and if r ∈ R, s ∈ S imply rs ∈ S; then S itself is an R-module. As S is a subgroup of A, we may form the factor group A/S, which is also an R-module, since r(b + S) = rb + S. The R-module A/S is called a quotient module.

If A and B are both R-modules, then α : A → B is an R-module homomorphism if

α(a + a′) = αa + αa′, α(ra) = r(α(a))

where r ∈ R and a ∈ A.

We define Hom_{R}(A, B) to be the set of all R-module homomorphisms from A into B. If f ∈ Hom_{R}(A, B) and t ∈ R, then we define tf : A → B by (tf)(a) = t(f(a)) for all a ∈ A. If the ring R is commutative, then we have that

(tf)(ra) = t(f(ra)) = tr(f(a)) = rt(f(a)) = r[(tf)(a)],

which shows that tf ∈ Hom_{R}(A, B) and so Hom_{R}(A, B) is an R-module. In particular, if A = B and R is a field, then Hom_{R}(A, A) is the space of linear functionals of the vector space A, that is the dual vector space.

1.5.2 Simplicial complexes, homology and cohomology

A simplicial complex Γ = (V, S) consists of a set V of vertices and a set S of finite non-empty subsets of V called simplexes such that

(a) Any set consisting of exactly one vertex is a simplex.

(b) Any non-empty subset of a simplex is a simplex.

A simplicial complex Γ′ = (V′, S′) is a subcomplex of Γ = (V, S) if V′ ⊆ V and S′ ⊆ S and a proper subcomplex if Γ′ ≠ Γ.

A simplex containing exactly q + 1 vertices is called a q-simplex. The dimension of a q-simplex is defined to be q. The dimension of the simplicial complex Γ, denoted by dimΓ, is defined to be equal to −1 if Γ is empty, equal to n if Γ contains an n-simplex but no (n + 1)- simplex and equal to ∞ if Γ contains n-simplexes for all n ≥ 0. The simplicial complex Γ is said to be finite if it contains only a finite number of simplexes.

If s and s′ are simplexes of Γ and s′ ⊆ s, then s′ is called a face of s (a proper face if s′ ≠ s) and if s′ is a p-simplex, it is called a p-face of s.

A simplex becomes oriented, or is given an orientation, when we specify a particular ordering of its vertices. Two orderings of the vertices which agree up to an even permutation of the vertices determine the same orientation. If σ is the simplex s with a particular orientation, then we represent s with the opposite orientation by −σ.

Let R be a finite ring and Γ a simplicial complex. Let S_{q}(Γ, R) be the R-module whose elements are the formal linear combinations ∑_{σ} r_{σ}σ, where σ runs through the oriented qsimplexes of Γ and r_{σ} is an element of R. The operations of the R-module are defined as follows:

∑_{σ} r_{σ}σ + ∑_{σ} r′_{σ}σ = ∑_{σ}(r_{σ} + r′_{σ})σ

r (∑_{σ} r_{σ}σ) = ∑_{σ}(rr_{σ})σ for r ∈ R.

Now let B be a list of the q-simplexes of Γ with each simplex given an orientation. Then since each oriented q-simplex is either σ or −σ for a unique element σ of B, it follows that each element of S_{q}(Γ, R) can be written uniquely in the form

∑_{σ∈B} r_{σ}σ + ∑_{σ∈B} r′_{σ}(−σ)

where r_{σ} and r′_{σ} are both elements of R.

Let S′_{q}(Γ, R) be the submodule of S_{q}(Γ, R) generated by elements of the form rσ + r(−σ), where r ∈ R and σ is an oriented q-simplex. We define C_{q}(Γ, R) to be the quotient module S_{q}(Γ, R)/S′_{q}(Γ, R). An element of the R-module C_{q}(Γ, R) is called a q-chain of Γ over R. Essentially, in C_{q}(Γ, R) we are allowed to replace rσ by −r(−σ) wherever we please. If Γ is non-empty, then we define C_{−1}(Γ, R) = 0 and if Γ has finite dimension n we define C_{q}(Γ, R) = 0 for q > n.

The q-th boundary operator is an R-homomorphism ∂_{q} : C_{q}(Γ, R) → C_{q−1}(Γ, R). Let s = (P_{0}P_{1} … P_{q}) be an oriented q-simplex of Γ (and so also a q-chain), then the action of ∂_{q} on s is

∂_{q}(s) = ∑_{i=0}^{q}(−1)^{i}(P_{0}P_{1} … P_{i−1}P_{i+1} … P_{q}).

The (q − 1)-simplex {P_{0}P_{1} … P_{i−1}P_{i+1} … P_{q}} is the face of s opposite P_{i}. Note that ∂_{q}(s) is independent of the ordering of the vertices we use to represent the orientation of s. If we take an even permutation, T, on the ordering of the vertices of s, we may consider the permutation as the product of a permutation, T_{1}, fixing the vertex P_{i} in the i-th position, and a permutation, T_{2}, shifting P_{i} r positions. In terms of the formula for ∂_{q}, T_{1} induces a permutation on the vertices of the face of s opposite P_{i}, while T_{2} multiplies the oriented face opposite P_{i} by a factor of (−1)^{r}. Since T is even, T_{1} and T_{2} are either both odd or both even. In either case, the orientation of the face of s opposite to P_{i}, as it appears in ∂_{q}(s) is preserved.

We extend ∂_{q} to act on q-chains of Γ over R by letting it act linearly, that is,

∂_{q}(r_{σ}σ) = r_{σ}∂_{q}(σ) and ∂_{q}(σ + σ′) = ∂_{q}(σ) + ∂_{q}(σ′).

where r_{σ} is an element of R and σ and σ′ are oriented q-simplexes of Γ. The (q − 1)-chain ∂_{q}(s) is called the boundary of s. Note that since ∂_{q}(σ) + ∂_{q}(−σ) = 0 for any q-simplex σ. The operator ∂_{q} does not depend on which representation of a q-chain we choose.

The following lemma is an exercise in algebra:

Lemma 1.5.1 ∂_{q} ∘ ∂_{q+1} = 0.

Proof: Let s = (P_{0}, P_{1}, …, P_{q+1}) be a oriented (q + 1)-simplex of Γ. Now

∂_{q} ⋯ ∂_{q+1}(s) = ∑_{0≤i<j≤q+1} k_{ij}(P_{0}P_{1} … P_{i−1}P_{i+1} … P_{j−1}P_{j+1} … P_{q+1}),

where k_{ij} is some element of R for each i, j, 0 ≤ i < j ≤ q + 1. We now show that every k_{ij} = 0. The oriented (q − 1)-simplex (P_{0}P_{1} … P_{i−1}P_{i+1} … P_{j−1}P_{j+1} … P_{q+1}) appears in ∂_{q} ⋯ ∂_{q+1}(s) either by ∂_{q+1} “removing” the vertex P_{i} and ∂_{q} removing the vertex P_{j}, or vice versa. Hence

k_{ij} = (−1)^{i}(−1)^{j−1} + (−1)^{j}(−1)^{i} = 0.

Thus ∂_{q} ∘ ∂_{q+1}(s) = 0 for any oriented (q + 1)-simplex s and it follows that ∂_{q} ∘ ∂_{q+1} = 0.

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A q-chain s such that ∂_{q}(s) = 0 is called a q-cycle and if s = ∂_{q+1}(s′) for some (q+1)-chain s′, then s is called a q-boundary. Two q-chains that differ by a boundary are called homologous and a q-cycle that is homologous to the zero q-chain is called null homologous. The set of q-cycles form an R-module (ker(∂_{q})) denoted by Z_{q}(Γ, R) and the set of q-boundaries also from an R-module (im(∂_{q+1})) which is denoted by B_{q}(Γ, R). If s is a q-boundary with s = ∂_{q+1}(s′), then ∂_{q}(s) = ∂_{q} ∘ ∂_{q+1}(s′) = 0 by Lemma 1.5.1 and so B_{q}(Γ, R) is a subgroup of Z_{q}(Γ, R). The quotient group Z_{q}(Γ, R)/B_{q}(Γ, R) is called the q-th homology group of Γ over R, and is denoted H_{q}(Γ, R). Note that H_{−1}(Γ, R) = 0 and if Γ has finite dimension n, then H_{q}(Γ, R) = 0 for q > n.

Let A be an R-module, then Hom_{R}(C_{q}(Γ, R), A) is the abelian group of R-homomorphisms from C_{q}(Γ, R) to A. We will denote Hom_{R}(C_{q}(Γ, R), A) by C^{q}(Γ, A), where A is understood to be an R-module. Any element of C^{q}(Γ, A) is called a q-cochain of Γ into A.

The q-th coboundary operator is a group homomorphism δ^{q} : C^{q}(Γ, A) → C^{q+1}(Γ, A). Let c be a q-cochain and s = (P_{0}P_{1} … P_{q+1}) an oriented q + 1-simplex of Γ. Then the action of δ^{q}c on s is defined as

δ^{q}c(s) = ∑_{i=0}^{q+1}(−1)^{i}c(P_{0}P_{1} … P_{i−1}P_{i+1} … P_{q+1}).

By linearity this determines the action of δ^{q}c on all q + 1-chains. The q + 1-cochain δ^{q}c is called the coboundary of c.

We have an analogous result to Lemma 1.5.1 for δ^{q}:

Lemma 1.5.2 δ^{q+1} ∘ δ^{q} = 0.

The following lemma links the boundary and coboundary operators:

Lemma 1.5.3 Let s be a (q + 1)-chain and c a q-cochain. Then

δ^{q}c(s) = c(∂_{q+1}s).

Proof: First let s = (P_{0}P_{1} … P_{q+1}) be an oriented (q + 1)-simplex of Γ. Then

∂_{q+1}(s) = ∑_{1=0}^{q+1}(−1)^{i}(P_{0}P_{1} … P_{i−1}P_{i+1} … P_{q+1}),

and so

c(∂_{q+1}(s)) = ∑_{1=0}^{q+1}(−1)^{i}c(P_{0}P_{1} … P_{i−1}P_{i+1} … P_{q+1})

since c is an R-homomorphism. Now by definition

δ^{q}c(s) = ∑_{1=0}^{q+1}(−1)^{i}c(P_{0}P_{1} … P_{i−1}P_{i+1} … P_{q+1})

and so δ^{q}c(s) = c(∂_{q+1}(s)) for any oriented (q + 1)-simplex s. From this it follows that δ^{q}c(s) = c(∂_{q+1}(s)) for any (q + 1)-chain s.

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A q-cochain c such that δ^{q}c = 0 is called a q-cocycle and if c = δ^{q−1}c′ for some (q − 1)- cochain c′, then c is called a coboundary. Two q-cochains that differ by a boundary are called cohomologous. Z^{q}(Γ, A) is the group of q-cocycles (ker(δ^{q})) and B^{q}(Γ, A) is the group of q-coboundaries (im(δ^{q−1})). Lemma 1.5.2 tells us that B^{q}(Γ, A) is a subgroup of Z^{q}(Γ, A). The quotient group Z^{q}(Γ, A)/B^{q}(Γ, A) is the q-th cohomology group of Γ over A, and is denoted H^{q}(Γ, A).

In the special case where Γ is finite and R = A = 𝔽 for some field 𝔽 we have the following theorem:

Theorem 1.5.4 Let Γ be a non-empty, finite simplicial complex and 𝔽 a field. Then

H_{q}(Γ, 𝔽) ≅ H^{q}(Γ, 𝔽) for all q ≥ −1.

Proof: Let the dimension of Γ be n, which is necessarily finite. If q = −1 or q > n then H_{q}(Γ, 𝔽) = H^{q}(Γ, 𝔽) = 0.

Now suppose that 0 ≤ q ≤ n. Since Γ is finite and 𝔽 is a field, C_{q}(Γ, 𝔽) is a vector space over 𝔽, of dimension k, say, and C^{q}(Γ, 𝔽) is the dual space of C_{q}(Γ, 𝔽). Z_{q}(Γ, 𝔽) and B_{q}(Γ, 𝔽) are subspaces of C_{q}(Γ, 𝔽). We will denote the respective dimensions of Z_{q}(Γ, 𝔽) and B_{q}(Γ, 𝔽) by k_{1} and k_{2}.

Now

annihilator(B_{q}(Γ, 𝔽)) = {c ∈ C^{q}(Γ, 𝔽) : c(s) = 0 for all s ∈ B_{q}(Γ, 𝔽)}
= {c ∈ C^{q}(Γ, 𝔽) : c(∂^{q+1}s′) = 0 for all s′ ∈ C_{q+1}(Γ, 𝔽)}
= {c ∈ C^{q}(Γ, 𝔽) : δ^{q}c(s′) = 0 for all s′ ∈ C_{q+1}(Γ, 𝔽)}
= Z^{q}(Γ, 𝔽)

Thus Z^{q}(Γ, 𝔽) is a subspace of C^{q}(Γ, 𝔽) with dimension k−k_{2}. By a similar argument we obtain that annihilator(B^{q}(Γ, 𝔽)) = Z_{q}(Γ, 𝔽) which implies that the dimension of B^{q}(Γ, 𝔽) is k − k_{1}. Now, the dimension of H_{q}(Γ, 𝔽) = k_{1} −k_{2} and the dimension of H^{q}(Γ, 𝔽) = (k −k_{2})−(k −k_{1}) = k_{1} − k_{2}. Since H_{q}(Γ, 𝔽) and H^{q}(Γ, 𝔽) are vector spaces over 𝔽 with the same dimension, we have that H_{q}(Γ, 𝔽) ≅ H^{q}(Γ, 𝔽).

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1.5.3 Cohomology groups of a simplicial complex

Recall, that in the previous section we defined C^{q}(Γ, A) to be the group Hom_{R}(C_{q}(Γ, A)) where A was understood to be an R-module. By linearity, if c is any element of Hom_{R}(C_{q}(Γ, A)), then c is determined by how it maps the oriented q-simplexes. If σ is an oriented q-simplex of Γ, then the only restriction on c(σ) is that c(σ) = −c(−σ). At this point we observe that the abelian group C^{q}(Γ, A) is independent of the ring R. So for any simplicial complex Γ and any abelian group A we will define the q-th cohomology group of Γ into A, without reference to any ring, or to A as an R-module. The group C^{q}(Γ, A) is defined to be the group of alternating functions from the oriented q-simplexes into A (where a function c is alternating if c(σ) = −c(−σ) for all oriented q-simplexes σ). We then define the coboundary operator and cohomology group as in Section 1.5.2.

1.5.4 The simplicial complex of a graph

Let G be a graph with vertex set V and edge set E. We define Γ_{G} to be the simplicial complex which has as 0-simplexes the vertices of G, 1-simplexes the edges of G, 2-simplexes the complete subgraphs on 3 vertices and in general, has as q-simplexes the complete subgraphs on q + 1 vertices.

If Γ is a simplicial complex, then the q-skeleton Γ^{q} of Γ is the simplicial complex consisting of all p-simplexes of Γ for p ≤ q. The 1-skeleton of Γ is the simplicial complex of 0-simplexes and 1-simplexes of Γ, which is a graph. The 1-skeleton of Γ_{G} is G.

1.5.5 Homology and cohomology over ℤ_{2}

Suppose that Γ is a simplicial complex and that R is a ring with even characteristic, that is −r = r for any r ∈ R and hence (−r)(−σ) = rσ for any r ∈ R and oriented q-simplex σ. Recall, from the construction of C_{q}(Γ, R), that if σ is an oriented q-simplex, then the q-chain represented by σ is the same as that represented by σ + (σ + (−σ)) = (1 + 1)(σ) + (−σ) = −σ. In other words, the orientation of the simplex is unimportant and C_{q}(Γ, R) may be considered as the R-module of formal linear sums of q-simplexes (unoriented) of Γ, over R.

In the case where R = ℤ_{2}, we may represent C_{q}(Γ, R) as follows. The elements of C_{q}(Γ, R) are the subsets of the set of q-simplexes and the abelian group operation is symmetric difference on the subsets. Given an element s of C_{q}(Γ, R), as a sum of q-simplexes, the subset of q-simplexes corresponding to s contains those q-simplexes whose coefficient in s is 1. The boundary of a q-simplex s is the set of (q − 1)-faces of s and the boundary of a q-chain σ is the symmetric difference of the sets of (q − 1)-faces of the q-simplexes of σ. Note, that from the definition of an R-module, we know that 1.σ = σ and 0.σ = 0 for any σ ∈ C_{q}(Γ, R).

If we consider the cohomology group C^{q}(Γ, A) where A is a ℤ_{2}-module, it should be noted that A has characteristic 2, since for a ∈ A we have that a+a = 1·a+1·a = (1+1)·a = 0·a = 0.

In particular, we now consider the vector space H_{1}(Γ, ℤ_{2}), for some finite simplicial complex Γ. Let σ be a 1-cycle of Γ, then ∂_{1}(σ) = 0 and so each 0-simplex of Γ is contained in an even number of 1-simplexes of σ (the number may be 0). Let σ′ be a 1-boundary of Γ with σ′ = ∂_{2}(σ′′), then σ′ is the symmetric difference of the sets of 1-faces of the 2-simplexes of σ′′.

Let σ = {s_{1}, s_{2}, …, s_{n}} be a 1-cycle of Γ. We say that σ is an elementary 1-cycle of Γ, if each vertex of Γ appears in exactly none or two 1-simplexes of σ. We say that σ is induced if for any 1-simplex s such that ∂_{1}(s) = {P, Q} with P ∈ ∂_{1}(s_{i}), Q ∈ ∂_{1}(s_{j}), then s ∈ σ. If σ is an induced 1-cycle that is not the boundary of a 2-simplex, we say that σ is a proper induced 1-cycle.

Lemma 1.5.5 A 1-cycle of Γ may be written as the sum of elementary 1-cycles, with no common 1-simplexes.

Proof: Let σ be a 1-cycle of Γ, which we represent as the set {s_{1}, s_{2}, …, s_{n}}, where the s_{i} are distinct 1-simplexes of Γ. Let i ∈ {1, 2, …, n} and let ∂_{1}(s_{i}) = {P_{1}, P_{2}}, where P_{1} and P_{2} are 0-simplexes of Γ. Since σ is a 1-cycle, there is an i_{1} such that 1 ≤ i_{1} ≤ n and ∂_{1}(s_{i_{1}}) = {P_{2}, P_{3}}, for some 0-simplex P_{3} of Γ. Similarly, there is an i_{2} such that 1 ≤ i_{2} ≤ n and ∂_{1}(s_{i_{2}}) = {P_{3}, P_{4}}, for some 0-simplex P_{4} of Γ. Continuing this process will eventually yield an i_{r} such that 1 ≤ i_{r} ≤ n and ∂_{1}(s_{i_{r}}) = {P_{r+1}, P_{1}}. The 1-cycle σ_{i} = {s_{i}, s_{i_{1}}, s_{i_{2}}, …, s_{i_{r}}} is an elementary 1-cycle. If we now choose s_{j} ∈ σ \ σ_{i}, we can generate a elementary 1-cycle contained in σ and disjoint to σ_{i}. Continuing this process gives σ as the sum of elementary 1-cycles.

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Now we prove a lemma showing that we may break down an elementary 1-cycle into induced 1-cycles.

Lemma 1.5.6 An elementary 1-cycle of Γ can be written as the sum of induced 1-cycles of Γ.

Proof: Let σ = {s_{1}, s_{2}, …, s_{n}} be an elementary 1-cycle of Γ such that ∂_{1}(s_{i}) = {P_{i}, P_{i+1}} for 1 ≤ i ≤ n − 1 and ∂_{1}(s_{n}) = {P_{n}, P_{1}}. If σ is an induced 1-circuit, then there is no more to show, so suppose that σ is not induced. Since σ is not induced we may assume, without loss of generality, that for some i, i ≠ 1, 2, n there exists a 1-simplex s with ∂_{1}(s) = {P_{1}, P_{i}}. Then σ may be written as the sum of the elementary 1-cycles {s_{1}, s_{2}, …, s_{i−1}, s} and {s_{i}, s_{i+1}, …, s_{n}, s}, both of which have fewer elements than σ. We continue this process until we have written σ as sum of induced 1-cycles.

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Combining Lemma 1.5.5 and Lemma 1.5.6 gives us the following result.

Corollary 1.5.7 A 1-cycle of Γ can be expressed as the sum of induced 1-cycles of Γ.

Thus we have the following theorem.

Theorem 1.5.8 H^{1}(Γ, ℤ_{2}) is trivial if and only if each induced 1-cycle of Γ is a 1-boundary.

The above definitions and results make more intuitive sense in the graph Γ^{1}, the 1-skeleton of Γ. Recall that a 2-simplex of Γ is a triangle in Γ^{1}, a 1-simplex an edge and a 0-simplex a vertex of Γ^{1}. In Γ^{1} a 1-cycle of Γ is a set of edges such that each vertex appears in an even number of edges, that is, the set of edges of a circuit of Γ^{1}. An elementary 1-cycle is the set of edges of a elementary circuit of Γ^{1} and an induced 1-cycle is the set of edges of an induced circuit of Γ^{1}. Lemma 1.5.5 says that every circuit may be decomposed into elementary circuits, while Lemma 1.5.6 says that any elementary circuit may be decomposed into induced circuits. Finally, Theorem 1.5.8 says that H^{1}(Γ, ℤ_{2}) is trivial if and only if each induced circuit of the graph Γ^{1} can be decomposed into triangles.

1.6 Covers of a graph and covers of a geometry

In this section we introduce the cover of a graph and the cover of a geometry, as in [12] (see also [56]).

Let G be a graph, then an m-fold cover of G is a pair (\overline{G}, p) where \overline{G} is a graph and p is a map from \overline{G} to G satisfying:

(i) For any vertex P ∈ G, p^{−1}(P) consists of m pairwise non-adjacent vertices

(ii) For any edge e of G, p^{−1}(e) consists of m disjoint edges

(iii) For any non-edge {P, Q} of G, p^{−1} ({P, Q}) is a graph with no edges.

If G is the point graph of a geometry S and (\overline{G}, p) satisfies

(iv) For any line ℓ of S, if P_{ℓ} = {P ∈ P : P I ℓ}, then p^{−1}(P_{ℓ}) consists of m disjoint complete graphs

then we can form a geometry \overline{S} with pointset the vertices of \overline{G} and lines (as sets of points) defined to be the complete graphs from (iv). The map p naturally induces a map from \overline{S} to S, which, introducing an abuse of notation, we also call p. The pair (\overline{S}, p) is called an mfold cover of S. The geometry \overline{S} will be called the covering geometry and the map p the covering map. We will often take the existence of the map p to be understood and call \overline{S} an m-fold cover of S. Any element of p^{−1}(P) will be called a cover of P and similarly, any element of p^{−1}(ℓ) a cover of ℓ,

Note that if (iv) is satisfied, and ℓ is a line of S, then (i) and (ii) imply that each line in the set p^{−1}(ℓ) has the same size as ℓ (as a set of points). Also, if P I ℓ, then each point in the set p^{−1}(P) is incident with exactly one element of p^{−1}(ℓ). This means that if ℓ (as a set of points) is {P_{1}, P_{2}, …, P_{s+1}}, then each line of the set p^{−1}(ℓ) has the form {P′_{1}, P′_{2}, …, P′_{s+1}}, where P′_{i} ∈ p^{−1}(P_{i}), for i = 1, 2, …, s + 1. That is, P′ incident with ℓ′ in \overline{S} implies that p(P′) is incident with p(ℓ′) in S.

If (\overline{S}, p) is an m-fold cover of the geometry S, S^{∧} is the dual geometry of S and \overline{S}^{∧} is the dual geometry of \overline{S}, then it is relatively straight-forward to show that (\overline{S}^{∧}, p) is an m-fold cover of \overline{S}^{∧}.

Let G be a graph, Γ_{G} the simplicial complex of G and A an additive abelian group of order m. To simplify matters, we will identify a vertex of Γ_{G} with the corresponding vertex of G, and a 1-simplex of Γ_{G} with the corresponding ordered pair of adjacent vertices of G. Recall that a 1-cochain c of Γ_{G} into A is a map from the set of ordered pairs of adjacent vertices of G to A, such that c(P, Q) = −c(Q, P) for {P, Q} any pair of adjacent vertices of G. An algebraic m-fold cover of G over A is an m-fold cover of G, (\overline{G}, p) where \overline{G} is the graph with

Vertex Set: {(P, α) : P ∈ G, α ∈ A}
Adjacency: (P, α) ∼ (Q, β) if P ∼ Q and c(P, Q) = α − β

and p is the map p((P, α)) = P. Any 1-cochain c defines an m-fold cover of G in the above way. If G is the point graph of a geometry S and (\overline{S}, p) is an m-fold cover of S, then we say that (\overline{S}, p) is an algebraic m-fold cover of S.

Let G be the point graph of a geometry S and let (\overline{G}, p) be an algebraic m-fold cover of G. We investigate conditions under which condition (iv) above is satisfied.

Let ℓ be a line of S such that P_{ℓ} = {P_{1}, P_{2}, …, P_{s+1}}, then (P_{1}, α) is collinear to the set of points {(P_{2}, α − c(P_{1}, P_{2})), (P_{3}, α − c(P_{1}, P_{3})), …, (P_{s+1}, α − c(P_{1}, P_{s+1})). Thus, p^{−1}(P_{ℓ}) consists of m disjoint complete graphs if and only if the m complete graphs have vertex sets

{(P_{1}, α), (P_{2}, α − c(P_{1}, P_{2})), …, (P_{s+1}, α − c(P_{1}, P_{s+1}))} for α ∈ A.

This is true if and only if

(P_{i}, α − c(P_{1}, P_{i})) ∼ (P_{j}, α − c(P_{1}, P_{j})) for all P_{i}, P_{j} where i ≠ j, i, j ≠ 1 and α ∈ A.

Writing δ for the first coboundary operator δ^{1}, we see that this is true if and only if

δc(P_{1}, P_{i}, P_{j}) = 0 for all P_{i}, P_{j} where i ≠ j and i, j ≠ 1.

Equivalently,

δc(P_{i}, P_{j}, P_{k}) = c(P_{i}, P_{j}) − c(P_{i}, P_{k}) + c(P_{j}, P_{k})
= (c(P_{1}, P_{j}) − c(P_{1}, P_{i})) − (c(P_{1}, P_{k}) − c(P_{1}, P_{i})) + (c(P_{1}, P_{k}) − c(P_{1}, P_{j}))
= 0,

for all P_{i}, P_{j} where i, j, k are distinct and i, j, k ≠ 1. Thus (\overline{G}, p) gives rise to an algebraic m-fold cover of S if and only if

δc(P, Q, R) = 0 for all distinct collinear points P, Q, R.

We will call (\overline{G}, p) and (\overline{S}, p) the algebraic m-fold covers of G and S respectively, defined by c, or say that c defines (\overline{G}, p) and (\overline{S}, p) respectively.

If (\overline{S}, p) and (\overline{S}′, p′) are two algebraic m-fold covers of the geometry S, defined by c and c′ respectively, such that \overline{S} and \overline{S}′ are isomorphic geometries, then we say that (\overline{S}, p) and (\overline{S}′, p′) are equivalent. Where \overline{S} and \overline{S}′ are understood to be covers of S, we say that c and c′ are equivalent. Note that if c and c′ are cohomologous, with c′ = c + δb, where δ is the first coboundary operator δ^{1}, then c and c′ are equivalent, and the map i : (P, α) ↦ (P, α + b(P)) is an isomorphism from \overline{S} to \overline{S}′.

The Kantor ovoid K1(σ)

In this chapter we investigate the K1(σ) ovoid of Kantor constructed in [30]. The K1(σ) ovoid exists for q = p^{h}, where p is an odd prime, h ≥ 2 and σ is a non-trivial automorphism of GF(q). For this chapter we will assume that q is of the above form. In Section 2.1 we investigate the connection between the K1(σ) ovoid and the q-clan (see Section 1.4.5) of Kantor ([31, (4.1) Theorem]), via the work of Thas on flocks ([63]). In Section 2.2 and Section 2.3 we consider the symmetry properties of the K1(σ) ovoid and the intersections of the K1(σ) ovoid with the elliptic quadric ovoids of Q(4, q). Section 2.5 contains some characterisations of the K1(σ) ovoid, while Section 2.6 is dedicated to constructing rosettes of Q(4, q) containing a K1(σ) ovoid.

Let the non-singular quadric Q^{+}(5, q) be given by the equation x_{0}x_{5} + x_{1}x_{4} + x_{2}x_{3} = 0 in PG(5, q). In section 5 of [30] Kantor constructs a family of ovoids of Q^{+}(5, q), given by the set of points

{(0, 0, 0, 0, 0, 1)} ∪ {(1, y, z, z^{τ} , −my^{σ}, −z^{τ+1} + my^{σ+1}) : y, z ∈ GF(q)}

where σ and τ are automorphisms of GF(q), not both trivial, and m is a fixed non-square element of GF(q).

In [30], Kantor makes the comment that if we let τ = 1, then the above ovoid of Q^{+}(5, q) is contained in a hyperplane section of Q^{+}(5, q) (with equation x_{2} = x_{3}) and hence is an ovoid of the GQ Q(4, q). In this case, Q(4, q) is given by the equation x_{0}x_{4} + x_{1}x_{3} + x_{2}^{2} = 0 and the ovoid comprises the following set of points

{(0, 0, 0, 0, 1)} ∪ {(1, y, z, −my^{σ}, −z^{2} + my^{σ+1}) : y, z ∈ GF(q)}

Any ovoid of Q(4, q) that is isomorphic to this set of points is a K1(σ) ovoid of Q(4, q).

2.1 Flocks and Ovoids

If we represent the above K1(σ) ovoid as the intersection of Q(4, q) with the variety defined by the equation mx_{1}^{σ} +x_{0}^{σ−1}x_{3} = 0, then it may be viewed as a perturbation of the classical elliptic quadric ovoid that is the intersection of Q(4, q) with the hyperplane with equation mx_{1}+x_{3} = 0. To put the K1(σ) ovoid into a broader geometrical context, we will consider the connection between ovoids of Q(4, q) and flocks of a quadratic cone of PG(3, q).

Let C be a conic of PG(2, q) and let K be the quadratic cone with vertex a point V of PG(3, q) \ PG(2, q) and base C. A flock of K is a set of q disjoint conics that partition the points of K \ {V } (see [21] and [19, p.254]). Note that we can equivalently define a flock to be the set of planes containing these q conics. We will swap between the two definitions according to which is more convenient. The following result, proved independently by Thas and Walker ([63] and [67], respectively), shows how we may construct an ovoid of Q^{+}(5, q) from a flock of PG(3, q).

Theorem 2.1.1 Let K be a quadratic cone of PG(3, q) and F a flock of K. Then F gives rise to an ovoid of the Klein quadric Q^{+}(5, q).

Proof: ([63]) Consider the flock F = {C_{1}, …, C_{q}} (of conics) of the quadratic cone K with vertex V. Embed K into Q = Q^{+}(5, q) and let ⊥ denote the polarity of Q. Note C_{i}^{⊥} is a conic on Q ([27, Theorem 22.7.2]).

If π_{i} is the plane containing C_{i} then π_{i} ∩π_{j} is an exterior line to Q, for i ≠ j, and so ⟨π_{i}^{⊥}, π_{j}^{⊥}⟩ is a non-singular three dimensional elliptic section of Q. Hence no two points of C_{i}^{⊥} ∪ C_{j}^{⊥} are on a common line of Q. Now ⟨π_{i}, π_{j}⟩ is the section containing K and so π_{i}^{⊥} ∩ π_{j}^{⊥} is a tangent to Q. Thus C_{i}^{⊥} ∩ C_{j}^{⊥} = {V }. So now the set C_{1}^{⊥} ∪ … ∪ C_{q}^{⊥} is a set of q^{2} + 1 points, no two on a common line of Q, which is an ovoid of Q.

□

Corollary 2.1.2 Let K be a quadratic cone in PG(3, q) and let F = {π_{1}, …, π_{q}} a flock (of planes) of K. If there is a point P ∈ PG(3, q) such that P is an exterior point to K and P ∈ π_{i} for i = 1, …, q, then F gives rise to an ovoid of Q(4, q).

Proof: By the construction outlined in Theorem 2.1.1, the flock F gives an ovoid O of Q^{+}(5, q). If the planes of F have a common exterior point P, then the conic planes that form O are contained in P^{⊥}; a common parabolic section of Q^{+}(5, q). Such a section is isomorphic to Q(4, q) and since O is a set of q^{2} + 1 points of Q(4, q), no 3 collinear, O is an ovoid of Q(4, q).

□

In [63, Section 2.5.3] Thas constructs a flock of a quadratic cone of PG(3, q) from a q-clan. We now consider the flock F_{σ} constructed in this manner from the q-clan of Kantor given in [31] and derive the corresponding ovoid of Q^{+}(5, q). We will show that the ovoid of Q^{+}(5, q) corresponding to the flock F_{σ} is an ovoid of Q(4, q), for Q(4, q) ⊂ Q^{+}(5, q) and, in fact, is a K1(σ) ovoid of Q(4, q). Let K be a quadratic cone of PG(3, q) given by the equation x_{0}x_{1} = x_{2}^{2} (and so with vertex V = (0, 0, 0, 1)), then the planes of the Kantor flock F_{σ} are

π_{t} : tx_{0} − mt^{σ}x_{1} + x_{3} = 0,

where σ ∈ Aut(GF(q)), σ ≠ identity, m is a fixed non-square of GF(q) and t ∈ GF(q).

Now embed the PG(3, q) containing K into the PG(5, q) containing Q^{+}(5, q), given by the equation x_{0}x_{1} + x_{2}x_{3} + x_{4}x_{5} = 0, so that K occurs as the intersection of Q^{+}(5, q) and

Σ = {(x_{0}, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}) : x_{2} + x_{3} = 0 and x_{5} = 0}.

Thus, in this representation, PG(3, q) is in coordinates (x_{0}, x_{1}, x_{2}, x_{4}), the cone K has equation x_{0}x_{1} − x_{2}^{2} = 0 and π_{t} is the intersection of Σ with the hyperplane of PG(5, q) with equation tx_{0} − mt^{σ}x_{1} + x_{4} = 0.

If we let ⊥ represent the polarity of the Q^{+}(5, q), then

π_{t}^{⊥} = ⟨(−mt^{σ}, t, 0, 0, 0, 1), (0, 0, 1, 1, 0, 0), (0, 0, 0, 0, 1, 0)⟩.

The point (0, 0, 1, −1, 0, 0) is on the plane π_{t} for all t ∈ GF(q) and so the ovoid constructed from F_{σ} is contained in the hyperplane (0, 0, 1, −1, 0, 0)^{⊥}, which has equation x_{2} = x_{3}. The corresponding section of Q^{+}(5, q) is the Q(4, q) with equation x_{0}x_{1} + x_{2}^{2} + x_{4}x_{5} = 0.

Since x_{2} = x_{3}, we suppress the x_{3} coordinate. In PG(4, q) with homogeneous coordinates (x_{0}, x_{1}, x_{2}, x_{4}, x_{5}); Q(4, q) has equation x_{0}x_{1} + x_{2}^{2} + x_{4}x_{5} = 0 and

π_{t}^{⊥} = ⟨(0, 0, 0, 1, 0), (0, 0, 1, 0, 0), (−mt^{σ}, t, 0, 0, 1)⟩
= {(x_{0}, x_{1}, x_{2}, x_{4}, x_{5}) : x_{0} + mt^{σ−1}x_{1} = 0 and x_{1} − tx_{5} = 0}.

Now π_{t}^{⊥} ∩ Q = {(−mt^{σ}, t, z, −z^{2} + mt^{σ+1}, 1) : z ∈ GF(q)} ∪ {(0, 0, 0, 1, 0)} is a conic, C_{t}, say, and so the ovoid is

θ_{σ} = {(−mt^{σ}, t, z, −z^{2} + mt^{σ+1}, 1) : t, z ∈ GF(q)} ∪ {(0, 0, 0, 1, 0)}.

This is exactly the form of the (canonical) K1(σ) ovoid of Kantor as given in Kantor [30].

The line ℓ_{σ} = ⟨(0, 0, 1, 0, 0), (0, 0, 0, 1, 0)⟩ is a tangent to the Q(4, q) and has the property that ℓ ⊂ π_{t}^{⊥} for i = 1, …, q. The line ℓ_{σ} is known as the special tangent line to θ_{σ}. The point X_{σ} = (0, 0, 0, 1, 0) has the property that X_{σ} ∈ π_{t}^{⊥} ∩ Q(4, q) for i = 1, …, q. In fact, X_{σ} = C_{t} ∩ C_{t′}, for t, t′ ∈ GF(q) and t ≠ t′. The point X_{σ} is known as the special point of the ovoid θ_{σ}. Any element of the set {π_{t}^{⊥} : t ∈ GF(q)} will be called a plane of θ_{σ}.

Now if F_{σ} is the Kantor flock defined by fixed non-square m and field automorphism σ and F′_{σ} is the Kantor flock defined by fixed non-square m′ and field automorphism σ, then by the proof of [31, Theorem (4.1)(i)] and the fundamental theorem of q-clan geometry ([47]), it follows that F_{σ} and F′_{σ} are projectively equivalent. That is, there is an automorphism of Σ that fixes K and maps F_{σ} onto F′_{σ}. This is the case if and only if there is an automorphism of PG(5, q) that fixes Q^{+}(5, q), Q(4, q) and maps the ovoid corresponding to F_{σ} onto the ovoid corresponding to F′_{σ}. Hence the ovoids are isomorphic in Q(4, q). This means that the isomorphism class of K1(σ) ovoids does not depend on the non-square m.

Similarly, if F_{σ} and F_{τ} are two Kantor flocks, defined by the same fixed non-square m and field automorphisms σ and τ, respectively, then by the proof of [31, Theorem (4.1)(i)] and the fundamental theorem of q-clan geometry ([47]), it follows that F_{σ} and F′_{σ} are projectively equivalent if and only if τ = σ or σ^{−1}. By a similar argument to above, the ovoids corresponding to F_{σ} and F_{τ}, respectively, are isomorphic in Q(4, q) if and only if τ = σ or σ^{−1}.

As we shall see in Chapter 3, the Kantor flock F_{σ} is interesting not only because it gives the K1(σ) ovoid but also because the q-clan GQ associated with F_{σ} contains a Q(4, q) subGQ, in which the subtended ovoids are K1(σ) ovoids.

2.2 Symmetry properties of the K1(σ) ovoid

Recall in Section 2.1 that θ_{σ} was an ovoid of Q(4, q) In this case Q(4, q) was the non-singular quadric in PG(4, q), in coordinates (x_{0}, x_{1}, x_{2}, x_{4}, x_{5}), defined by the equation x_{0}x_{1}+x_{2}^{2}+x_{4}x_{5} = 0, lying on the Klein quadric Q^{+}(5, q) in PG(5, q). Since from this section onwards we will not be using the embedding of Q(4, q) in Q^{+}(5, q), we define Q(4, q) to be the non-singular quadric in PG(4, q), in coordinates (x_{0}, x_{1}, x_{2}, x_{3}, x_{4}), defined by the equation x_{0}x_{1} + x_{2}^{2} + x_{3}x_{4} = 0. In this representation we let

θ_{σ} = {(−mt^{σ}, t, z, −z^{2} + mt^{σ+1}, 1) : t, z ∈ GF(q)} ∪ {(0, 0, 0, 1, 0)}
X_{σ} = (0, 0, 0, 1, 0)
ℓ_{σ} = ⟨(0, 0, 0, 1, 0), (0, 0, 1, 0, 0)⟩
π_{t} : x_{0} + mt^{σ−1}x_{1} = 0
: x_{1} − tx_{4} = 0, for t ∈ GF(q).

Note that we are now using π_{t} for the plane π_{t}^{⊥} in Section 2.1, and so we will let C_{t} denote the conic that is the intersection of π_{t} and Q(4, q). As in Section 2.1 any plane π_{t} for t ∈ GF(q) (that contains the special line ℓ_{σ} and meets θ_{σ} in a conic) is called a plane of θ_{σ}.

In this section we will investigate particular elements of the stabiliser of θ_{σ}, in the group of Q(4, q) and what they tell us about the symmetry properties of θ_{σ}. By the construction of the K1(σ) ovoid from the flock F_{σ}, we can see that any element of the stabiliser of θ_{σ}, in the group of Q(4, q), must necessarily fix the point X_{σ} and the line ℓ_{σ}.

For the calculations that follow it will be useful to consider the point P_{t} ∈ π_{t} with the coordinates (−mt^{σ}, t, 0, 0, 1), so π_{t} = ⟨ℓ_{σ}, P_{t}⟩. For t ∈ GF(q) \ {0}, let S_{t} be the automorphism of PG(4, q) defined by

S_{t}(x_{0}, x_{1}, x_{2}, x_{3}, x_{4}) = (t^{σ}x_{0}, tx_{1}, \sqrt{t^{σ+1}}x_{2}, t^{σ+1}x_{3}, x_{4}).

Note that S_{t} fixes Q(4, q), X_{σ} and ℓ_{σ}. Also

S_{t}(P_{t′}) = (−mt^{σ}t′^{σ}, t′, 0, 0, 1)
= (−m(tt′)^{σ}, tt′, 0, 0, 1)
= P_{tt′},

which means that S_{t} : π_{t′} ⟼ π_{tt′} and thus S_{t} fixes θ_{σ}.

For t ∈ GF(q) let T_{t} be the automorphism of PG(4, q) defined by

T_{t}(x_{0}, x_{1}, x_{2}, x_{3}, x_{4}) = (−mt^{σ}x_{4} + x_{0}, tx_{4} + x_{1}, x_{2}, −tx_{0} + mt^{σ}x_{1} + x_{3} + mt^{σ+1}x_{4}, x_{4}).

The automorphism T_{t} fixes Q(4, q), X_{σ} and ℓ_{σ}. Also

T_{t}(P_{t′}) = (−mt^{σ} − mt′^{σ}, t + t′, 0, −, 1)
= (−m(t + t′)^{σ}, t + t′, 0, −, 1)
∈ π_{t+t′},

which means that T_{t} : π_{t′} ⟼ π_{t+t′} and thus T_{t} fixes θ_{σ}.

From the automorphisms T_{t} it is immediately obvious that the group of θ_{σ} is transitive on the planes π_{t′}. However, by using both the S and T automorphisms we can say more than this, since

T_{t}S_{t′−t}(π_{0}, π_{1}) = T_{t}(π_{0}, π_{t′−t})
= (π_{t}, π_{t′})

Thus the group of θ_{σ} is transitive on the ordered pairs of planes (π_{t}, π_{t′}), t, t′ ∈ GF(q) and t ≠ t′. This is equivalent to saying that the Kantor flock F_{σ} is doubly transitive.

Now we consider the orbits of the group of the canonical K1(σ) ovoid θ_{σ} on the points of θ_{σ}. One orbit consists of the special point X_{σ}. Suppose that P and P′ are two points of the set C_{t} \ {X_{σ}}. Let the point Q be the intersection of the line ℓ_{σ} with the line ⟨P, P′⟩. Recall from Section 1.2 the symmetry μ_{Q}, where for R ∈ Q(4, q) we have

μ_{Q}(R) = \{ \begin{matrix}R & if ⟨R, Q⟩ is tangent to Q(4, q) \\ R′ & if ⟨R, Q⟩ meets Q(4, q) at R′ ≠ R, \end{matrix}

is the automorphism of PG(4, q) that fixes Q(4, q). In fact fixes each subspace of Q(4, q) containing Q. Thus μ_{Q} fixes ℓ_{σ}, X_{σ}, π_{t′} for t′ ∈ GF(q) and hence the ovoid θ_{σ}. We also have that μ_{Q}(P) = (P′). Now suppose that P and P′ are two distinct points of θ_{σ}, where ⟨P, ℓ_{σ}⟩ = π_{t} and ⟨P′, ℓ_{σ}⟩ = π_{t′} for some t, t′ ∈ GF(q) with t ≠ t′. Since we have transitivity on the planes of θ_{σ} there exists U, an element of the stabiliser of θ_{σ}, that maps π_{t} to π_{t′}. Hence, U(P) is an element of C_{t′} \ {X_{σ}} and by the above may be mapped onto P′ by an element of the stabiliser of θ_{σ} (possibly 0). We may summarise the above in the following theorem.

Theorem 2.2.1 Let θ_{σ} be the canonical K1(σ) ovoid of Q(4, q). Let X_{σ} be the special point of θ_{σ} and let {π_{t} : t ∈ GF(q)} be the set of planes of θ_{σ} through the special line ℓ_{σ} meeting θ_{σ} in a conic. The group of θ_{σ} is 2-transitive on the planes of θ_{σ} and has two orbits on the points of θ_{σ}. The two orbits on the points of θ_{σ} are {X_{σ}} and θ_{σ} \ {X_{σ}}.

2.3 Intersections of θ_{σ} and elliptic quadrics on Q(4, q), containing X_{σ}

In this section we investigate the intersection of the canonical K1(σ) ovoid θ_{σ} and the elliptic quadric ovoids of Q(4, q), given that the special point of θ_{σ}, X_{σ}, is contained in the intersection. Let Σ be a three dimensional projective subspace of PG(4, q) that intersects the Q(4, q) in a non-singular elliptic quadric O . Suppose that X_{σ} ∈ O, then we calculate the intersection of θ_{σ} and O. We separate this intersection problem into the case where ℓ_{σ} ⊂ Σ and the case where ℓ_{σ} ⊄ Σ.

2.3.1 Case 1: ℓ_{σ} ⊂ Σ

Now since ℓ_{σ} ⊂ Σ it follows that if Σ contains a point P of θ_{σ}, then it contains the whole plane of θ_{σ} containing P. Thus the intersection of θ_{σ} and the elliptic quadric Σ ∩ Q(4, q) is the union of a number of the conics C_{1}, C_{2}, …, C_{q} (possibly 0).

Now let Σ_{t,t′} = ⟨π_{t}, π_{t′}⟩, then Σ_{t,t′} is a three dimensional projective subspace that intersects Q(4, q) in an elliptic quadric. The equation of Σ_{t,t′} is

(t − t′)x_{0} + m(t^{σ} − t′^{σ})x_{1} + (mt′^{σ}t − mt^{σ}t′)x_{4} = 0.

Using the 2-transitivity on the planes of θ_{σ}, we need only consider the elliptic quadric O_{0,1} on Q(4, q) that is the intersection of Q(4, q) and the three-dimensional subspace Σ_{0,1}, which has equation x_{0} + mx_{1} = 0. We now ask how many of the planes π_{t} are contained in Σ_{0,1}, which will in turn tell us the intersection size of the θ_{σ} and O_{0,1}. Now

π_{t} ⊂ Σ_{0,1} ⟺ P_{t} ∈ Σ_{0,1}
⟺ −mt^{σ} + mt = 0
⟺ t^{σ} = t,

and so π_{t} ⊂ Σ_{0,1} if and only if t ∈ F ix(σ), where F ix(σ) is the fixed field of the automorphism σ. Let |F ix(σ)| = k, then |Σ_{0,1} ∩ K1(σ)| = kq + 1 and consequently

|Σ_{t,t′} ∩ θ_{σ}| = kq + 1,

where t, t′ ∈ GF(q) with t ≠ t′. That is, if a three-dimensional subspace of PG(4, q) contains two elements of the set {π_{t} : t ∈ GF(q)} (and so necessarily meets Q(4, q) in an elliptic quadric), then it contains exactly k. Thus there are \binom{q}{2}/\binom{k}{2} = q(q − 1)/k(k − 1) elliptic quadric ovoids of Q(4, q) intersecting the θ_{σ} in k conic planes.

For a given π_{t} there are (q − 1)/(k − 1) three-dimensional subspaces of PG(4, q) containing π_{t} and containing k elements of the set {π_{t} : t ∈ GF(q)}. In total there are (q − 1)/2 threedimensional subspaces of PG(4, q) containing π_{t} and meeting Q(4, q) in an elliptic quadric and so there are (q − 1)(1/2 − 1/(k − 1)) elliptic quadrics ovoids of Q(4, q) meeting θ_{σ} in exactly the conic C_{t} = π_{t} ∩ Q(4, q), that is, with an intersection size of q + 1.

This leaves

\frac{q(q − 1)}{2} − \frac{q(q − 1)}{k(k − 1)} − q(q − 1)(\frac{1}{2} − \frac{1}{k − 1})

= q(q − 1) [\frac{1}{2} − \frac{1}{k(k − 1)} − \frac{1}{2} + \frac{1}{k − 1}]
= \frac{q(q − 1)}{k}

elliptic quadrics ovoids of Q(4, q) which intersect θ_{σ} in exactly X_{σ}.

2.3.2 Case 2: ℓ_{σ} ⊄ Σ but X_{σ} ∈ Σ

Now suppose that X_{σ} ∈ O but that ℓ_{σ} ⊄ Σ. For t ∈ GF(q), let ℓ_{t} = π_{t} ∩ Σ. Now X_{σ} ∈ ℓ_{t} and ℓ_{t} ≠ ℓ_{σ} so it follows that ℓ_{t} is a secant line to the conic C_{t} = π_{t} ∩ Q(4, q). Thus we have that |θ_{σ} ∩ O| = q + 1. There are q^{2}(q − 1)/2 elliptic quadric ovoids of Q(4, q) containing X_{σ} and q(q − 1)/2 of these sections have an ambient three space containing ℓ_{σ}. Thus there are

\frac{q^{2}(q − 1)}{2} − \frac{q(q − 1)}{2} = \frac{q(q − 1)^{2}}{2}

elliptic quadric ovoids of Q(4, q) intersecting θ_{σ} in the above way.

We summarise the intersection sizes of the two cases above in the following theorem Theorem 2.3.1 Let θ_{σ} be the canonical K1(σ) ovoid of Q(4, q) and let X_{σ} be the special point of θ_{σ}. Suppose that Σ is a hyperplane of PG(4, q) that intersects Q(4, q) in a non-singular elliptic quadric O and that X_{σ} ∈ O. If ℓ_{σ} ⊂ Σ, then |θ_{σ} ∩ O| = q + 1 or k(q + 1), where k is the size of the fixed field of σ. If ℓ_{σ} ⊄ Σ, then |θ_{σ} ∩ O| = q + 1.

2.4 More properties of the Σ_{t,t′}

In this section we investigate some more properties of the set of three dimensional subspaces {Σ_{t,t′} : t, t′ ∈ GF(q) with t ≠ t′}, where Σ_{t,t′} has equation (t − t′)x_{0} + m(t^{σ} − t′^{σ})x_{1} + (mt′^{σ}t − mt^{σ}t′)x_{4} = 0.

Now in Section 2.3.1 we saw that

π_{t′′} ∈ Σ_{0,1} ⟺ t′′ ∈ F ix(σ),

and since S_{1/(t′−t)}T_{−t}(Σ_{t,t′}) = Σ_{0,1}, it follows that

π_{t′′} ∈ Σ_{t,t′} ⟺ S_{1/(t′−t)}T_{−t}(π_{t′′}) ∈ Σ_{0,1}.

Thus we have

π_{t′′} ∈ Σ_{t,t′} ⟺ \frac{t′′ − t}{t′ − t} ∈ F ix(σ).

First we determine when the intersection of Σ_{t,t′} and Σ_{t′′,t′′′} is a plane of θ_{σ} (recall π is a plane of θ_{σ} if it is an element of the set {π_{t} : t ∈ GF(q)}). To do this we first consider the conditions under which Σ_{0,1} ∩ Σ_{t,t′} is a plane of θ_{σ}. Now by the above

Σ_{0,1} ∩ Σ_{t,t′} = π_{t′′} ⟺ t′′ ∈ F ix(σ) and \frac{t′′ − t}{t′ − t} ∈ F ix(σ).

Now

t′′, \frac{t′′ − t}{t′ − t} ∈ F ix(σ) ⟺ (\frac{t′′ − t}{t′ − t})^{σ} = \frac{t′′ − t}{t′ − t} and t′′ ∈ F ix(σ),
⟺ (t′′ − t^{σ})(t′ − t) = (t′^{σ} − t^{σ})(t′′ − t) and t′′ ∈ F ix(σ),
⟺ t′′ = \frac{tt′^{σ} − t′t^{σ}}{t′^{σ} − t^{σ} − t′ + t} ∈ F ix(σ).

Thus it follows that Σ_{0,1} ∩ Σ_{t,t′} is a plane of θ_{σ} if and only if

\frac{tt′^{σ} − t′t^{σ}}{t′^{σ} − t^{σ} − t′ + t} ∈ F ix(σ).

The conditions under which the intersection Σ_{t,t′} and Σ_{t′′,t′′′} is a plane of θ_{σ} can be determined by using the transitivity of the group of θ_{σ} on the set {Σ_{s,s′} : s, s′ ∈ GF(q) and s ≠ s′}.

Note that for |σ| = 2, (tt′^{σ} −t′t^{σ})/(t′^{σ} −t^{σ} −t′ +t) is fixed by σ and so Σ_{t,t′} ∩Σ_{t′,t′′} is always a plane of θ_{σ}.

Second we investigate the elliptic quadric ovoids of Q(4, q) whose ambient three space contains k planes of θ_{σ}. Since π_{t} ⊂ Σ_{0,1} if and only if t ∈ F ix(σ) it follows that π_{t} ⊂ T_{t′}(Σ_{0,1}) if and only if t ∈ F ix(σ) + t′. Thus the additive cosets of F ix(σ) partition the planes of θ_{σ} into q/k sets of size k, each set contained in a distinct three-dimensional subspace of PG(4, q).

Now Σ_{0,1} : x_{0} + mx_{1} = 0 meets Q(4, q) in the elliptic quadric with the equation −mx_{1}^{2} + x_{2}^{2} + x_{3}x_{4} = 0. The tangent plane at X_{σ} to this elliptic quadric is the plane π′ : x_{4} = 0. Thus π′ has equations x_{0} + mx_{1} = 0 and x_{4} = 0, and π′ = ⟨(−m, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0)⟩. Since T_{t′}(−m, 1, 0, 0, 0) = (−m, 1, 0, mt^{σ} + mt, 0), which is an element of π′, it follows that T_{t′} fixes π′. Thus for t, t′ ∈ GF(q), with t ≠ t′, the hyperplanes T_{t}(Σ_{0,1}) and T_{t′}(Σ_{0,1}) meet in the plane π′. We generalise this set of hyperplanes to give the following definition.

Definition 2.4.1 Let θ be a K1(σ) ovoid of Q(4, q). Let τ be a set of q/k distinct hyperplanes of PG(4, q), each meeting Q(4, q) in a non-singular elliptic quadric and each containing the special tangent line of θ. Suppose that any two elements of τ intersect in a fixed plane π′, that is tangent to Q(4, q). If each plane of θ is a subspace of an element of τ, then we say that the set τ is a trade of θ. The plane π′ is called the base plane of the trade τ.

Note that it follows from Theorem 2.3.1 that if τ is a trade of the K1(σ) ovoid θ, then each element of τ contains exactly k planes of θ.

Note also that since any two elements of a trade intersect in a fixed plane tangent to Q(4, q), it follows that the set of q/k elliptic quadric ovoids of Q(4, q) formed by the intersection of the trade with Q(4, q) meet pairwise in a fixed point. Suppose that ρ is a set of q/k K1(σ) ovoids meeting pairwise in a fixed point and with a common trade τ. In Section2.6 we shall see that if R is a rosette containing ρ, then we may take R, “trade” ρ for the set of elliptic quadric ovoids corresponding to τ, and form a new rosette.

Our original model for the definition of a trade of a K1(σ) ovoid of Q(4, q) was to take the canonical ovoid θ_{σ} and consider the set τ = {T_{t}(Σ_{0,1}) : t ∈ GF(q)} and the plane π′ with equations x_{4} = 0 and x_{0} + mx_{1} = 0. By transitivity on the planes of θ_{σ} it follows that every hyperplane of the form Σ_{t,t′} is contained in a trade of θ_{σ}. Now consider the point (−m, 1, 0, 0, 0) ∈ π′ \ ℓ_{σ}. We have that (−m, 1, 0, 0, 0) ∈ Σ_{tt′} if and only if t − t′ ∈ F ix(σ) and so (−m, 1, 0, 0, 0) is contained in exactly q/k of the Σ_{t,t′}. Consequently, the plane π′ is contained in exactly q/k of the Σ_{t,t′}, and so is the base plane for exactly one trade of θ_{σ}, namely the trade τ = {T_{t}(Σ_{0,1}) : t ∈ GF(q)}. Now since for any element T_{t}(Σ_{0,1}) of τ, π′ is the unique plane of T_{t}(Σ_{0,1}) that is tangent to Q(4, q) at X_{σ}, it follows that T_{t}(Σ_{0,1}) is contained in exactly one trade of θ_{σ}. Hence it follows that every hyperplane of the form Σ_{t,t′} is contained in exactly one trade of θ_{σ} (the unique trade with base plane the tangent plane to Σ_{t,t′} ∩ Q(4, q) at X_{σ}).

We conclude this section with a summary theorem.

Theorem 2.4.2 Let θ_{σ} = {(−mt^{σ}, t, z, −z^{2} + mt^{σ+1}, 1) : t, z ∈ GF(q)} ∪ {(0, 0, 0, 1, 0)} be the canonical K1(σ) ovoid of Q(4, q), where Q(4, q) is defined by the equation x_{0}x_{1} +x_{2}^{2} +x_{3}x_{4} = 0, with special point X_{σ} = (0, 0, 0, 1, 0). Let {Σ_{t,t′} : t, t′ ∈ GF(q)} be the set of hyperplanes of PG(4, q) where Σ_{t,t′} : (t − t′)x_{0} + m(t^{σ} − t′^{σ})x_{1} + (mt′^{σ}t − mt^{σ}t′)x_{4} = 0. Each hyperplane Σ_{t,t′} meets Q(4, q) in a non-singular elliptic quadric and contains k = |F ix(σ)| planes of θ_{σ}. Further

(i) If t − t′ ∉ F ix(σ) then the plane Σ_{0,1} ∩ Σ_{t,t′} is a plane of θ_{σ} if and only if

\frac{tt′^{σ} − t′t^{σ}}{t′^{σ} − t^{σ} − t′ + t} ∈ F ix(σ).

(ii) Each Σ_{t,t′} is contained in a unique trade of θ_{σ}, with base plane X_{σ}^{⊥} ∩ Σ_{t,t′}. The plane X_{σ}^{⊥} ∩ Σ_{t,t′} is the base plane for exactly one trade of θ_{σ}.

2.5 Characterisations of the K1(σ) ovoid

In this section we give three characterisations of the K1(σ) ovoid. The first two are in terms of the conic structure of the ovoid. The third is a characterisation in terms of the trade structure of the K1(σ) ovoid.

Theorem 2.5.1 ([63]) Let θ be a set of points of Q(4, q) consisting of q conics, C_{1}, …, C_{q} intersecting pairwise in a point X, that is, C_{i} ∩ C_{j} = {X} for i, j = 1, 2, …, q, i ≠ j. Let π_{1}, …, π_{q} be the planes of PG(4, q) such that C_{i} ⊂ π_{i}, for i = 1, …, q. Suppose that ℓ is a fixed line of PG(4, q) such that π_{i} ∩ π_{j} = ℓ for i, j ∈ GF(q), i ≠ j and that no hyperplane of PG(4, q) contains all the π_{i}. Suppose also that for each pair i, j ∈ GF(q), i ≠ j the hyperplane ⟨π_{i}, π_{j}⟩ intersects Q(4, q) in a non-singular elliptic quadric. Then the set θ is isomorphic to the K1(σ) ovoid.

Proof: Suppose that we have such a set of points θ. Embed Q(4, q) in Q^{+}(5, q), which has polarity denoted by ⊥. Then π_{1}^{⊥}, …, π_{q}^{⊥} forms a flock F (of planes) of the cone ℓ^{⊥} ∩ Q^{+}(5, q). Since θ is not contained in a hyperplane of PG(4, q), and so it follows that F is non-linear (that is, the planes of the F do not intersect pairwise in a fixed line). Each plane π_{i}^{⊥} contains the exterior point PG(4, q)^{⊥} and so by [63, Section 1.5.6] F is a Kantor flock F_{σ} for some field automorphism σ and so θ is a K1(σ) ovoid.

□

Note that the automorphism is only determined up to the size of its fixed field, which is equal to the number of elements of the set {π_{1}, π_{2} …, π_{q}} contained in the space ⟨π_{i}, π_{j}⟩.

Theorem 2.5.2 ([63] and [22], see [6, Lemma 15]) Let θ be a non-classical ovoid of Q(4, q) consisting of the q conics C_{1}, …, C_{q}, that meet pairwise in a fixed point X. Then θ is isomorphic to the K1(σ) ovoid, for some σ ∈ aut(GF(q)).

Proof: In general let π_{i} be the plane containing C_{i}. Then for any i, j ∈ GF(q) with i ≠ j, C_{i} ∪ C_{j} is a set of 2q + 1 points of Q(4, q), no two collinear. Suppose that π_{i} ∩ π_{j} = {X} for some i, j ∈ GF(q) with i ≠ j.

If ⊥ represents the polarity of Q(4, q), then π_{j}^{⊥} is a line of PG(4, q), which is either a secant to Q(4, q) or external to Q(4, q). Let H_{j} be the set of three-dimensional subspaces of PG(4, q) containing π_{j} and intersecting Q(4, q) in a non-singular hyperbolic quadric. If π_{j}^{⊥} is a secant line to Q(4, q), then |H_{j}| = (q − 1)/2 and if π_{j}^{⊥} is an external to Q(4, q), then |H_{j}| = (q + 1)/2. Now let Σ and Σ′ be two distinct elements of H_{j}. Since the intersection of π_{i} and π_{j} is a single point, it follows that ⟨π_{i}, π_{j}⟩ is PG(4, q) and that neither Σ nor Σ′ contains π_{i}. Thus ℓ = Σ ∩ π_{i} and ℓ′ = Σ ∩ π_{j} are both lines of π_{i}, and in fact distinct lines of π_{i} since if ℓ = ℓ′, then Σ = Σ′. Now since X is incident with both ℓ and ℓ′ and π_{i} contains a unique line that is tangent to C_{i} at X, it follows that at least one of ℓ and ℓ′ is a secant to C_{i}. Thus, without loss of generality we may suppose that Σ contains a point of C_{i} \ {X}, say Y. Since Y ∉ C_{j} and ⟨π_{j}, Y ⟩ meets Q(4, q) in a non-singular hyperbolic quadric, it follows that Y is collinear with two points of C_{j}, which gives a contradiction. Thus π_{i} ∩ π_{j} is a line tangent to Q(4, q) at X for all i, j ∈ GF(q), with i ≠ j. Further if we let ℓ_{1} be the (unique) tangent to C_{1} at X, it follows by the above that π_{1} ∩ π_{j} = ℓ_{1} for i = 2, …, q. Thus ℓ is contained in all of the planes π_{i} and so is the pairwise intersection of any two of them. Since C_{i} ∪ C_{j} is a set of points with no two collinear it follows that ⟨π_{i}, π_{j}⟩ intersects Q(4, q) in a non-singular elliptic quadric. Thus by Theorem 2.5.1, the ovoid θ is isomorphic to the K1(σ) ovoid for some σ ∈ aut(GF(q)).

□

Theorem 2.5.3 Let Q = Q(4, q), let ℓ be a line tangent to Q and let X = ℓ ∩ Q. Let σ ∈

aut(GF(q)) and k = |F ix(σ)|. Let S = \{τ_{1,1}, τ_{1,2}, …, τ_{1, \frac{q}{k}} , τ_{2,1}, …, τ_{2, \frac{q}{k}} , …, τ_{\frac{q−1}{k−1} , \frac{q}{k}} \} be a set

of q(q − 1)/k(k − 1) hyperplanes of PG(4, q), each element of which contains ℓ and meets Q in a non-singular elliptic quadric. Suppose that the following two conditions hold:

(i) τ_{i,j} ∩ τ_{i,j′} = α_{i} is a tangent plane to Q at X and α_{i} ≠ α_{k} if i ≠ k.

(ii) For each τ_{i,j} there are k sets of the following form:

\{τ_{i,j}, τ_{i_{2},j_{2}}, …, τ_{i_{\frac{q−1}{k−1}} ,j_{\frac{q−1}{k−1}}} \}

such that elements of each set intersect pairwise in the same plane of PG(4, q), which intersects Q(4, q) in a conic and is contained in no other τ_{i′,j′}.

Then the points of the conics in (ii) form an ovoid of Q isomorphic to the K1(σ) ovoid.

Proof: Since each pair of sets of hyperplanes in (ii) intersect in exactly one element, it follows that the span of any two planes in (ii) is a hyperplane intersecting Q in an elliptic quadric. Hence if we can show that there are q conics in (ii), then by Theorem 2.5.2 we have a K1(σ) ovoid. If we let the number of planes be a then counting the incident (plane, τ_{i,j}) pairs for the planes of the conics in (ii) yields

a · \frac{q − 1}{k − 1} = \frac{q}{k} · \frac{q − 1}{k − 1} · k

and so a = q.

□

Note that in Theorem 2.5.3 the trades of the K1(σ) ovoid are the sets {τ_{i,j} : j = 1, 2, …, q/k} for i = 1, 2, …, q/k.

2.6 Rosettes containing K1(σ) ovoids

In this section we will construct a number of types of rosettes of Q(4, q) that contain K1(σ) ovoids. We will consider the two types of elation rosettes of K1(σ) ovoids (where the basepoint is a non-special point and a special point, respectively), non-elation rosettes consisting entirely of K1(σ) ovoids and rosettes containing both K1(σ) ovoid and elliptic quadric ovoids.

We will conclude the section by constructing all of such rosettes of the GQ Q(4, 9), with the aid of a computer search conducted by Gordon Royle ([54]).

2.6.1 Elation rosettes of K1(σ) ovoids

Let θ be an ovoid of Q(4, q), q odd, and P a fixed point of θ. We recall the construction of the q + 1 elation rosettes containing θ and with base point P, from Section 1.4.6. In the dual of Q(4, q), W (q), θ is a spread S of W (q) and P a line ℓ of S. If Q is any point of W (q) (and so of PG(3, q)) incident with ℓ, then W (q) admits the elations of PG(3, q) with axis Q^{⊥} and centre Q (where ⊥ is the polarity of W (q)). The image of S under this group of elations is a rosette of spreads of W (q). Since there are q + 1 points of W (q) incident with ℓ, it follows that there are q + 1 such rosettes. Dualising, these are the q + 1 elation rosettes containing θ and with base point P. Now for the case where θ is an elliptic quadric ovoid there is a unique rosette R of elliptic quadric ovoids containing θ and with basepoint P. If π is the tangent plane to θ at P, then R is the set of elliptic quadric ovoids for which π is also a tangent plane. Thus the q + 1 elation rosettes containing θ and based at P coincide. In Chapter 4 (Lemma 4.1.7) we will see that a similar result holds if θ is a Tits-type ovoid (that is, if the image of θ under an isomorphism from Q(4, q) to W (q) is a Tits ovoid). This, however, is not true in general for elation rosettes of Q(4, q), since if θ is a fixed K1(σ) ovoid and P a non-special point of θ, then elation rosettes containing θ and with base point P are not all identical:

Theorem 2.6.1 Suppose that θ is a K1(σ) ovoid of Q(4, q) and that P is a point of Q(4, q) such that P ∈ θ, but that P is not the special point of θ. Then the q + 1 elation rosettes containing θ and with base point P are distinct. Further, any two such rosettes intersect in exactly θ.

Proof: We will work in W (q) the dual of Q(4, q). Let S be the dual of θ, ℓ be the dual of P, ℓ_{∞} the dual of the special point of θ and let ⊥ represent the polarity (symplectic) of W (q). If Q is a point of ℓ, then let E_{Q} be the group of elations of PG(3, q) with axis Q^{⊥} and centre Q. Let R_{Q} be the elation rosette of spreads {T (S) : T ∈ E_{Q}}, containing S and with base line ℓ.

Now Q^{⊥} is a plane of PG(3, q) not containing ℓ_{∞} and so Q^{⊥} ∩ ℓ_{∞} is a point of PG(3, q), Q_{∞} say. Since Q_{∞} is contained in the axis of the elation group E_{Q}, it is fixed by the elements of E_{Q} and so the special line of each element of R_{Q} contains Q_{∞}. Further, since ℓ_{∞}, the special line of S, is not contained in the plane Q^{⊥}, the axis of the elations in E_{Q}, it follows that each element of R_{Q} has a distinct special line. Thus the set of special lines of spreads of R_{Q} is the set of q lines of W (q), not ℓ, incident with Q_{∞} (note that the lines in this set are all contained in the plane Q_{∞}^{⊥}). If Q and Q′ are two distinct point of ℓ, then clearly the sets of special lines of the corresponding elation rosettes R_{Q} and R_{Q′} are concurrent at different points of W (q), and so R_{Q} and R_{Q′} are distinct.

Now suppose that Q and Q′ are two distinct points of ℓ such that R_{Q} and R_{Q′} contain a common spread S′. Let the intersection of Q′^{⊥} and ℓ_{∞} be the point Q′_{∞}. Now if Q_{∞} = Q′_{∞}, then {Q_{∞}, Q, Q′} is a triangle in W (q), and so Q_{∞} ≠ Q′_{∞}. By the above, since S′ is an element of both R_{Q} and R_{Q′}, it follows that the special line of S′ is incident with both Q_{∞} and Q′_{∞}. This means that the special line of S′ is ⟨Q_{∞}, Q′_{∞}⟩, but this line is ℓ_{∞}, the special line of S. Thus S≅ S′.

□

Now we consider the elation rosettes containing a fixed K1(σ) ovoid θ and with base point P, where P is the special point of θ. We will start with elations in W (q) the dual of Q(4, q) and dualise (via the Klein quadric) to get the elation rosettes of the K1(σ) on its special point. The first result we prove is that all of the elation rosettes coincide.

Theorem 2.6.2 Let θ be a K1(σ) ovoid of Q(4, q) and P the special point of θ. Then the q + 1 elation rosettes of Q(4, q) containing θ and with base point P, coincide.

Proof: Let Q′ = Q^{+}(5, q) be the Klein quadric with equation x_{0}x_{1} + x_{2}x_{3} + x_{4}x_{5} = 0. We will consider the case where Q(4, q) is the intersection of Q′ and the hyperplane of PG(5, q) with equation x_{2} = x_{3}. In this setting we abuse notation by saying the canonical K1(σ) ovoid θ_{σ} has the form

θ_{σ} = {(−mt^{σ}, t, z, z, −z^{2} + mt^{σ+1}, 1) : t, z ∈ GF(q)} ∪ {(0, 0, 0, 0, 1, 0)}

with special point X_{σ} = (0, 0, 0, 0, 1, 0) and special tangent line ℓ_{σ} = ⟨(0, 0, 0, 0, 1, 0), (0, 0, 0, 1, 0, 0)⟩. Let S_{θ_{σ}} be the dual spread of θ_{σ} in W (q) and ℓ_{X_{σ}} the dual line of X_{σ} in W (q). We will first establish the form of the elations of PG(3, q), with centre Q ∈ ℓ_{X_{σ}} and axis Q^{⊥} (where ⊥ denotes the polarity of W (q)), as automorphisms of Q(4, q).

If ℓ = ⟨(x_{0}, x_{1}, x_{2}, x_{3}), (y_{0}, y_{1}, y_{2}, y_{3})⟩ is a line of PG(3, q), then the Klein correspondence λ, mapping lines of PG(3, q) to points of Q′, sends ℓ to the point

(ℓ_{01}, ℓ_{23}, ℓ_{02}, ℓ_{31}, ℓ_{03}, ℓ_{12}) where ℓ_{ij} = x_{i}y_{j} − x_{j}y_{i}

(see [25, Chapter 15]). The Klein correspondence defines a duality from W (q), defined by the equation x_{0}y_{2} − x_{2}y_{0} − x_{3}y_{1} + x_{1}y_{3} = 0, to Q(4, q) ⊂ Q′. We shall denote the bilinear form associated with W (q) by β((x_{0}, x_{1}, x_{2}, x_{3}), (y_{0}, y_{1}, y_{2}, y_{3})) = x_{0}y_{2} − x_{2}y_{0} − x_{3}y_{1} + x_{1}y_{3}.

Now the special line of the spread S_{θ_{σ}} is

ℓ_{X_{σ}} = λ^{−1}(X_{σ})
= λ^{−1}(0, 0, 0, 0, 1, 0)
= ⟨(1, 0, 0, 0), (0, 0, 0, 1)⟩.

If U is a point of W (q), then the group of elations with centre U and axis U^{⊥} is {t_{a,U} : V ↦ V + aβ(U, V)U|a ∈ GF(q)}. The automorphism of Q(4, q) equivalent to t_{a,U} is the restriction of the automorphism λt_{a,U}λ^{−1} of Q to Q(4, q). If we let U_{s} = (s, 0, 0, 1) and U_{∞} = (1, 0, 0, 0), then ℓ_{X_{σ}} = {U_{s} : s ∈ GF(q)} ∪ {U_{∞}}.

Now t_{a,U_{s}}(V) = V + a(−sv_{2} + v_{1})U_{s} = (v_{0} − as^{2}v_{2} + asv_{1}, v_{1}, v_{2}, v_{3} − asv_{2} + av_{1}) which acts on the Plucker coordinates by

ℓ_{01} ↦ (x_{0} − as^{2}x_{2} + asx_{1})y_{1} − x_{1}(y_{0} − as^{2}y_{2} + asy_{1})
= ℓ_{01} + as^{2}ℓ_{12},
ℓ_{02} ↦ (x_{0} − as^{2}x_{2} + asx_{1})y_{2} − x_{2}(y_{0} − as^{2}y_{2} + asy_{1})
= ℓ_{02} + asℓ_{12},
ℓ_{03} ↦ (x_{0} − as^{2}x_{2} + asx_{1})(y_{3} − asy_{2} + ay_{1}) − (x_{3} − asx_{2} + ax_{1})(y_{0} − as^{2}y_{2} + asy_{1}),
= ℓ_{03} − asℓ_{02} + aℓ_{01} − as^{2}ℓ_{23} − asℓ_{31}
ℓ_{12} ↦ x_{1}y_{2} − x_{2}y_{1}
= ℓ_{12},
ℓ_{31} ↦ (x_{3} − asx_{2} + ax_{1})y_{1} − x_{1}(y_{3} − asy_{2} + ay_{1})
= ℓ_{31} + asℓ_{12},
ℓ_{23} ↦ x_{2}(y_{3} − asy_{2} + ay_{1}) − (x_{3} − asx_{2} + ax_{1})y_{2}
= ℓ_{23} − aℓ_{12}.

Thus

λt_{a,U_{s}}λ^{−1}(x_{0}, x_{1}, x_{2}, x_{3}, x_{4}, x_{5})
= (x_{0} + as^{2}x_{5}, x_{1} − ax_{5}, x_{2} + asx_{5}, x_{3} + asx_{5}, x_{4} + ax_{0} − as^{2}x_{1} − asx_{2} − asx_{3}, x_{5}),

When we restrict this to to Q(4, q) we have the automorphism

E_{a,s}(x_{0}, x_{1}, x_{2}, x_{2}, x_{4}, x_{5})
= (x_{0} + as^{2}x_{5}, x_{1} − ax_{5}, x_{2} + asx_{5}, x_{2} + asx_{5}, x_{4} + ax_{0} − as^{2}x_{1} − 2asx_{2}, x_{5}).

We now repeat the above for the elations t_{a,U_{∞}}.

Now t_{a,U_{∞}}(V) = V +a(−v_{2})U_{∞} = (v_{0}−av_{2}, v_{1}, v_{2}, v_{3}) and so acts on the Plucker coordinates by

(ℓ_{01}, ℓ_{02}, ℓ_{03}, ℓ_{12}, ℓ_{31}, ℓ_{23}) ↦ (ℓ_{01} + aℓ_{12}, ℓ_{02}, ℓ_{03} − aℓ_{23}, ℓ_{12}, ℓ_{31}, ℓ_{23}),

that is

λt_{a,U_{∞}}λ^{−1}(x_{0}, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}) = (x_{0} + ax_{5}, x_{1}, x_{2}, x_{3}, x_{4} − ax_{1}, x_{5}).

When restricted to to Q(4, q) we have the automorphism

E_{a,∞}(x_{0}, x_{1}, x_{2}, x_{2}, x_{4}, x_{5}) = (x_{0} + ax_{5}, x_{1}, x_{2}, x_{2}, x_{4} − ax_{1}, x_{5}).

The q + 1 elation rosettes of Q(4, q) containing θ_{σ} and with special point X_{σ} = (0, 0, 0, 0, 1, 0) are

R_{s} = {E_{a,s}(θ_{σ}) : a ∈ GF(q)} for s ∈ GF(q)
R_{∞} = {E_{a,∞}(θ_{σ}) : a ∈ GF(q)}.

We now show that R_{∞} = R_{s} for any s ∈ GF(q). Consider the ovoid E_{a,s}(θ_{σ}) ∈ R_{s}, then

E_{a,s}(θ_{σ}) = {(−mt^{σ} + as^{2}, t − a, z + as, z + as, −z^{2} + mt^{σ+1} − mat^{σ} − as^{2}t − 2asz, 1) :
t, z ∈ GF(q)} ∪ {(0, 0, 0, 0, 1, 0)}

Now let a′ = as^{2} − ma^{σ}, then consider

E_{a′,∞}(θ_{σ}) = {(−mt^{σ} + a′, t, z, z, −z^{2} + mt^{σ+1} − a′t, 1) : z, t ∈ GF(q)} ∪ {(0, 0, 0, 0, 1, 0)}

= {(−mt^{σ} + ma^{σ} + a′, t − a, z, z, −z^{2} + ma^{σ+1} − mat^{σ} − ma^{σ}t + mt^{σ+1} + aa′ − a′t,
1) : z, t ∈ GF(q)} ∪ {(0, 0, 0, 0, 1, 0)} by replacing the parameter t by t − a,
= {(−mt^{σ} + as^{2}, t − a, z + as, z + as, −z^{2} − 2asz − a^{2}s^{2} + ma^{σ+1} − mat^{σ} − ma^{σ}t +
mt^{σ+1} − aa′ − a′t, 1) : z, t ∈ GF(q)} ∪ {(0, 0, 0, 0, 1, 0)} by replacing z by z + as,
= {(−mt^{σ} + as^{2}, t − a, z + as, z + as, −z^{2} − 2asz − mat^{σ} + mt^{σ+1} − as^{2}t, 1)
: z, t ∈ GF(q)} ∪ {(0, 0, 0, 0, 1, 0)} by substituting a′ = as^{2} − ma^{σ},
= E_{a,s}(θ_{σ}).

Thus each ovoid of R_{s} is contained in R_{∞}, and so R_{s} = R_{∞} for all s ∈ GF(q). From this it follows that the q + 1 elation rosettes coincide.

□

We now consider Q(4, q) to be the quadric in PG(4, q) defined by the equation x_{0}x_{1} + x_{2}^{2} + x_{3}x_{4} = 0 and the canonical K1(σ) ovoid θ_{σ} to be

θ_{σ} = {(−mt^{σ}, t, z, −z^{2} + mt^{σ+1}, 1) : t, z ∈ GF(q)} ∪ {(0, 0, 0, 1, 0)}

with special point X_{σ} = (0, 0, 0, 1, 0) and special tangent line ℓ_{σ} = ⟨(0, 0, 0, 1, 0), (0, 0, 1, 0, 0)⟩. In this setting the elations derived in the proof of Theorem 2.6.2 have the form

E_{a,s}(x_{0}, x_{1}, x_{2}, x_{3}, x_{4}) = (x_{0} + as^{2}x_{4}, x_{1} − ax_{4}, x_{2} + asx_{4}, x_{3} + ax_{0} − as^{2}x_{1} − 2asx_{2}, x_{4}),
E_{a,∞}(x_{0}, x_{1}, x_{2}, x_{3}, x_{4}) = (x_{0} + ax_{4}, x_{1}, x_{2}, x_{3} − ax_{1}, x_{4})

for a, s ∈ GF(q).

Let R_{σ} be the elation rosette of Q(4, q) containing θ_{σ} and with base point X_{σ}, the special point of θ_{σ}. In the following we investigate the structure of R_{σ} by calculating the action of the elation group {E_{a,0} : a ∈ GF(q)} on R_{σ}.

Theorem 2.6.3 Let θ_{1} be a K1(σ) ovoid of Q(4, q) with special point P. Let R = {θ_{1}, θ_{2}, …, θ_{q}} be the (unique) elation rosette of Q(4, q) containing θ_{1} and based at P. If Ω_{i} is the set {π′ : π′ is a base plane of a trade of θ_{i}}, then Ω_{i} = Ω_{j}, for all 1 ≤ i, j ≤ q with i ≠ j. Further, if τ_{i} is any trade of θ_{i} and τ_{j} is any trade of θ_{j}, then τ_{i} and τ_{j} are either equal or disjoint.

Proof: We prove the result for the case where Q(4, q) is defined by the equation x_{0}x_{1} + x_{2}^{2} + x_{3}x_{4} = 0 and θ_{1} = θ_{σ}, the canonical K1(σ) ovoid.

The rosette R is generated as the set of images of θ_{σ} under the group of elations {E_{a,0} : a ∈ GF(q)}, that is R = {E_{a,0}(θ_{σ}) : a ∈ GF(q)}. Let B_{a} be the set of base planes of trades of the ovoid E_{a,0}(θ_{σ}). We first show that B_{a} = B_{a′} for all a, a′ ∈ GF(q) with a ≠ a′.

Recall from Theorem 2.4.2(ii) that the three-dimensional subspace of PG(4, q) Σ_{t,t′} (which contains k = |F ix(σ)| planes of θ_{σ}) is contained in a unique trade of θ_{σ} and the base plane of this trade is X_{σ}^{⊥} ∩Σ_{t,t′}. Now since the group {S_{η} : η ∈ GF(q)^{∗}} (where S_{η} is the automorphism of Q(4, q) fixing θ_{σ}, introduced in Section 2.2), also fixes X_{σ}^{⊥} and is transitive on the set of Σ_{t,t′}, it follows that the set of base planes of trades of θ_{σ} is

B_{0} = \{\overline{π}_{η} = S_{η}(Σ_{0,1} ∩ X_{σ}^{⊥}) = Σ_{0,η} ∩ X_{σ}^{⊥} : η ∈ GF(q)^{∗}\}.

Note that the size of this set is (q − 1)/(k − 1) and that the plane \overline{π}_{η} has equations

\overline{π}_{η} : ηx_{0} + mη^{σ}x_{1} = 0
x_{4} = 0.

Now let A_{η} ∈ \overline{π}_{η} \ℓ_{σ} be the point with coordinates (−mη^{σ−1}, 1, 0, 0, 0), and so \overline{π}_{η} = ⟨ℓ_{σ}, A_{η}⟩. The point E_{a,0}(A_{η}) = (−mη^{σ−1}, 1, 0, −maη^{σ−1}, 0) ∈ \overline{π}_{η} and E_{a,0}(ℓ_{σ}) = ℓ_{σ} from which it follows that E_{a,0}(\overline{π}_{η}) = \overline{π}_{η}. Since B_{a} is the set of base planes of the ovoid E_{a,0}(θ_{σ}) ∈ R, it follows that

B_{a} = E_{a,0}(B_{0})
= \{E_{a,0}(\overline{π}_{η}) : η ∈ GF(q)^{∗}\}
= \{\overline{π}_{η} : η ∈ GF(q)^{∗}\}
= B_{0},

and so B_{a} = B_{a′} for all a, a′ ∈ GF(q) with a ≠ a′.

Now let τ be a trade of E_{a,0}(θ_{σ}), with base plane \overline{π} and τ′ a trade of E_{a′,0}(θ_{σ}), with base plane \overline{π}′. We show that τ and τ′ are either equal or disjoint. If Σ is an element of τ and τ′, then it contains both \overline{π} and \overline{π}′. Now \overline{π} and \overline{π}′ are both planes tangent to Q(4, q) at the point X_{σ}, but Σ contains a unique plane tangent to Σ ∩ Q(4, q) at X_{σ}, thus \overline{π} = \overline{π}′. So if τ and τ′ have different base planes, then they are disjoint. We now consider the case where τ and τ′ share the same base plane, that is \overline{π} = \overline{π}′. Since B_{a} = B_{0} for all a ∈ GF(q), it follows that \overline{π} = \overline{π}′ ∈ B_{0}, that is \overline{π} = \overline{π}_{η} ∈ B_{0} for some η ∈ GF(q)^{∗}. Now by the above E_{a,0}(\overline{π}_{η}) = E_{a′,0}(\overline{π}_{η}) = \overline{π}_{η}, and so if τ_{η} is the unique trade of θ_{σ} with base plane \overline{π}_{η}, then τ = E_{a,0}(τ_{η}) and τ′ = E_{a′,0}(τ_{η}). Recall from Section 2.2 that the trade of θ_{σ} with base plane \overline{π}_{1} (that is, the trade containing Σ_{0,1}) is τ_{1} = {T_{α}(Σ_{0,1}) : α ∈ GF(q)} where T_{α} is the automorphism of Q(4, q) fixing θ_{σ} introduced in Section 2.2. Now since \overline{π}_{η} = S_{η}(\overline{π}_{1}) it follows that

τ_{η} = S_{η}(τ_{1})
= {S_{η}T_{α}(Σ_{0,1}) : α ∈ GF(q)}
= {Σ_{ηα,η(α+1)} : α ∈ GF(q)} (see Section 2.2)
= {[η, mη^{σ}, 0, 0, mη^{σ+1}(α − α^{σ})] : α ∈ GF(q)}

Now E_{a,0}([η, mη^{σ}, 0, 0, mη^{σ+1}(α − α^{σ})]) = [η, mη^{σ}, 0, 0, mη^{σ}(ηα − ηα^{σ} + a)], and so

τ = E_{a,0}(τ_{η}) = {[η, mη^{σ}, 0, 0, mη^{σ}(ηα − ηα^{σ} + a)] : α ∈ GF(q)}.

Similarly

τ′ = E_{a,0}(τ_{η}) = {[η, mη^{σ}, 0, 0, mη^{σ}(ηα − ηα^{σ} + a′)] : α ∈ GF(q)}.

Consequently τ = τ′ if a − a′ = η(β − β^{σ}) for some β ∈ GF(q) and τ and τ′ are disjoint if a − a′ ≠ η(β − β^{σ}) for all β ∈ GF(q).

□

If π′ is a base plane of a trade of an ovoid of R, then π′ is called a base plane of R and any trade of an ovoid of R is called a trade of R.

Corollary 2.6.4 Let θ_{1} be a K1(σ) ovoid of Q(4, q) with special point P. Let R = {θ_{1}, θ_{2}, …, θ_{q}} be the (unique) elation rosette of Q(4, q) containing θ_{1} and based at P. If π′ is a base plane of R, then the set of trades of R, with base plane π′ partitions the set of hyperplanes that contain π′ and intersect Q(4, q) in a non-singular elliptic quadric.

If π′ is a base plane of R, then the set T of trades of R with base plane π′ as base plane is called a tower of R and π′ is called the base of the tower T .

2.6.2 Rosettes, trades and towers

In this section we will use the ideas of a trade of a K1(σ) ovoid and a tower of an elation rosette of K1(σ) ovoids to construct a variety of rosettes of Q(4, q), each of which contains K1(σ) ovoids.

The general idea of the constructions is as follows. Let S be a GQ of order s. Let R = {θ_{1}, θ_{2}, …, θ_{s}} and R′ = {θ′_{1}, θ′_{2}, …, θ′_{s}} be two rosettes of ovoids of S, that have the same base point P. Suppose that as sets of points

θ′_{1} ∪ θ′_{2} … ∪ θ′_{r} = θ_{1} ∪ θ_{2} ∪ … ∪ θ_{r},

then both {θ′_{1}, θ′_{2}, …, θ′_{r}, θ_{r+1}, θ_{r+2}, …, θ_{s}} and {θ_{1}, θ_{2}, …, θ_{r}, θ′_{r+1}, θ′_{r+2}, …, θ′_{s}} are rosettes of S with base point P.

Now we consider the particular case where S = Q(4, q) and R is an elation rosette of K1(σ) ovoids of Q(4, q) with base point P, where P is the special point of the rosettes of R. Let T be a tower of R with base π and let R′ = {θ′_{1}, θ′_{2}, …, θ′_{q}} be the rosette of elliptic quadric ovoids with tangent plane π. Let τ = {θ_{1}, θ_{2}, …, θ_{q/k}} ∈ T be a trade of R such that, without loss of generality, the elements of τ intersect the Q(4, q) in the elliptic quadric ovoids θ′_{1}, θ′_{2}, …, θ′_{k} (where k = |F ix(σ)|) and the k ovoids of R that have τ as a trade are θ_{1}, θ_{2}, …, θ_{k}. Then the sets of ovoids {θ′_{1}, θ′_{2}, …, θ′_{k}, θ_{k+1}, θ_{k+2}, …, θ_{q}} and {θ_{1}, θ_{2}, …, θ_{k}, θ′_{k+1}, …, θ′_{q}}, are both a rosette of ovoids of Q(4, q) with base point P, containing both K1(σ) ovoids and elliptic quadric ovoids.

We can generalise these construction in the following way. Let the elements of the tower T of R be τ_{1}, τ_{2}, …, τ_{q/k}. Let the set of k ovoids of R that have τ_{i} as a trade be denoted \overline{τ_{i}}. Then the set of ovoids ρ_{1} ∪ ρ_{2} ∪ … ∪ ρ_{q/k}, where ρ_{i} = τ_{i} or \overline{τ_{i}} for each i, is a rosette of ovoids of Q(4, q) with base point P. Note that this construction includes both R and R′. If π and π′ are two distinct tangent planes to the Q(4, q) at P and θ and θ′ are two elliptic quadric ovoids with π and π′ as tangent planes, respectively, then θ and θ′ intersect in a conic. Thus, in the above construction all the trades τ_{i} must come from the same tower of R.

A natural extension of the above construction provides a construction of rosettes of Q(4, q) containing all K1(σ) ovoids, but that are not elation rosettes.

Theorem 2.6.5 Let R be an elation rosette of K1(σ) ovoids of Q(4, q) and let T = {τ_{1}, τ_{2}, …, τ_{q/k}} be a tower of R. Let ρ_{i} be a set of q/k K1(σ) ovoids contained in a common elation rosette (not necessarily R), and sharing the trade τ_{i}. Then the set of ovoids ρ_{1} ∪ ρ_{2} ∪ … ∪ ρ_{q/k} is a rosette of Q(4, q).

We are interested in determining whether rosettes constructed as in Theorem 2.6.5 exist. We do this by looking for K1(σ) ovoids, not contained in R that share a trade with R. This is the direction we pursue in the next two sections. To complete this section we summarise all of the above constructions of rosettes.

Theorem 2.6.6 Let R be an elation rosette of K1(σ) ovoids of Q(4, q) and let T = {τ_{1}, τ_{2}, …, τ_{q/k}} be a tower of R. Let ρ_{i} be either

1. a set of q/k K1(σ) ovoids contained in a common elation rosette (not necessarily R), and sharing the trade τ_{i}; or

2. the set of q/k elliptic quadric ovoids of Q(4, q) whose elements are the intersection of the elements of τ_{i} with Q(4, q).

Then the set of ovoids ρ_{1} ∪ ρ_{2} ∪ … ∪ ρ_{q/k} is a rosette of Q(4, q).

K1(σ) ovoids sharing a trade and with the same special tangent line

Our aim in this section is to find examples of K1(σ) ovoids that have the same special tangent line as the canonical K1(σ) ovoid θ_{σ} and share a trade with θ_{σ}. From Section 2.6.1 we have the example of the ovoids of the elation rosette R_{σ} containing θ_{σ} and with base point X_{σ}. Here we calculate whether there are any examples of K1(σ) ovoids sharing a trade and the special tangent line with θ_{σ}, that are not in R_{σ}.

We first consider collineations of PG(4, q) that fix Q(4, q) (with equation x_{0}x_{1} +x_{2}^{2} +x_{3}x_{4} = 0) and ℓ_{σ}, the special tangent line of θ_{σ}, which map θ_{σ} onto K1(σ) ovoids with special tangent line ℓ_{σ}. Now suppose that θ is a K1(σ) ovoid with special tangent line ℓ_{σ} and such that θ and θ_{σ} share the trade τ. Since by Theorem 2.2.1 the group of θ_{σ} is 2-transitive on the planes of θ_{σ}, we may assume that τ is the trade of θ_{σ} containing Σ_{0,1}. Also, the group of θ is 2-transitive on the planes of θ, so we consider collineations of PG(4, q) that fix Q(4, q), Σ_{0,1} and ℓ_{σ}.

Let Υ = T α be the collineation of PG(4, q) consisting of the automorphic collineation α (we will use α to represent both the automorphic collineation of PG(4, q) and the automorphism of GF(q)) and the homography T, which has matrix [t_{ij}] where 0 ≤ i, j ≤ 4. From this point we will associate the homography T with the matrix [t_{ij}]. Now we require that Υ fixes Σ_{0,1}, ℓ_{σ} and X_{σ}, and hence fixes Σ_{0,1}^{⊥} = (m, 1, 0, 0, 0), X_{σ} = (0, 0, 0, 1, 0) and ℓ_{σ}; so

\begin{aligned}Υ(0, 0, 0, 1, 0) & = & (0, 0, 0, 1, 0), \\ Υ(m, 1, 0, 0, 0) & = & (m, 1, 0, 0, 0) and \\ Υ(0, 0, 1, 0, 0) & = & (0, 0, 1, a, 0) \end{aligned}

for some a ∈ GF(q). Now we also require that Υ fixes ℓ_{σ}^{⊥} = [0, 0, 1, 0, 0] ∩ [0, 0, 0, 0, 1]. The intersection of ℓ_{σ} with Q(4, q) is the line pair ℓ_{1} = ⟨(0, 0, 0, 1, 0), (1, 0, 0, 0, 0)⟩ and ℓ_{2} = ⟨(0, 0, 0, 1, 0), (0, 1, 0, 0, 0)⟩. We will first consider the case where Υ(ℓ_{1}) = ℓ_{1} and Υ(ℓ_{2}) = ℓ_{2}. Thus

\begin{aligned}Υ(1, 0, 0, 0, 0) & = & (1, 0, 0, b, 0) and \\ Υ(0, 1, 0, 0, 0) & = & (0, 1, 0, c, 0) \end{aligned}

for some b, c ∈ GF(q). Thus T has the form:

T = ( \begin{matrix}t_{00} & 0 & 0 & 0 & t_{04} \\ 0 & t_{00}m^{α−1} & 0 & 0 & t_{14} \\ 0 & 0 & t_{22} & 0 & t_{24} \\ bt_{00} & ct_{00}m^{α−1} & at_{22} & t_{33} & t_{34} \\ 0 & 0 & 0 & 0 & t_{44} \end{matrix} ) , such that mb + c = 0.

We also require that Υ fixes Q(4, q). Since (x_{0}x_{1} +x_{2}^{2} +x_{3}x_{4})^{α} = x_{0}^{α}x_{1}^{α} +(x_{2}^{α})^{2} +x_{3}^{α}x_{4}^{α}, it follows that the automorphic collineation α fixes Q(4, q). Thus to find the conditions under which Υ fixes Q(4, q) we find the conditions under which T fixes Q(4, q). These conditions are:

\begin{aligned}m^{α−1}t_{00}^{2} = t_{22}^{2} & = & t_{33}t_{44}, \\ t_{00}t_{14} + bt_{00}t_{44} & = & 0, \\ t_{00}t_{04}m^{α−1} + t_{00}t_{44}m^{α−1}c & = & 0, \\ 2t_{22}t_{24} + at_{22}t_{44} & = & 0, \\ t_{04}t_{14} + t_{24}^{2} + t_{34}t_{44} & = & 0. \end{aligned}

We normalise by letting t_{44} = 1 and let t_{00} = s ∈ GF(q) \ {0}. Thus, T has the form

T = ( \begin{matrix}s & 0 & 0 & 0 & −c \\ 0 & sm^{α−1} & 0 & 0 & −b \\ 0 & 0 & ±γs & 0 & −a/2 \\ bs & csm^{α−1} & ±aγs & m^{α−1}s^{2} & −bc − a^{2}/4 \\ 0 & 0 & 0 & 0 & 1 \end{matrix} )  (2.6.1)

for b, c ∈ GF(q) such that mb + c = 0, s ∈ GF(q) \ {0} and γ = m^{(α−1)/2}.

If G_{σ} is the group fixing Q(4, q), Σ_{0,1} and ℓ_{σ}, then the subgroup of G_{σ} consisting of the elements of G_{σ} that fix ℓ_{1} (and hence also fix ℓ_{2}), is a subgroup of index 2 in G_{σ}. We will denote this subgroup of G_{σ} by G′_{σ}. The group G′_{σ} consists of collineations of the form T α where T is of the form in 2.6.1. If we define

W (x_{0}, x_{1}, x_{2}, x_{3}, x_{4}) = (mx_{1}, x_{0}/m, x_{2}, x_{3}, x_{4}),

then W is a collineation of PG(4, q) fixing Q(4, q), Σ_{0,1} and ℓ_{σ} and swapping ℓ_{1} and ℓ_{2}. Thus we have that

G_{σ} = G′_{σ} ∪ W G′_{σ}.

We are now interested in elements of G_{σ} that fix the trade τ of θ_{σ} that contains Σ_{0,1}, since the image of θ_{σ} under such a collineation is a K1(σ) ovoid for which τ is also a trade. The group of such elements is the stabiliser of τ in G_{σ} which we will denote by (G_{σ})_{τ}. Using the description of τ in the proof of Theorem 2.6.3 we may easily show that τ consists of the hyperplanes of PG(4, q) with coordinates [1, m, 0, 0, u] where u = m(t^{σ} − t) for some t ∈ GF(q). First we consider the action of Υ ∈ G′_{σ} on the polar point of the hyperplane [1, m, 0, 0, u] which is given by

Υ(m, 1, 0, u, 0) = T (m^{α}, 1, 0, u^{α}, 0)
= (sm^{α}, sm^{α−1}, 0, m^{α−1}s^{2}u^{α}, 0)
≡ (m, 1, 0, su^{α}, 0)

Thus the collineation Υ = T α ∈ G′_{σ} (where T has the form of 2.6.1) fixes the trade τ if and only if for each t ∈ GF(q) there is a t′ ∈ GF(q) such that s(m(t^{σ} − t))^{α} = m(t′^{σ} − t′). That is, for each t ∈ GF(q) there is a t′ ∈ GF(q) such that

sm^{α}(t^{ασ} − t^{α}) = m(t′^{σ} − t′)  (2.6.2)

This is equivalent to the single condition

sm^{α−1}{t^{σ} − t : t ∈ GF(q)} = {t^{σ} − t : t ∈ GF(q)}.  (2.6.3)

Solving this problem in general is messy so we will instead consider a subset of the solutions, which will correspond to a subgroup of the stabiliser of τ in G′_{σ}.

We consider the elements Υ = T α of G′_{σ} (so T is of the form of 2.6.1) such that sm^{α−1} ∈ F ix(σ) from which it follows that Υ satisfies 2.6.3. Let two such collineations be Υ = T α and Υ′ = T′α′. Now (T α)(T′α′) = (T T′^{α})(αα′) (where T′^{α} is the matrix obtained by operating on each element of T′ with α) and

(ss′^{α})m^{αα′−1} = (sm^{α−1})(s′m^{α′−1})^{α}.

Now F ix(σ) is the unique subfield of GF(q) of order k, so F ix(σ)^{α} = F ix(σ) which implies that (s′m^{α′−1})^{α} is an element of F ix(σ). Thus (sm^{α−1})(s′m^{α′−1})^{α} ∈ F ix(σ) and

H′_{σ} = {Υ = T α : Υ ∈ G′_{σ} and sm^{α−1} ∈ F ix(σ)}

is a subgroup of G′_{σ}. Now for any automorphism α of GF(q) there are k − 1 possibilities for s such that sm^{α−1} ∈ GF(q), which gives a total of h(k − 1) pairs (s, α). Further in the matrix T b is any element of GF(q), c is dependent on b and a may be any element of GF(q). Hence the order of G′_{σ} is 2(k − 1)hq^{2}.

Note that the elements of the set {t^{σ} − t : t ∈ GF(q)} are in a one-to-one correspondence to the additive cosets of F ix(σ) . Thus the set has size q/k and the maximum number of solutions, in x, to x{t^{σ} − t : t ∈ GF(q)} = {t^{σ} − t : t ∈ GF(q)} is q/k. But if x ∈ F ix(σ), then x is clearly a solution so there are at least k solutions. In the case where k = q/k the maximum number of solutions in x is equal to the number of solutions where x ∈ F ix(σ). Thus every Υ ∈ G′_{σ} that satisfies 2.6.3 is an element of H′_{σ}, that is, H′_{σ} is the full stabiliser of τ in G′_{σ}. Note that k = q/k if and only if σ^{2} is the identity automorphism of GF(q).

Now W (m, 1, 0, 0, 0) = (m, 1, 0, 0, 0) and so the trade τ = {[1, m, 0, 0, u] : u = m(t^{σ} − t) for some t ∈ GF(q)} is fixed by W. Since H′_{σ} is a subgroup of G′_{σ} each element of H′_{σ} fixes ℓ_{1} and it follows that the set W H′_{σ} is a set of elements of G_{σ} that swap ℓ_{1} and ℓ_{2}. We now show that H_{σ} = H′_{σ} ∪ W H′_{σ} is a subgroup of G_{σ} by displaying that multiplication amongst elements of H_{σ} is closed.

Let h and h′ be two elements of H_{σ}. If h, h′ ∈ H′_{σ}, then hh′ ∈ H′_{σ} ⊂ H_{σ}. If h ∈ W H′_{σ}, say h = W T α and h ∈ H′_{σ}, then hh′ = W (T αh′) ∈ W H′_{σ} ⊂ H_{σ}. If h, h′ ∈ W H′_{σ}, say h = W T α and h′ = W T′α′ where

T = ( \begin{matrix}s & 0 & 0 & 0 & −c \\ 0 & sm^{α−1} & 0 & 0 & −b \\ 0 & 0 & ±γs & 0 & −a/2 \\ bs & csm^{α−1} & ±aγs & m^{α−1}s^{2} & −bc − a^{2}/4 \\ 0 & 0 & 0 & 0 & 1 \end{matrix} ) where mb + c = 0 and γ = m^{(α−1)/2},

then

W T W^{α} = ( \begin{matrix}s & 0 & 0 & 0 & −mb \\ 0 & sm^{α−1} & 0 & 0 & −c/m \\ 0 & 0 & ±γs & 0 & −a/2 \\ cs/m & bsm^{α} & ±aγs & m^{α−1}s^{2} & −bc − a^{2}/4 \\ 0 & 0 & 0 & 0 & 1 \end{matrix} )

and so (W T W^{α})α is an element of H′_{σ}. Thus hh′ = (W T α)(W T′α′) = (W T W^{α})(αT′α′) = (W T W^{α})α(T′α′) ∈ H′_{σ} ⊂ H_{σ}. Finally, if h ∈ H′_{σ}, h = T α say and h′ ∈ W H′_{σ}, h′ = W T′α′ say, then

hh′ = (T α)(W T′α′)
= (W^{2}T α)(W T′α′) since W^{2} is the identity
= W (W T α)(W T′α′)
∈ W H′_{σ} ⊂ H_{σ}.

We have shown that multiplication of elements of H_{σ} is closed and so H_{σ} is a subgroup of (G_{σ})_{τ}. The order of H_{σ} is 4h(k − 1)q^{2} where q = p^{h}.

Note that in the case where σ^{2} is the identity H_{σ} is the full stabiliser of τ in G_{σ}, that is, H_{σ} = (G_{σ})τ.

Now H_{σ} is a group of collineations of PG(4, q) fixing Q(4, q), Σ_{0,1}, ℓ_{σ} and the trade τ of θ_{σ} that contains Σ_{0,1}. Thus the image of θ_{σ} under any element of H_{σ} is a K1(σ) ovoid of Q(4, q) that share the trade τ with θ_{σ}. Now if θ_{1} and θ_{2} are K1(σ) ovoids with special tangent line ℓ_{σ} and with the common trade τ, then there exists an element of (G_{σ})_{τ} mapping θ_{1} to θ_{2} (this follows from the 2-transitivity of the group of a K1(σ) ovoid on its planes). Thus we may use the orbit-stabiliser theorem on (G_{σ})_{τ} to determine the number of distinct K1(σ) ovoids with special tangent line ℓ_{σ} and with τ as a trade.

In the case where σ^{2} is the identity, since H_{σ} = (G_{σ})_{τ} it follows that

# distinct ovoids with special tangent line ℓ_{σ} and trade τ

= \frac{|H_{σ}|}{|Stabiliser of θ_{σ} in H_{σ}|}  (2.6.4)

In the case where σ^{2} is not the identity, since H_{σ} is a subgroup of (G_{σ})_{τ}, it follows that

# distinct ovoids with special tangent line ℓ_{σ} and trade τ

= \frac{|(G_{σ})_{τ} |}{|Stabiliser of θ_{σ} in (G_{σ})_{τ} |}

≥ \frac{|H_{σ}|}{|Stabiliser of θ_{σ} in (G_{σ})_{τ} |}  (2.6.5)

Now if Υ ∈ G_{σ} fixes θ_{σ}, then since Υ fixes Σ_{0,1} it follows that Υ also fixes τ, that is, Υ ∈ (G_{σ})_{τ}. Thus the stabiliser of θ_{σ} in G_{σ} is equal to the stabiliser of θ_{σ} in (G_{σ})_{τ}. Consequently we calculate the stabiliser of θ_{σ} in G_{σ}.

First we consider Υ = T α ∈ G′_{σ}. Now Υ fixes θ_{σ} if and only if it fixes the set of planes of θ_{σ}. Recall from Section 2.2 that the plane π_{t} of θ_{σ} may be written as ⟨ℓ_{σ}, (−mt^{σ}, t, 0, 0, 1)⟩, and so since Υ fixes ℓ_{σ}, it follows that

Υ(π_{t}) = Υ(⟨ℓ_{σ}, (−mt^{σ}, t, 0, 0, 1)⟩)
= ⟨ℓ_{σ}, (−m^{α}t^{σα}s − c, sm^{α−1}t^{α} − b, −, −, 1)⟩

Now recall from Section 2.2 that the plane π_{η} of θ_{σ} is defined by the equations x_{0}+mη^{σ−1}x_{1} = 0 and x_{1} − ηx_{4} = 0, and so T α(−mt^{σ}, t, 0, 0, 1) ∈ π_{t} if and only if

η = sm^{α−1}t^{α} − b and − m^{α}t^{σα}s − c + m(sm^{α−1}t^{α} − b)^{σ} = 0.

Combined, these equations imply that the Υ(π_{t}) is a plane of θ_{σ} if and only if

t^{σα}(−m^{α}s + s^{σ}m^{(α−1)σ+1}) − c − mb^{σ} = 0  (2.6.6)

Now, Υ fixes the set of planes of θ_{σ} if and only if 2.6.6 is satisfied for all t ∈ GF(q). This is equivalent to the polynomial x^{σα}(−m^{α} + s^{σ}m^{(α−1)σ+1}) − c − mb^{σ} being identically zero. This in turn holds, if and only if the following two equations are satisfied

−m^{α}s + s^{σ}m^{(α−1)σ+1} = 0  (2.6.7)
−c − mb^{σ} = 0  (2.6.8)

Since b and c are related by mb+c = 0, 2.6.8 is equivalent to b^{σ} = −b. If b^{σ} = −b, then b^{2σ} = b^{2} and so b^{2} ∈ F ix(σ). Now suppose that x is a non-zero element of GF(q), such that x^{σ} = x. If y^{2} = x, then y^{2σ} = y^{2} and so either y^{σ} = y or y^{σ} = −y. Since the other square root of x is −y, it follows that if y^{σ} = y, then (−y)^{σ} = (−y) and if y^{σ} = −y, then (−y)^{σ} = −(−y). Thus, for x a square such that x^{σ} = x, either both square roots of x are in F ix(σ), or both are in the set {b : b^{σ} = −b} ⊂ GF(q) \ F ix(σ). Since F ix(σ) is a field of order k, there are (k − 1)/2 elements of F ix(σ) that are square in GF(q) and have square roots in F ix(σ). If q = p^{h} and k = p^{r}, then the other (k − 1)/2 non-zero elements of F ix(σ) are square in GF(q) if and only if 2 divides (h/r). When 2 divides (h/r) there are (k − 1) non-zero solutions to b^{σ} = −b and including the solution b = 0 gives a total of k. If 2 does not divide (h/r), then the only solution to the equation b^{σ} = −b is b = 0.

Now solving equation 2.6.7,

−m^{α}s + s^{σ}m^{(α−1)σ+1} = 0
⟺ −m^{α}s + (sm^{α−1})^{σ}m = 0
⟺ (sm^{α−1})^{σ} = sm^{α−1}.

Thus, 2.6.7 is satisfied if and only if sm^{α−1} ∈ GF(q). Hence if 2 divides h/r then there are 2qk(h(k − 1)) elements of G′_{σ} that fix θ_{σ} and if 2 does not divide h/r, then there are 2qh(k − 1) elements of G′_{σ} that fix θ_{σ}.

Having calculated which elements of G′_{σ} fix θ_{σ}, we now calculate which elements of W G′_{σ} fix θ_{σ}.

Now

W (π_{t}) = W (⟨ℓ_{σ}, (−mt^{σ}, t, 0, 0, 1)⟩)
= ⟨ℓ_{σ}, (mt, −t^{σ}, 0, 0, 1)⟩

and so W fixes θ_{σ} if and only if

t − t^{σ^{2}} = 0 for all t ∈ GF(q).

Thus, in the case where σ^{2} is the identity automorphism of GF(q), W fixes θ_{σ} and so the set of elements of W G′_{σ} that fix θ_{σ} is {W Υ : Υ ∈ G′_{σ} and Υ(θ_{σ}) = θ_{σ}}. Hence if σ^{2} is the identity (and so 2 divides h/r) there are 4qk(h(k − 1)) elements of G_{σ} that fix θ_{σ}.

In the case where σ^{2} is not the identity, if Υ = T α ∈ G′_{σ}, then

W Υ(π_{t}) = W T α(⟨ℓ_{σ}, (−mt^{σ}, t, 0, 0, 1)⟩)
= ⟨ℓ_{σ}, (sm^{α}t^{α} − mb, −m^{α−1}t^{σα}s − c/m, −, −, 1)⟩

and so W Υ fixes θ_{σ} if and only if

sm^{α}t^{α} − mb + m(−m^{α−1}t^{σα}s − \frac{c}{m})^{σ} = 0 for all t ∈ GF(q)

⟺ sm^{α}t^{α} − mb + m (−m^{α−1}st^{σ^{2}α} − \frac{c^{σ}}{m^{σ}} ) = 0 for all t ∈ GF(q)

⟺  −m^{α}st^{σα} + m^{α}st^{α} − mb − \frac{c^{σ}}{m^{σ−1}} = 0 for all t ∈ GF(q).

Thus, W Υ fixes θ_{σ} if and only if the polynomial

−m^{α}sx^{σα} + m^{α}sx^{α} − mb − \frac{c^{σ}}{m^{σ−1}}

is identically zero. This, in turn, yields the equations

−m^{α}s = 0 and − mb − \frac{c^{σ}}{m^{σ−1}} .

Since s ≠ 0 these equations are never satisfied.

Hence if σ^{2} is not the identity and 2 divides h/r, then there are 2qkh(k − 1) elements of G_{σ} that fix θ_{σ} and if 2 does not divide h/r, then there are 2qh(k − 1) elements of G_{σ} that fix θ_{σ}

Theorem 2.6.7 Let θ be a K1(σ) ovoid of Q(4, q) with special tangent line ℓ and let τ be a trade of θ. The number of K1(σ) ovoids, with special tangent line ℓ, that have τ as a trade is:

at least  2q  if 2 does not divide h/r,
at least 2(q/k) if 2 | (q/k) and σ^{2} ≠ identity,
exactly  q/k if σ^{2} = identity.

Proof: Recall that the stabiliser of θ_{σ} in (G_{σ})_{τ} is equal to the stabiliser of θ_{σ} in G_{σ}. Thus using 2.6.4 and 2.6.5, |H_{σ}| = 4h(k − 1)q^{2} and the size of the stabiliser of θ_{σ} in G_{σ} calculated above we have the result.

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We are now able to determine the conditions under which there exists two K1(σ) ovoids with the same special tangent line and a common trade but not in a common elation rosette.

Corollary 2.6.8 Let σ be a non-identity automorphism of GF(q), q odd. If σ^{2} is the identity, then any two K1(σ) ovoids of Q(4, q) with the same special tangent line and sharing a trade are contained in a common elation rosette. If σ^{2} is not the identity, then for any K1(σ) ovoid of Q(4, q) θ and trade τ of θ, there exists at least one K1(σ) ovoid θ′, not equal to θ, such that θ′ has the same special tangent line as θ, has τ as a trade but is not contained in a common elation rosette with θ.

Proof: Let R be the rosette of Q(4, q) containing θ and with base point the special point of θ. Then there are q/k ovoids of R (including θ) with τ as a trade (and necessarily with the same special tangent line as θ). Applying Theorem 2.6.7 gives the result.

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K1(σ) ovoids sharing a trade but with different special tangent lines

In this section we find K1(σ) ovoids that share the trade τ = {[1, m, 0, 0, u] : u = m(t^{σ} − t) for some t ∈ GF(q)} (the trade of θ_{σ} containing Σ_{0,1}, as in Section 2.6.2) with θ_{σ} but not the special tangent line with θ_{σ}. We do this by finding automorphisms of Q(4, q) that fix τ, but do not fix ℓ_{σ}.

If A is a point of π \ {X_{σ}} then recall the symmetry μ_{A} of Q(4, q) from Section 1.2 and Section 2.2. The symmetry μ_{A} fixes X_{σ} and every subspace of PG(4, q) that contains A. Thus it follows that μ_{A}(θ_{σ}) is a K1(σ) ovoid with special point X_{σ} with τ as a trade and with special tangent line μ_{A}(ℓ_{σ}). Now ℓ_{σ} is such that ℓ_{σ}^{⊥} is a plane intersecting Q(4, q) in a pair of lines (as observed in Section 2.6.2). There are (q + 1)/2 such tangents, containing X_{σ}, in π and we may easily show that the set of symmetries {μ_{A} : A ∈ π\{X_{σ}}} is transitive on these tangents. Thus there are (at least) (q − 1)/2 K1(σ) ovoids (not including θ_{σ}) of the form μ_{A}(θ_{σ}), A ∈ π \ {X_{σ}}, that share the trade τ with θ_{σ}. If R_{A} is the elation rosette containing μ_{A}(θ_{σ}) and with base point X_{σ}, then there are q/k ovoids in R_{A} with τ as a trade, each with special tangent line μ_{A}(ℓ_{σ}). Thus there are (at least) (q − 1)q/(2k) K1(σ) ovoids that share the trade τ with θ_{σ}, but not the special tangent line ℓ_{σ}. Thus we have the following theorem.

Theorem 2.6.9 Let θ be a K1(σ) ovoid of Q(4, q) with special tangent line ℓ. Let τ be a trade of θ. Then there are (at least) (q − 1)q/(2k) K1(σ) ovoids of Q(4, q) that share the trade τ with θ, but not the special tangent line ℓ.

2.6.3 Rosettes of Q(4, 9)

In this section we consider rosettes containing K1(σ) ovoids of Q(4, 9), where σ is the nonidentity automorphism of GF(9) (that is, x^{σ} = x^{3} and σ^{2} is the identity). Penttila and Royle have shown ([53]) that there are only two isomorphism classes of ovoids of Q(4, 9), the elliptic quadric ovoids and the K1(σ) ovoids. Hence if {θ_{1}, θ_{2}, …, θ_{9}} is a rosette of Q(4, 9), then for each i = 1, 2, …, 9 the ovoid θ_{i} is either and elliptic quadric ovoid or a K1(σ) ovoid. We know that if θ_{i} is an elliptic quadric ovoid for i = 1, 2, …, 9, then the rosette is an elation rosette. Otherwise the rosette must contain a K1(σ) ovoid, without loss of generality the ovoid θ_{1}. A computer search by Gordon Royle ([54]) gives us the following result on such rosettes.

Theorem 2.6.10 [54] Let θ_{1} be a fixed K1(σ) ovoid of Q(4, 9) with special point X. Then every rosette {θ_{1}, θ_{2}, …, θ_{9}} is included in the following table.

Isomorphism  Size of orbits
Base point  Type  # rosettes (K1(σ), EQ) on ovoids
Fixed non-  I  2  (9,0)  9
special point  II  8  (9,0)  9
A  32  (6,3)  3,6
B  48  (9,0)  3,6
C  48  (9,0)  3,6
D  4  (3,6)  3,6
E  8  (6,3)  3,6

Special point

F  1  (9,0)  9

In the above table the first column indicates whether the base point of the rosette is the special point X of θ_{1}, or not. The second column indicates the different isomorphism classes of rosettes containing θ_{1}, while the third gives the number of rosettes in each isomorphism class. The fourth column gives the number of K1(σ) and elliptic quadric ovoids in a rosette of a particular isomorphism type. The final column gives the orbit sizes for the action of the group of a rosette (in a particular isomorphism class) on the ovoids of the rosette.

We will see that all of these rosettes may be constructed either as in Theorem 2.6.1 or by the method shown in Theorem 2.6.6 and the work in Section 2.6.2.

By Theorem 2.6.1, if P is a fixed non-special point of θ_{1}, then there are q + 1 = 10 elation rosettes containing θ_{1} with base point P. These are precisely the rosettes of type I and II.

In the current context Theorem 2.6.6 says the following: let X be the special point of the K1(σ) ovoid θ_{1} and R the elation rosette containing θ_{1} and with base point X. Let T = {τ_{1}, τ_{2}, τ_{3}} be a tower of R (see Section 2.6.1) such that τ_{1} is a trade of θ_{1}. Let ρ_{1} be the set of ovoids of R that have τ_{1} as a trade and ρ_{i}, for i = 2, 3 be either

1. a set of 3 K1(σ) ovoids contained in a common elation rosette (not necessarily R), and sharing the trade τ_{i}; or

2. the set of 3 elliptic quadric ovoids of Q(4, 9) whose elements are the intersection of the elements of τ_{i} with Q(4, 9).

Then the set of ovoids ρ_{1} ∪ ρ_{2} ∪ ρ_{3} is a rosette of Q(4, 9). We will call these different types of sets of ovoids sets of type 1 or 2, as enumerated above.

Note that R has (q − 1)/(k − 1) = 4 towers and that the group of θ_{1} is transitive on these towers.

The rosette of type F is the (unique) elation rosette R, containing θ_{1} and with base point X, the special point of θ_{1}.

Recall in Section 2.6.2 that we found (q+1)/2 = 5 possible sets of type 1 for ρ_{2}, including the set of 3 ovoids of R for which τ_{2} is a trade. Each of these sets has the property that the ovoids it contains have a common special tangent line. Also the special tangent line corresponding to each of these sets is distinct (the special tangent lines are those tangent lines in the base plane of T that have a polar with hyperbolic character). A similar result holds for sets of type 1 for ρ_{3}.

Now the rosette ρ_{1} ∪ τ_{2} ∪ τ_{3} is of type D and the 4 towers give the 4 rosettes of type D. If ρ_{2} is the set of type 1 contained in R, then ρ_{1} ∪ ρ_{2} ∪ τ_{3} is a rosette of type E. Similarly,

the rosette ρ_{1} ∪ τ_{2}ρ_{3} with ρ_{2} ⊂ R. The 4 towers give us the 8 rosettes of type E.

If ρ_{2} is a set of type 1 not contained in R, then ρ_{1} ∪ ρ_{2} ∪ τ_{3} is a rosette of type A and there are 4 such rosettes. Similarly, there are 4 rosettes of the form ρ_{1} ∪ τ_{2} ∪ ρ_{3} of type A and the 4 towers give us the 32 rosettes of type A.

Now suppose that ρ_{2} and ρ_{3} are sets of type 1. Let ℓ_{i} be the special tangent line of the ovoids in ρ_{i}. There are two possibilities: either two elements of {ℓ_{1}, ℓ_{2}, ℓ_{3}} coincide, or they are all distinct (if all three coincide then we have the elation rosette of type F). These 2 possibilities correspond to the rosettes of type B and C.

2.7 Remarks

With a little more computational effort it may be possible to classify the rosettes of Q(4, q) such that each ovoid of the rosette is either an elliptic quadric ovoid of a K1(σ) ovoid (for fixed σ). Also of interest is considering rosettes of Q(4, q) containing other ovoids of Q(4, q), to try and determine whether there is a GQ of order (q, q^{2}) containing Q(4, q) as a subquadrangle and subtending rosettes of these ovoids. Similarly for q even since the q-clan GQs all possess subquadrangles of order q isomorphic to T_{2}(O) for some oval O of PG(2, q), it would be interesting to investigate rosettes of the GQ T_{2}(O).

SPGs and GQs of order (r, r^{2})

3.1 SPGs from GQs of order (r, r^{2})

Consider the SPG (constructed by Metz see [18] and by Hirschfeld and Thas [26]) with parameters s = q − 1, t = q^{2}, α = 2, μ = 2q(q − 1), where q is a prime power. The construction due to Metz as follows: let Q = Q(4, q) and let P be the set of three dimensional, non-singular elliptic quadrics contained in Q. Let a bundle of Q be a set of q elements of P that meet pairwise in a common point. Let B be the set of bundles of Q. Define incidence I ⊆ (P × B) ∪ (B × P) to be containment. Since each bundle is a set of q elliptic quadric ovoids of Q sharing a common tangent plane at the point where the elliptic quadrics intersect, and two elements of P that are tangent are incident with exactly one common bundle, it follows that the structure T = (P, B, I ) is a SPG with the above parameters.

Now consider the GQ S = Q(5, q). S contains Q as a subGQ and the subtended ovoids are exactly the elements of P. Moreover, the subtended rosettes are the bundles and T is the incidence structure obtained by taking subtended ovoids as points and subtended rosettes as lines. This relation between Q(5, q), Q(4, q) and the SPG T depends on the combinatorics of the situation, rather than the specific geometry. In this section we generalise the Metz/ Hirschfeld and Thas construction of an SPG by using a GQ of order (r, r^{2}) and a subquadrangle of order r (with particular combinatorial properties) in the place of Q(5, q) and Q(4, q), resulting in Theorem 3.1.7. In Section 3.1.1 we construct an algebraic 2-fold cover of the SPG constructed in Theorem 3.1.7 and use this in Section 3.2 to solve the isomorphism problem for such SPGs. In Section 3.3 we construct a new SPG from a GQ constructed by Kantor in [31]. In Section 3.4 we investigate the q-clan GQs, q even and the subquadrangles of order q constructed by Payne in [45]. We determine the conditions under which a q-clan GQ, with such a subquadrangle, gives rise to an SPG by the method of Theorem 3.1.7.

To begin with we state and prove a lemma which is implicit in [49, Chapter 2]. The lemma proves important for the work in this chapter and in Chapter 4. Recall from Section 1.4.1 that if S is a GQ with a subGQ S′, then each line of S is a line of S′, is incident with exactly one point of S′ or is incident with no point of S′. In the last two cases the line is called tangent and external (to S′), respectively. Dually we define tangent points and external points.

Lemma 3.1.1 Let S = (P, B, I) be a GQ of order (r, r^{2}) and S′ = (P′, B′, I′) a subquadrangle of S of order r. Then each point of P \ P′ is an external point to S′ and each line of B \ B′ is a tangent line to S′.

Proof: If ℓ ∈ B′, then it follows that each point of S incident with ℓ is a point of S′. Thus a point of P \ P′ is incident with no line of S′ and hence is external to S′. By [49, 2.2.1] each external point P is collinear with exactly 1 + r^{2} points of S′, that is, each line incident with P is incident with exactly one point of S′. Thus each line of B \ B′ is a tangent line to S′.

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Corollary 3.1.2 Let S = (P, B, I) be a GQ of order (r, r^{2}) and S′ = (P′, B′, I′) a subquadrangle of S of order r. Then each point of P \P′ subtends an ovoid of S′ and each line of B\B′ subtends a rosette of S′ (as in Section 1.4.1).

Now we are interested in determining the intersection of subtended ovoids. To this end we recall a result of Bose and Shrikhande, interpreted in the GQ context.

Lemma 3.1.3 ([8], see [49, 1.2.4]) If S is a GQ of order (r, r^{2}) and {X, Y, Z} is a triad of S, then |{X, Y, Z}^{⊥}| = r + 1.

Corollary 3.1.4 Let S be a GQ of order (r, r^{2}) and S′ a subGQ of order r. A subtended ovoid of S′ is subtended by at most two points of S. Further, if an ovoid θ is subtended by two points X, X′, then the size of the intersection of θ with any other subtended ovoid θ_{Y}, Y ≠ X, X′, is determined: if Y ∼ X or Y ∼ X′, then |θ ∩ θ_{Y} | = 1 and if Y ≁ X, X′, then |θ ∩ θ_{Y} | = r + 1.

Proof: Suppose that an ovoid θ is subtended by three points X, Y, Z. These three points are necessarily pairwise non-collinear and so form a triad of S. Since |{X, Y, Z}^{⊥}| ≥ r^{2} + 1 we have a contradiction of Lemma 3.1.3. Thus any ovoid may be subtended by at most two points.

Now suppose that θ is subtended by exactly two points X and X′. Let θ_{Y} be the ovoid subtended by the point Y, Y ≠ X, X′. Suppose that Y ∼ X or Y ∼ X′; without loss of generality we may suppose that Y ∼ X. Thus θ_{X} and θ_{Y} are contained in the rosette subtended by the line ⟨X, Y ⟩ and so θ_{X} ∩ θ_{Y} = {P } for some point P and |θ ∩ θ_{Y} | = |{X, X′, Y }^{⊥}| = 1. Suppose now that Y ≁ X, X′, then {X, X′, Y } is a triad of S and so |θ ∩ θ_{Y} | = |{X, X′, Y }^{⊥}| = r + 1.

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If a GQ S of order (r, r^{2}) has a subGQ S′ ⊂ S of order r such that each subtended ovoid of S′ is subtended by exactly two points of S, then we say that S′ is doubly subtended in S. In some sense a doubly subtended subquadrangle is an ‘extreme’ subquadrangle, so it is not surprising that we get some nice geometry from it. At this stage we introduce a slight abuse of notation. If X is a point of S, then we denote this by X ∈ S. If X is a point of S but not a point of S′, then we denote this X ∈ S \ S′.

We might now ask how many subtended rosettes can two subtended ovoids have in common. Let θ_{X} and θ_{Y} be two subtended ovoids; subtended by X and Y, respectively, where θ_{X} ≠ θ_{Y}.

There are three cases to consider: (i) neither θ_{X} nor θ_{Y} is subtended by a second point, (ii) θ_{X} is subtended by another point X′ and θ_{Y} is not subtended by a second point, (iii) θ_{X} and θ_{Y} are each subtended by a second point, say X′ and Y′ respectively. For case (i) θ_{X} and θ_{Y} are contained in exactly one common subtended rosette if X ∼ Y and in none otherwise. In case (ii) Y may be collinear with at most one of X, X′ since otherwise Y ∈ {X, X′}^{⊥} ⊂ S′ which contradicts Y ∈ S \ S′. Thus for case (ii) θ_{X} and θ_{Y} have one common rosette if Y ∼ X or Y ∼ X′ (not both) and none otherwise. For case (iii), to each unordered, incident pair taken from {X, X′, Y, Y′}, there corresponds a subtended rosette containing θ_{X} and θ_{Y}. So there may be none, one or two subtended rosettes containing θ_{X} and θ_{Y}. Note that it is possible that two distinct lines subtend the same rosette.

We have already seen that a GQ S of order (r, r^{2}) with a subGQ S′ of order r that has all subtended ovoids of S being subtended twice is a special and also extremal case of a subGQ of S. We now give a result which shows the relationship between double subtending and the existence of a particular type of involution of the GQ S. The idea for the construction of the involution comes from Thas [62]. For the following we will denote the ovoid subtended by a point X by θ_{X}.

Lemma 3.1.5 Let S be a GQ of order (r, r^{2}) and S′ a subGQ of order r. Then S′ is doubly subtended in S if and only if there exists a non-identity involution of S that fixes S′ pointwise.

Proof: First, let τ be an involution of S that fixes S′ pointwise. We first show that τ fixes no point of S \ S′. Suppose that X ∈ S \ S′ and τ(X) = X. Consider a line ℓ such that X ∈ ℓ and let ℓ ∩ S′ = P, say. Now τ(P) = P and so τ(ℓ) = ℓ. Now let R ∈ ℓ, R ≠ P. Then θ_{R} = θ_{τ(R)} but τ(R) ∈ ℓ and so R = τ(R). Thus ℓ is fixed pointwise by τ. Now consider a point Y ∈ S \S′, Y ≁ X. If θ_{Y} = θ_{X} then since an ovoid may only be subtended twice and τ(Y) subtends θ_{Y} we must have that τ(Y) = Y. If θ_{Y} ≠ θ_{X}, then there exists a point R′ such that R′ ∈ ℓ, R′ ∈ S \ S′ and Y ∼ R′. However, R′ is fixed by τ as it is collinear with X and so every point collinear with R′ is fixed, so τ also fixes Y. Thus τ fixes every point of S and so is the identity, which is a contradiction. Hence τ fixes no point in S \ S′.

Thus we can now say that for any point X ∈ S \ S′ we have τ(X) = X′ ≠ X and so θ_{X} is subtended by the distinct points X, X′.

Now, suppose that S′ is doubly subtended in S. If X ∈ S \ S′ then let X′ be the second point subtending θ_{X}. Now define τ to be the following map:

τ : X ↦ X′ X ∈ S \ S′
X ↦ X X ∈ S′.

Consider points P, Q ∈ S, with P ∼ Q. If P, Q ∈ S′ then P = τ(P) ∼ τ(Q) = Q. If P ∈ S′, Q ∈ S \ S′ then P ∈ θ_{Q} = θ_{Q′} = θ_{τ(Q)} and so P ∼ τ(Q), that is τ(P) ∼ τ(Q).

Now if P, Q ∈ S \ S′ then |θ_{P} ∩ θ_{Q}| = |θ_{τ(P)} ∩ θ_{τ(Q)}| = 1; so by Corollary 3.1.4 we have that τ(Q) ∼ P or τ(Q) ∼ τ(P). Since Q ∼ P it must be that τ(Q) ≁ P and so τ(P) ∼ τ(Q). Thus τ is an automorphism of S and clearly an involution.

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Corollary 3.1.6 If S is a GQ of order (r, r^{2}) that has a doubly subtended subGQ S′ of order r, then for each incident point, subtended ovoid pair (X, θ) there exists a unique subtended rosette R, containing θ and with basepoint X.

Proof: If θ is subtended by the points Y and Y′, then the only subtended rosettes containing θ and with base point X, are those subtended by the lines ⟨X, Y ⟩ and ⟨X, Y′⟩. However ⟨X, Y ⟩ is the image of ⟨X, Y′⟩ under the involution constructed in Lemma 3.1.5, and vice-versa. Since the involution in Lemma 3.1.5 fixes the subquadrangle S′ pointwise (and linewise), the rosette subtended by ⟨X, Y ⟩ and the rosette subtended by ⟨X, Y′⟩ are the same.

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Now we show that if a subGQ is doubly subtended, then we get an SPG from its subtended ovoid/rosette structure.

Theorem 3.1.7 Let S be a GQ of order (r, r^{2}) containing a subGQ S′ of order r, such that S′ is doubly subtended in S. Consider the following incidence structure T :

Points : Subtended ovoids of S′. Lines : Subtended rosettes of S′. Incidence : Containment.

Then T is a SPG with parameters s = r − 1, t = r^{2}, α = 2 and μ = 2r(r − 1).

Proof: A rosette contains r ovoids, thus s = r−1. By Corollary 3.1.6 there are r^{2}+1 subtended rosettes containing a subtended ovoid θ_{X}, that is, t = r^{2}.

Now consider a subtended rosette R with basepoint P and not containing the ovoid θ_{X}. Recall that ovoids of R partition the points of S′ that are not collinear with P. Suppose that P ∈ θ_{X}. Then θ_{X} ⊂ S′ \ P^{⊥}. Let n_{1} and n_{r+1} be the number of ovoids of R that meet θ_{X} in 1 and r + 1 points, respectively. Then we have the following equations:

n_{r+1} · r + n_{1} · 0 = r^{2}
n_{r+1} + n_{1} = r.

Solving simultaneously gives n_{r+1} = r and n_{1} = 0, that is, θ_{X} meets each ovoid in R in r + 1 points.

Suppose now that P ∉ θ_{X}. Then θ_{X} has r^{2} − r points non-collinear with P and so we have the following equations:

n_{r+1} · (r + 1) + n_{1} = r^{2} − r
n_{r+1} + n_{1} = r

Solving simultaneously we have n_{r+1} = r − 2 and n_{1} = 2.

In terms of T the above means that if we have a non-incident point/line pair (A, ℓ) in T there are 0 or 2 point/line pairs (B, m) such that A I m I B I l.

Now consider two subtended ovoids of S′, say θ_{X} and θ_{Y}, such that |θ_{X} ∩ θ_{Y} | = r + 1. Let Ω = θ_{X} \ θ_{Y}. By Corollary 3.1.6, for each Q ∈ Ω there exists exactly one subtended rosette R, with base point Q and containing θ_{X}. By the above, we see that there are two subtended rosettes containing θ_{Y} and an ovoid in R, and so two subtended ovoids that are contained in a subtended rosette with θ_{X} and contained in a distinct subtended rosette with θ_{Y}. This is true for all points in Ω and so there are 2(r^{2} − r) subtended ovoids that are contained in a rosette with both θ_{X} and θ_{Y}. In T , this means that given two non-collinear points A and B there are 2r(r − 1) points collinear to both A and B.

Thus T is an SPG with the parameters as required.

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Corollary 3.1.8 Let S be a GQ of order (r, r^{2}) containing a subGQ S′ of order r such that there exists a non-identity involution of S that fixes S′ pointwise. Then there is an associated SPG with parameters s = r − 1, t = r^{2}, α = 2 and μ = 2r(r − 1).

3.1.1 Algebraic 2-fold covers of SPGs and the GQ condition

Suppose that S is a GQ of order (r, r^{2}) and S′ is a subquadrangle of order r that is doubly subtended. Let T be the SPG constructed from S and S′ as in Theorem 3.1.7. In this section we show that if \overline{T} is the geometry of S \ S′ (that is, the subgeometry of S consisting of the points and lines not in S′), then there exists a map p from \overline{T} to T such that (\overline{T} , p) is a 2-fold cover of T . In fact the 2-fold cover is an algebraic 2-fold cover. We then show that we may reconstruct S from S′ and \overline{T} . Generalising this construction we construct a GQ of order (r, r^{2}) from S′, \overline{T} and an algebraic 2-fold cover of \overline{T} , provided the cover has a property called the GQ condition.

Let S, S′ and T be as above and let (\overline{T} , p) be an algebraic 2-fold cover of T over ℤ_{2} (where ℤ_{2} is the unique abelian group of order 2), defined by the 1-cochain c. Then (\overline{T} , p) is said to satisfy the GQ condition if for each set {P, Q, R} of pairwise collinear points of T

δc(P, Q, R) = c(P, q) − c(P, R) + c(Q, R) = 0 ⟺ P, Q, R are collinear points of T . (3.1.1)

Theorem 3.1.9 Let S = (P, B, I) be a GQ of order (r, r^{2}) and S′ = (P′, B′, I) a doubly subtended subGQ of order r. Let T be the SPG constructed from S and S′, as in Theorem 3.1.7 and let \overline{T} be the geometry (P \ P′, B \ B′, I). Let Θ be the set of subtended ovoids of S′ and represent P \ P′ as the set {(θ, 0), (θ, 1) : θ ∈ Θ}. Let c be the 1-cochain (of the simplicial complex of the point graph of T ) defined by

c(θ_{i}, θ_{j}) = \{ \begin{matrix}0 & if |θ_{i} ∩ θ_{j}| = 1 and (θ_{i}, 0) and (θ_{j}, 0) are collinear. \\ 1 & if |θ_{i} ∩ θ_{j}| = 1 and (θ_{i}, 0) and (θ_{j}, 0) are not collinear. \end{matrix}

Then c defines an algebraic 2-fold cover of T with covering geometry \overline{T} . Furthermore, c satisfies the GQ condition.

Proof: Let θ_{1} and θ_{2} be two collinear points of T and so |θ_{1} ∩ θ_{2}| = 1. Clearly, c(θ_{1}, θ_{2}) = c(θ_{2}, θ_{1}) and so c is a 1-cochain. Since they are collinear points of T , θ_{1} and θ_{2} are two subtended ovoids of S′ contained in a common subtended rosette, that is subtended by two lines of S. The point labelled (θ_{1}, 0) is incident with one of these lines and (θ_{1}, 1) is incident with the other, and similarly for (θ_{2}, 0) and (θ_{2}, 1). Thus (θ_{1}, α) is collinear with (θ_{2}, β) if and only if c(θ_{1}, θ_{2}) = α + β. Thus c defines an algebraic 2-fold cover of the point graph of T .

To show that c defines an algebraic 2-fold cover of the geometry T we need to show that δc(θ_{1}, θ_{2}, θ_{3}) = 0 whenever θ_{1}, θ_{2} and θ_{3} are collinear points of T . Let θ_{1}, θ_{2} and θ_{3} be three collinear points of T , then they are contained in a common subtended rosette R of S′. Now (θ_{1}, 0) is collinear with (θ_{2}, c(θ_{1}, θ_{2})) and (θ_{1}, 0) with (θ_{3}, c(θ_{1}, θ_{3})). Since ⟨(θ_{1}, 0), (θ_{2}, c(θ_{1}, θ_{2}))⟩ and ⟨(θ_{1}, 0), (θ_{3}, c(θ_{1}, θ_{3}))⟩ both subtend the rosette R, it follows that (θ_{2}, c(θ_{1}, θ_{2})) and (θ_{3}, c(θ_{1}, θ_{3})) are collinear and so δc(θ_{1}, θ_{2}, θ_{3}) = 0. Thus c defines a cover of the geometry T .

If θ_{1}, θ_{2} and θ_{3} are pairwise collinear but not incident with a common line of T , then it follows that they are not contained in a common subtended rosette of S′. Thus (θ_{1}, 0), (θ_{2}, c(θ_{1}, θ_{2})) and (θ_{3}, c(θ_{1}, θ_{3})) are not incident with a common line of \overline{T} and so (θ_{2}, c(θ_{1}, θ_{2})) and (θ_{3}, c(θ_{1}, θ_{3})) are not collinear since this would be a triangle in S. Hence, δc(θ_{1}, θ_{2}, θ_{3}) = 1 and c satisfies the GQ condition.

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Given the notation of Theorem 3.1.9, consider the following description of S.

Points (i)  Points of S′.
(ii)  Points of \overline{T} .
Lines (a)  Lines of S′.
(b) ℓ ∪ P where ℓ is a line of \overline{T} and

P the basepoint of the subtended rosette covered by ℓ.

Incidence (i),(a) as in S′.
(i),(b) A point P of type (i) is incident with a line ℓ ∪ Q of type (b)

if and only if P = Q.

(ii),(a) None.
(ii),(b) A point P of type (ii) is incident with a line ℓ ∪ Q of type (b)

if and only if P is incident with ℓ in \overline{T}

Now suppose that in the above incidence structure instead of using the algebraic 2-fold cover of T from Theorem 3.1.9 we use an arbitrary algebraic 2-fold cover of T . The following theorem specifies the conditions under which this new incidence structure is a GQ.

Theorem 3.1.10 Let S = (P, B, I) be a GQ of order (r, r^{2}) and S′ = (P′, B′, I) a doubly subtended subGQ of order r. Let T be the SPG constructed from S and S′, as in Theorem 3.1.7 and let (\overline{T} , p) be an algebraic 2-fold cover of T defined by the 1-cochain c. Define W to be the following incidence structure:

Points (i)  Points of S′.
(ii)  Points of \overline{T} .
Lines (a)  Lines of S′.
(b) ℓ ∪ P where ℓ is a line of \overline{T} and

P the basepoint of the subtended rosette covered by ℓ.

Incidence (i),(a) as in S′.
(i),(b) A point P of type (i) is incident with a line ℓ ∪ Q of type (b)

if and only if P = Q.

(ii),(a) None.
(ii),(b) A point P of type (ii) is incident with a line ℓ ∪ Q of type (b)

if and only if P is incident with ℓ in \overline{T}

Then S is a GQ of order (r, r^{2}) if and only if c satisfies the GQ condition. In this case S contains S′ as a subquadrangle and S′ is doubly subtended by S. The SPG constructed from S and S′ as in Theorem 3.1.7 is T .

Proof: Any line of S′ is incident with r + 1 points of S′ and so r + 1 points of W. A line ℓ ∪ P of type (b) is incident with P and with the r points of \overline{T} incident with ℓ. Thus each line of S is incident with r + 1 points.

Let Q be a point of type (i), then Q is incident with r + 1 lines of S′. There are (r^{2} − r)/2 subtended rosettes that have Q as a basepoint and so there are r^{2} − r lines of S of type (b) that are incident with Q. Thus Q is incident with r^{2} + 1 lines of S. By Corollary 3.1.6 each subtended ovoid of S′ is contained in r^{2} + 1 subtended rosettes and so each type (ii) point of S is incident with r^{2} + 1 lines of W.

We check the third GQ axiom for each non-incident point/line pair, (P, ℓ) of S. If P is of type (i) and ℓ is of type (a), then since S′ is a GQ the property holds. Let P be of type (i) and ℓ ∪ Q of type (b). If P and Q are collinear then there is no ovoid of S′ containing both P and Q. Thus, Q is the unique point of ℓ ∪ Q that is collinear with P. If P is not collinear with Q then P is contained in a unique ovoid in the rosette subtended by ℓ. There is a unique subtended rosette containing this ovoid and with basepoint P.

Let P be of type (ii) and ℓ of type (a). The ovoid θ, corresponding to P meets ℓ in exactly one point X. There is a unique subtended rosette containing θ and with basepoint X and thus a unique line of type (b) containing P and X.

Let P be of type (ii) and let ℓ ∪ Q of type (b). Let θ be the ovoid of S′ corresponding to P and R = {θ_{1}, …, θ_{r}} the subtended rosette of S′ corresponding to ℓ. Without loss of generality suppose that P = (θ, 0). There are two possibilities for ℓ, either

ℓ = {(θ_{1}, 0), (θ_{2}, c(θ_{1}, θ_{2})), …, (θ_{r}, c(θ_{1}, θ_{r}))} or

ℓ = {(θ_{1}, 1), (θ_{2}, c(θ_{1}, θ_{2}) + 1), …, (θ_{r}, c(θ_{r}, θ_{1}) + 1)}. Suppose that θ ∈ R and that without loss of generality θ = θ_{1}. Then since (θ, 0) is not incident with ℓ we have that ℓ = {(θ_{1}, 1), (θ_{2}, c(θ_{1}, θ_{2}) + 1), …, (θ_{r}, c(θ_{1}, θ_{r}) + 1)} and (θ, 0) is collinear with none of the points on ℓ. Thus Q is the unique point on ℓ ∪ Q that is collinear with P. Now suppose that θ ∉ R and that without loss of generality ℓ = {(θ_{1}, 0), (θ_{2}, c(θ_{1}, θ_{2})), …, (θ_{r}, c(θ_{r}, θ_{1}))}. If Q ∈ θ then θ meets each of the θ_{i} in r + 1 points and is contained in a unique subtended rosette with Q as the basepoint, which gives a unique line incident with P and a point of ℓ ∪ Q. If Q ∉ θ, then there are two ovoids of R that meet θ in precisely one point. Without loss of generality let these ovoids be θ_{1} and θ_{2}. Now (θ_{1}, 0) is collinear to (θ_{2}, c(θ_{1}, θ_{2})) (on ℓ) and (θ_{1}, 1) is collinear to (θ_{2}, c(θ_{1}, θ_{2}) + 1), while (θ, 0) is collinear to exactly one point of the form (θ_{1}, −) and one of the form (θ_{2}, −). So (θ, 0) is collinear to exactly to one point on ℓ ∪ Q if and only if either (θ, 0) is collinear to (θ_{1}, 0) and (θ_{2}, c(θ_{1}, θ_{2}) + 1) or (θ, 0) is collinear to (θ_{1}, 1) and (θ_{2}, c(θ_{1}, θ_{2})). This occurs if and only if c(θ, θ_{2}) = c(θ, θ_{1}) + c(θ_{1}, θ_{2}) + 1. That is, if and only if

δc(θ, θ_{1}, θ_{2}) = c(θ, θ_{1}) + c(θ, θ_{2}) + c(θ_{1}, θ_{2})
= c(θ, θ_{1}) + (c(θ, θ_{1}) + c(θ_{1}, θ_{2}) + 1) + c(θ_{1}, θ_{2})
= 1.

This is precisely the GQ condition. Thus W is a GQ of order (r, r^{2}) if and only if c satisfies the GQ condition.

Now suppose that W is a GQ of order (r, r^{2}). If X is a point of type (ii) of S, then let θ_{X} be the ovoid of S′ that is covered by X. The set of lines of W incident with X meets S′ in the set of basepoints of subtended rosettes containing θ_{X}. So, in W, X subtends the ovoid θ_{X} in S′. It then follows that a line ℓ ∪ P of W subtends the rosette that is covered by the line ℓ of \overline{T} . Thus S′ is doubly subtended in W and subtended ovoid/rosette structure is T .

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We will study this construction in greater depth in Chapter 4.

Note that in Theorem 3.1.10 since c defines an algebraic 2-fold cover of T , it follows from Section 1.6 that c satisfies one half of the GQ condition

δc(θ_{1}, θ_{2}, θ_{3}) = c(θ_{1}, θ_{2}) − c(θ_{1}, θ_{3}) + c(θ_{2}, θ_{3}) = 0 if θ_{1}, θ_{2}, θ_{3} are distinct collinear points of T .

In fact by Theorem 3.1.10 any 1-cochain satisfying the GQ condition, defines an algebraic 2-fold cover of T and the construction in Theorem 3.1.10 is a GQ.

3.2 Isomorphisms of SPGs

In this section we determine when two SPGs, constructed from the double subtending process, are isomorphic. We also calculate the group of such an SPG.

Theorem 3.2.1 Let W and S be two GQs of order (r, r^{2}) and let W′ and S′ be subGQs of W and S respectively, of order r. Let W′ and S′ be doubly subtended in W and S and let the SPGs constructed as in Theorem 3.1.7 be T_{W} and T_{S}. The SPGs T_{W} and T_{S} are isomorphic if and only if there exists an isomorphism from W′ to S′ that induces an isomorphism from T_{W} to T_{S}.

Proof: First, let c_{S} define an algebraic 2-fold cover of T_{S}, as in Theorem 3.1.9, and let i : T_{W} → T_{S} be an isomorphism. If θ and θ′ are two points of T_{W}, then we may easily show that the function c_{W} = c_{S}(i(θ), i(θ′)) defines an algebraic 2-fold cover of T_{W}, that satisfies the GQ condition. Let (T_{W}^{c_{W}} , p_{W}) be the algebraic 2-fold cover of T_{W} defined by c_{W}, (T_{W}^{c_{S}}, p_{S}) the algebraic 2-fold cover of T_{W} defined by c_{S} and \overline{S} the GQ of order (r, r^{2}) constructed from (T_{W}^{c_{W}} , p_{W}) and W′ as in Theorem 3.1.10. Now, let \overline{i} be the map from the pointset of T_{W}^{c_{W}} to the pointset of T_{W}^{c_{S}}, which acts by (θ, α) ↦ (i(θ), α)), for θ a point of T_{W} and α ∈ ℤ_{2}. If the lines of T_{W} are considered as sets of points of T_{W}, then \overline{i} induces an isomorphism from T_{W}^{c_{W}} to T_{W}^{c_{S}}, which we also denote by \overline{i}. We show that \overline{i} may be extended to an isomorphism from \overline{S} to S.

Let ℓ and m be two skew (that is, non-concurrent) lines of \overline{S} that are tangent to W′. A line of \overline{S} that is tangent to W′ is said to be a transversal to ℓ and m if it is concurrent to both ℓ and m.

Let P be a point of W′ incident with ℓ, then P is the unique point of W′ that is incident with ℓ. Let Q be the unique point of W′ that is incident with m. Now if P and Q are not collinear, then by the third GQ axiom one point of the set ℓ \ {P } is collinear with Q and each of the remaining r − 1 points of ℓ \ {P } is collinear with a unique point of m \ {Q}. Thus there are r − 1 transversals to ℓ and m in this case. If P is collinear to Q (with the line incident with P and Q necessarily a line of W′), then each point of ℓ \ {P } is collinear with a unique point of m \ {Q}. Thus there are r transversals to ℓ and m.

Now if ℓ and m are not skew, but meet in a point of W′, that is P = Q, then the third GQ axiom decrees that there are no transversals to ℓ and m.

Let ℓ′ and m′ be the lines of T_{W}^{c_{W}} such that ℓ = ℓ′ ∪ {P } and m = m′ ∪ {Q}. Let a line of T_{W}^{c_{W}} be a transversal to ℓ′ and m′ if it is concurrent to both ℓ′ and m′. Consider the incidence structure which has pointset {L : L is a set of q^{2} − q pairwise skew lines of T_{W}^{c_{W}} } and points L_{1} and L_{2} collinear if ℓ_{1} and ℓ_{2} have exactly r transversals, for all ℓ_{1} ∈ L_{1} and ℓ_{2} ∈ L_{2} (lines are the maximal sets of pairwise collinear points). The above calculations show that this incidence structure is W′.

Thus \overline{i} may be extended to an isomorphism from \overline{S} to S. The restriction of \overline{i} to W′ is an isomorphism from W′ to S′ that induces i from T_{W} to T_{S}.

Now suppose that there exists an isomorphism from W′ to S′ that takes T_{W} to T_{S}. Since such an isomorphism maps ovoids to ovoids, rosettes to rosettes and preserves inclusion of an ovoid in a rosette, it induces an isomorphism from T_{W} to T_{S}.

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As a corollary of Theorem 3.2.1 we state the automorphism group of an SPG arising from the double subtending process.

Corollary 3.2.2 Let S be a GQ of order (r, r^{2}) and S′ a subGQ of order r. Let S′ be doubly subtended in S, with SPG T constructed as in Theorem 3.1.7. The automorphism group of T is the stabiliser of T in the automorphism group of S′.

Proof: From the proof of Theorem 3.2.1, if T = T_{W} = T_{S} and i is an automorphism of T , then there is an automorphism of S′ that induces i. Also, any automorphism of S′ that fixes T induces an automorphism of T . Since any point of S′ may be expressed as the intersection of two ovoids that are points of T , any automorphism of S′ that induces the identity on T must be the identity. So, if we consider the group of S′ that fixes T acting on T by the automorphism it induces, the action is faithful and so it is the group of T .

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3.3 A GQ of Kantor, an ovoid of Kantor and a new SPG

In [31] Kantor constructed the q-clan C_{σ}, which has associated GQ S(C_{σ}) of order (q^{2}, q) (see Section 1.4.5 for details and references). In this section we investigate the connection between the GQ S(C_{σ}) and the ovoid θ_{σ} constructed by Kantor in [30] and studied in Chapter 2 of this document. The dual GQ S(C_{σ}) possesses a subquadrangle isomorphic to Q(4, q) and we show that each subtended ovoid of the Q(4, q) subquadrangle is isomorphic to θ_{σ}. We also show that the Q(4, q) subquadrangle is doubly subtended in S(C_{σ}). Since the subtended ovoids are isomorphic to θ_{σ}, Theorem 3.2.1 shows that the SPG constructed as in Theorem 3.1.7 is distinct from the Metz/Hirschfeld and Thas SPG constructed in the classical case. Hence we have a new SPG. Note that the work in this section relies heavily on the work of Payne and Rogers in [48].

Consider the q-clan

C_{σ} = \{A_{t} = ( \begin{matrix}t & 0 \\ 0 & −mt^{σ} \end{matrix} ) : t ∈ GF(q)\}

where q is an odd prime power, m a fixed non-square of GF(q) and σ ∈ Aut(GF (q)), as constructed in [31]. Recall from Section 1.4.5 that if G is the group with elements {(α, c, β) : α, β ∈ GF(q)^{2}, c ∈ GF(q)} and operation

(α, c, β) · (α′, c′, β′) = (α + α′, c + c′ + β(α′)^{T} , β + β′),

then the family of subgroups of G (of order q^{2})

A(∞) = \{(\overline{0}, 0, β) : β ∈ GF(q)^{2}\}
A(t) = {(α, αA_{t}α^{T} , αK_{t}) : α ∈ GF(q)^{2}} for t ∈ GF(q), where K_{t} = A_{t} + A_{t}^{T}
= {(α, αA_{t}α^{T} , 2αA_{t}) : α ∈ GF(q)^{2}} for t ∈ GF(q)

is a 4-gonal family for G (see Section 1.4.5) which we denote by F(C_{σ}). The family of subgroups

A^{⋆}(∞) = \{(\overline{0}, c, β) : c ∈ GF(q), α ∈ GF(q)^{2}\}
A^{⋆}(t) = {(α, c, 2αA_{t}) : c ∈ GF(q), α ∈ GF(q)^{2}} for t ∈ GF(q)

is denoted by F^{⋆}(C_{σ}). The GQ of order (q^{2}, q) constructed from the 4-gonal family F(C_{σ}) is denoted by S(C_{σ}). Note that Kantor observes that S(C_{σ}) ≅ S(C_{σ^{−1}}).

In [31] Kantor observes that if Q_{1} = (GF(q)×0)×GF(q)×(GF(q)×0) and Q_{2} = (0×GF(q))× GF(q) × (0 × GF(q)) are subgroups of G, then for i = 1 or 2, F_{i} = {A_{i}(t) = A(t) ∩ Q_{i} : t ∈ GF(q) ∪ {∞}} is a 4-gonal family for Q_{i} giving rise to an Sp(4, q) subquadrangle, that is, a subquadrangle of S(C_{σ}) isomorphic to W (q). We saw in Lemma 1.4.1 that if W is a GQ of order (r, r^{2}) with W′ a subquadrangle of order r, then each point of W, external to W′, subtends an ovoid of W′. Dually, W^{∧} is a GQ of order (r^{2}, r) with (W′)^{∧} a subquadrangle of order r and each line external to (W′)^{∧} subtends a spread of (W′)^{∧}. If W′ is doubly subtended in W, then we say that (W′)^{∧} is doubly subtended in W^{∧}. So, we are interested in the subtended spreads of the GQ determined by F_{1} and F_{2}. It can be shown using Theorem 1.4.7 ([48, IV.1.]) that there exists an isomorphism of S(C_{σ}) that maps the subquadrangle determined by F_{1} to the subquadrangle determined by F_{2}. Since this is the case, we will consider only the subquadrangle determined by F_{1}, which we will referred to as W (q) from here on.

Firstly, we show that the subtended spreads of W (q) are pairwise isomorphic. We do this by showing that the group of S(C_{σ}) that fixes W (q) is transitive on the lines of S(C_{σ}) external to W (q). The lines of S(C_{σ}) that are external to W (q) are A(∞)g, for g ∈ G, such that A(∞)g∩Q_{1} is empty, and A(t)g, for t ∈ GF(q) and g ∈ G, such that A(t)g ∩ Q_{1} is empty. The following lemma deals with the external lines of the form A(∞)g.

Lemma 3.3.1 The stabiliser of W (q) in S(C_{σ}) is transitive on lines external to W (q) that have the form A(∞)g.

Proof: Let A(∞)g and A(∞)g′ be two external lines of S(C_{σ}) external to W (q). So g = ((g_{1}, g_{2}), g_{3}, g_{4}, (g_{4}, g_{5})) with g_{2} ≠ 0 and g = ((g′_{1}, g′_{2}), g′_{3}, g_{4}, (g′_{4}, g′_{5})) with g′_{2} ≠ 0. Our aim is to find an automorphism of S(C_{σ}) that fixes W (q) and maps A(∞)g to A(∞)g′.

Recall from Theorem 1.4.6 that an automorphism of the group G naturally induces an automorphism of the GQ S(C_{σ}). We first find an automorphism of G that maps g to an element of G of the form ((g′_{1}, g′_{2}), −, (−, −)), such that the corresponding induced automorphism of S(C_{σ}) fixes W (q). For z, k ∈ GF(q) let

T_{z} : (α, c, β) ↦ ((α_{1} + z, α_{2}), c, (β_{1}, β_{2})) where α = (α_{1}, α_{2}) and β = (β_{1}, β_{2})
S_{k} : (α, c, β) ↦ (kα, k^{2}c, kβ)

Now T_{z} and S_{k} are automorphisms of G; hence induce automorphisms of S(C_{σ}) which we also denote by T_{z} and S_{k}. Clearly, T_{z} and S_{k} both fix Q_{1} and hence fix W (q). Choosing z = g′_{1}g_{2}/g′_{2} and k = g′_{2}/g_{2}, we have that

S_{k}T_{z} : g ↦ ((g′_{1}, g′_{2}), −, (−, −)),

as required. So, without loss of generality we may assume that g = ((g′_{1}, g′_{2}), g_{3}, (g_{4}, g_{5})). Let x = g′_{3} + (g_{4} − g′_{4})g′_{1} + (g_{5} − g′_{5})g′_{2}, then the automorphism Θ_{x} of G defined by

Θ_{x} : (α, c, β) ↦ (α, c + x − g_{3}, β)

maps g to ((g′_{1}, g′_{2}), x, (g_{4}, g_{5})). A straight-forward calculation shows that Θ_{x}g and g′ are in the same coset of A(∞), and so A(∞)Θx = A(∞)g′. Thus, if Θ_{x} induces the automorphism Θ_{x} of S(C_{σ}), then Θ_{x} maps A(∞)g to A(∞)g′.

□

Lemma 3.3.2 The stabiliser of W (q) in S(C_{σ}) is transitive on lines external to W (q).

Proof: First we show that the stabiliser of W (q) in S(C_{σ}) is transitive on the set {[A(t)] : t ∈ GF(q)\}. Let s ∈ GF(q) and let π be a permutation of GF(q), π : t ↦ \overline{t} = t + s. Then

A\overline{t} = A_{t+s} = ( \begin{matrix}t + s & 0 \\ 0 & −m(t + s)^{σ} \end{matrix} )

= ( \begin{matrix}t & 0 \\ 0 & −mt^{σ} \end{matrix} ) + ( \begin{matrix}s & 0 \\ 0 & −ms^{σ} \end{matrix} )
= A_{t} + A_{s} = A_{t} + A\overline{0}

So by Theorem 1.4.7 there is an automorphism Θ of S(C_{σ}) with λ = 1, σ the identity, D = I (see Theorem 1.4.7 for notation) and associated permutation π. By Theorem 1.4.7 we have that

Θ : T_{s}([A(t)]) ↦ [A(t + s)]
: T_{s}(α, c, β) ↦ (α, c + αA_{s}α^{T} , β + 2αA_{s})

Now Θ fixes W (q) and if we let s vary over GF(q), then we have the desired transitivity on {[A(t)] : t ∈ GF(q)}.

From this it can be shown that any line external to W (q), of the form A(t)g may be mapped to an external line of the form A(0)g′, for some g′. We now wish to show to construct an automorphism of S(C_{σ}) that fixes W (q) and swaps [A(∞)] and [A(0)] (and hence maps a line of the form A(0)g to one of the form A(∞)g′ and vice versa). Consider the q-clan C′_{σ} = {A_{t}^{−1} : t ∈

GF(q) \ \{0\}\} ∪ \{A_{0}\}. Now if S = ( \begin{matrix}1 & 0 \\ 0 & m \end{matrix} ) then the automorphism of G given by (α, c, β) ↦

(αS^{−1}, c, βS) induces an isomorphism from S(C′_{σ}) to S(C_{σ}) which maps ([A(t)], [A(0)], [A(∞)]) to ([A(t^{−1})], [A(0)], [A(∞)]) for t ∈ GF(q) \ {0}. Composing this isomorphism with that in Theorem 1.4.8 yields an automorphism of S(C_{σ}) mapping [A(∞)] ↔ [A(0)] and fixing W (q).

Thus any line of S(C_{σ}) external to W (q), that has the form A(t)g may be mapped to a line of the form A(∞)g′ by an automorphism of S(C_{σ}) fixing W (q). Lemma 3.3.1 then implies that the stabiliser of W (q) in S(C_{σ}) is transitive on the lines of S(C_{σ}) external to W (q).

□

We now show that each spread of W (q) subtended by a line of S(C_{σ}) external to W (q), is dual to a K1(σ) ovoid of Q(4, q). That is, under the duality from W (q) to Q(4, q) any subtended spread of W (q) is mapped to a K1(σ) ovoid of Q(4, q). Given Lemma 3.3.2, if one subtended spread of W (q) is dual to a K1(σ) ovoid, then all subtended spreads must be.

The K1(σ) ovoid of Kantor ([30]) was studied in detail in Chapter 2, but for our present purposes we require only the canonical form of the K1(σ) ovoid. We will let Q(4, q) be defined by the equation x_{0}x_{4} + x_{1}x_{3} + x_{2}^{2} = 0 and let ϕ_{σ} be the canonical K1(σ) ovoid, where

ϕ_{σ} = {(0, 0, 0, 0, 1)} ∪ {(1, y, z, −my^{σ}, −z^{2} + my^{σ+1}) for y, z ∈ GF(q)}

Here m is the same fixed non-square and σ the same automorphism of GF(q) used in the definition of C_{σ}. The ovoid ϕ_{σ} may also be written as the intersection of Q(4, q) with the variety defined by the equation mx_{1}^{σ} + x_{0}^{σ−1}x_{3} = 0.

Note that this form is slightly different to the canonical form for Q(4, q) and the K1(σ) ovoid presented in Chapter 2.

The GQ W (q) may be represented as the set of absolute points and lines of a symplectic polarity in PG(3, q). The canonical form of W (q) in PG(3, q) is given by the polarity which has associated bilinear form x_{0}y_{1}−x_{1}y_{0}+x_{2}y_{3}−x_{3}y_{2} = 0. In the following lemma we give an explicit isomorphism between the W (q) as a subquadrangle of S(C_{σ}) and the canonical representation in PG(3, q).

Lemma 3.3.3 Let F_{1} = {A_{1}(t) = A(t)∩Q_{1} : t ∈ GF(q)∪{∞}} be the 4-gonal family for W (q) as above. Let W (q)′ be the GQ arising in PG(3, q) as the set of absolute points and absolute lines of the symplectic polarity with associated bilinear form x_{0}y_{1} − x_{1}y_{0} + x_{2}y_{3} − x_{3}y_{2} = 0. Then the map ρ is an isomorphism from W (q) to W (q)′ where ρ acts as follows:

\begin{matrix}(∞) & ↦ & (0, 1, 0, 0) \\ [A_{1}(t)] & ↦ & ⟨(0, 0, 2t, 1), (0, 1, 0, 0)⟩ \\ [A_{1}(∞)] & ↦ & ⟨(0, 1, 0, 0), (0, 0, 1, 0)⟩ \\ ((g_{1}, 0), g_{2}, (g_{3}, 0)) & ↦ & (1, 2g_{2} − g_{1}g_{3}, g_{3}, g_{1}) \\ A_{1}(t)((g_{1}, 0), g_{2}, (g_{3}, 0)) & ↦ & ⟨(1, 2g_{2} − g_{1}g_{3}, g_{3}, g_{1}), (0, 2g_{1}t − g_{3}, 2t, 1)⟩ \\ A_{1}(∞)((g_{1}, 0), g_{2}, (g_{3}, 0)) & ↦ & ⟨(1, 2g_{2} − g_{1}g_{3}, g_{3}, g_{1}), (0, g_{1}, 1, 0)⟩ \\ A_{1}^{⋆}(t)((g_{1}, 0), g_{2}, (g_{3}, 0)) & ↦ & (0, 2tg_{1} − g_{3}, 2t, 1) \\ A_{1}^{⋆}(∞)((g_{1}, 0), g_{2}, (g_{3}, 0)) & ↦ & (0, 2g_{1}, 1, 0) \end{matrix}

where t, g_{1}, g_{2}, g_{3} ∈ GF(q).

Theorem 3.3.4 Let W (q) be the subquadrangle of S(C_{σ}) given by F_{1}. Then, each spread of W (q) subtended by a line of S(C_{σ}) external to W (q) is dual to a K1(σ) ovoid of Q(4, q).

Proof: Given Lemma 3.3.2, we may take our favourite fixed line of S(C_{σ}) external to W (q) and find its subtended spread S. Let ℓ = A(∞)g be the external line where g = ((0, 1), 0, (0, 0)). Then the points on A(∞)g are

{((0, 1), u_{2}, (u_{1}, u_{2})) : u_{1}, u_{2} ∈ GF(q)} ∪ {A^{⋆}(∞)g}

where A^{⋆}(∞)g = {((0, 1), u_{3}, (u_{4}, u_{5})) : u_{3}, u_{4}, u_{5} ∈ GF(q)}.

For each point on A(∞)g there is a unique line of W (q) incident with it, which is a line of the subtended spread. For the point A^{⋆}(∞) the corresponding line is [A(∞)]. Now let h(u_{1}, u_{2}) = ((0, 1), u_{2}, (u_{1}, u_{2})) for u_{1}, u_{2} ∈ GF(q), then for each pair (u_{1}, u_{2}) we need to find t ∈ GF(q) such that A(t)h(u_{1}, u_{2}) is a line of W (q). Now

A(t)h(u_{1}, u_{2}) = {((v_{1}, v_{2} + 1), v_{1}^{2}t − mv_{2}^{2}t^{σ} + u_{2} − 2mv_{2}t^{σ}, (2v_{1}t + u_{1}, −2mv_{2}t^{σ} + u_{2}))
: v_{1}, v_{2} ∈ GF(q)}

and is a line of W (q) if it contains any point of W (q) (in which case it contains q + 1 points of W (q)). This occurs if, as a coset of A(t) in G, A(t)h(u_{1}, u_{2}) contains an elements of Q_{1} (in which case it contains q elements of Q_{1}). So v_{2} = −1 and −2mv_{2}t^{σ} = −u_{2} which implies

t = − (\frac{u_{2}}{2m})^{σ^{−1}}.

Thus the spread of W (q) subtended by S(C_{σ}) has the form:

S = \{ \begin{matrix}[A(∞)] \\ A (− (\frac{u_{2}}{2m})^{σ^{−1}}) h(u_{1}, u_{2}) & u_{1}, u_{2} ∈ GF(q) \end{matrix}

To express the spread in the group coset representation of W (q) without reference to S(C_{σ}),

we need a representative of the coset A (− (\frac{u_{2}}{2m})^{σ^{−1}}) h(u_{1}, u_{2}) that is in Q_{1}, say ((0, 0), u_{2} +

2mt^{σ}, (u_{1}, 0)) = ((0, 0), u_{2}/2, (u_{1}, 0)). Thus the spread is

S = \{ \begin{matrix}[A(∞)] \\ A (− (\frac{u_{2}}{2m})^{σ^{−1}}) ((0, 0), \frac{u_{2}}{2} , (u_{1}, 0)) & u_{1}, u_{2} ∈ GF(q). \end{matrix}

By using the isomorphism in Lemma 3.3.3, S has the following form in W (q)′:

S = \{ \begin{matrix}\{x_{0} = 0; x_{3} = 0\} = ⟨(0, 1, 0, 0), (0, 0, 1, 0)⟩ \\ ⟨(1, u_{2}, u_{1}, 0), (0, −u_{1}, 2t, 1)⟩, & t = − (\frac{u_{2}}{2m})^{σ^{−1}} for u_{1}, u_{2} ∈ GF(q). \end{matrix}

We use Plucker coordinates and the Klein correspondence (see [25, Chapter 15]) to give a duality from W (q)′ to Q(4, q). Thus, in Q(4, q) the spread S becomes an ovoid θ, say, which has the form:

θ = \{ \begin{matrix}(0, 0, 0, 0, 1) \\ (1, 2t, −u_{1}, u_{2}, −2u_{2}t − u_{1}^{2}), & t = − (\frac{u_{2}}{2m})^{σ^{−1}} for u_{1}, u_{2} ∈ GF(q). \end{matrix}

Now since

m(2t)^{σ} + 1^{σ−1}u_{2} = m (2 (\frac{−u_{2}}{2m} )^{σ^{−1}})^{σ} + u_{2}
= m (\frac{−u_{2}}{m} ) + u_{2}
= 0,

it follows that every point of θ satisfies the equation mx_{1}^{σ} + x_{0}^{σ−1}x_{3} = 0. Thus θ is the canonical K1(σ) ovoid ϕ_{σ}.

□

Theorem 3.3.5 There is an involution of S(C_{σ}) that fixes W (q).

Proof: Consider the automorphism Θ_{D} = Θ(π, λ, ρ, D) of S(C_{σ}), using the notation of Theorem 1.4.7, where π is the identity permutation on GF(q), λ = 1, ρ the identity automorphism

of GF(q) and D = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) ∈ GL(2, q). Then

Θ_{D} : (α, c, β) ↦ ((α_{1}, −α_{2}), c, (β_{1}, −β_{2})) where α = (α_{1}, α_{2}) and β = (β_{1}, β_{2}).

Thus Θ_{D} is an involution of S(C_{σ}) that fixes W (q).

□

Corollary 3.3.6 The SPG constructed by the GQ S(C_{σ})^{∧} and the subGQ isomorphic to Q(4, q), as in Theorem 3.1.7, is not isomorphic to the known SPG with parameters s = q − 1, t = q^{2}, α = 2 and μ = 2q(q − 1).

Proof: The GQ S(C_{σ})^{∧} of order (q, q^{2}) has a subquadrangle of order q isomorphic to Q(4, q) such that there is a collineation of S(C_{σ})^{∧} that fixes this subquadrangle pointwise. Thus, by Corollary 3.1.8 we have an SPG and since each subtended ovoid of the subquadrangle is nonclassical, by Theorem 3.2.1 it is not isomorphic to the known SPG of that order.

□

It would be interesting to determine the configuration in Q(4, q) of the special points of the K1(σ) ovoids that are points of the SPG (that is, subtended in Q(4, q) by S(C_{σ})^{∧}). One possible conjecture is that the special points form a line ℓ of Q(4, q) and that the set of rosettes with base point P ∈ ℓ are the elation rosettes with special tangent line in the polar plane of ℓ (with respect to Q(4, q)). This sort of information would be important in using the techniques of Section 4.2 to try to characterise the GQ S(C_{σ})^{∧} by the Q(4, q) subquadrangle.

3.4 SPG from q-clan GQs, q even

Let S be a GQ of order (q^{2}, q), q even, constructed from a q-clan, as in Section 1.4.5 and reprised in Section 3.3. For any such GQ S Payne ([45]) has constructed a family of subquadrangles, {S_{α} : α ∈ GF(q)}, of order q. In this section we investigate whether any of the subquadrangles S_{α} is doubly subtended in S and so leads to an SPG via the construction in Theorem 3.1.7 (note that we are considering the dual of the situation in Theorem 3.1.7). In Section 3.4.1 we determine the algebraic conditions on the q-clan under which S_{α} is doubly subtended in S. In Section 3.4.2 we use the algebraic conditions generated in Section 3.4.1 to show that of the known q-clan GQs, q even, only in the classical case S ≅ Q(5, q), is any S_{α} doubly subtended in S.

3.4.1 Algebraic conditions

In this section we outline the construction of the subquadrangles S_{α} and then look for an automorphism of S that fixes S_{α}. By Lemma 3.1.5 such an automorphism exists if and only if S_{α} is doubly subtended in S.

The following derivation of the subquadrangles S_{α} is taken from [46].

For q = 2^{e} a q-clan C may be normalised to a q-clan C′ giving a GQ isomorphic to that given by C, such that C′ has the form {A_{t} : t ∈ GF(q)}, where

A_{0} = ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} ); A_{1} = ( \begin{matrix}1 & 1 \\ 0 & δ \end{matrix} ) (with trace(δ) = 1); and A_{t} = ( \begin{matrix}x_{t} & t \\ 0 & y_{t} \end{matrix} ) , t ∈ GF(q).

From this point we will assume that a q-clan C has the above normalised form.

We now introduce a group isomorphic to G where the subgroups generating the subGQs will be easier to recognise. In Section 1.4.5 4-gonal families are constructed for the group

G = {(α, c, β) : α, β ∈ GF(q)^{2}, c ∈ GF(q)}

with the group operation

(α, c, β) · (α′, c′, β′) = (α + α′, c + c′ + β(α′)^{T} , β + β′).

The subgroups of the 4-gonal family are then:

A(∞) = {(0, 0, β) : β ∈ GF(q)^{2}}
A(t) = (α, αA_{t}α^{T} , αK_{t}) : α ∈ GF(q)^{2}}
A^{⋆}(∞) = \{(0, c, β) ∈ \overline{G} : c ∈ GF(q), β ∈ GF(q)^{2}\}
A^{⋆}(t) = {α, c, αK_{t}) : α ∈ GF(q)^{2}}
where K_{t} = A_{t} + A_{t}^{T} .

Now, let E = GF(q^{2}), such that E = GF(q)(μ) where μ^{2} + μ + δ (where δ is an element of GF(q) with trace(δ) = 1). We will associated the element a + bμ of E with the element (a, b) of GF(q^{2}). Let x ↦ \overline{x} = x^{q} and define α ∘ β = α\overline{β} + \overline{α}β. We now define a new group

\overline{G} = \{(α, c, β) : α, β ∈ E, c ∈ GF(q)\}

with operation

(α, c, β) · (α′, c′, β′) = (α + α′, c + c′ + \sqrt{β ∘ α′}, β + β′).

If P = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ), then ζ : G → \overline{G} : (α, c, β) ↦ (α, \sqrt{c}, βP) is an isomorphism with ζ^{−1} : \overline{G} →

G : (α, c, β) ↦ (α, c^{2}, βP).

Thus the q-clan C gives the following 4-gonal family \overline{F}.

\begin{aligned}\overline{A}(∞) & = & \{(0, 0, β) : β ∈ E\} \\ \overline{A}(t) & = & (α, \sqrt{αA_{t}α^{T}} , y_{t}α) : α ∈ E\} \\ \overline{A}^{⋆}(∞) & = & \{(0, c, β) : c ∈ GF(q), β ∈ E\} \\ \overline{A}^{⋆}(t) & = & \{α, c, y_{t}α) : α ∈ E\} \end{aligned}

Consider the following subgroup of \overline{G}:

\overline{G}_{α} = \{(aα, c, bα) ∈ G : a, b, c ∈ GF(q)\} where α ∈ E

The group operation of \overline{G} restricts to \overline{G}_{α} in the following way:

(aα, c, bα) · (a′α, c′, b′α) = (aα + a′α, c + c′ + \sqrt{bα ∘ a′α}, bα + b′α)
= ((a + a′)α, c + c′, (b + b′)α)

So \overline{G}_{α} is an elementary abelian group of order q^{3}, and in fact \overline{G}_{α} ≅ GF(q)^{3} under componentwise addition.

Now consider the following subgroups of \overline{G}_{α}:

\overline{A}_{α}(∞) = A(∞) ∩ G_{α} = \{(0, 0, bα) : b ∈ GF(q)\}
\overline{A}_{α}(t)  = A(t) ∩ G_{α} = \{(aα, a\sqrt{αA_{t}α^{T}} , atα) : a ∈ GF(q)\}
\overline{A}_{α}^{⋆}(∞) = A^{⋆}(∞) ∩ G_{α} = \{(0, c, bα) : b, c ∈ GF(q)\}
\overline{A}_{α}^{⋆}(t) = A^{⋆}(t) ∩ G_{α} = \{(aα, c, atα) : a, c ∈ GF(q)\}.

The family \overline{F}_{α} = \{\overline{A}_{α}(t) : t ∈ GF(q) ∪ \{(∞)\}\} is a 4-gonal family for a subGQ, which we will denote by S_{α}.

At this point we make the following definition from [48]

Definition 3.4.1 Let A, B ∈ GL(2, q) such that A = ( \begin{matrix}x & y \\ z & w \end{matrix} ) and B = ( \begin{matrix}a & b \\ c & d \end{matrix} ). Then

A ≡ B if x = a, y = d, z + y = b + c.

Given this definition it can be shown that, for q even, [48, IV.1] is equivalent to the following theorem.

Theorem 3.4.2 Let C = \{A_{t} : t ∈ GF(q)\} be a q-clan with A_{0} = ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} ). Let Θ be a

collineation of the GQ S = S(G, F(C)) derived from C which fixes the point (∞), the line [A(∞)] and the point (0, 0, 0). Then the following must exist:

(i) A permutation π : t ↦ t′ of F.

(ii) λ ∈ GF(q), λ ≠ 0.

(iii) σ ∈ Aut(GF(q)).

(iv) D ∈ GL(2, q) for which A_{t′} ≡ λD^{T} A_{t}^{σ}D − A_{0′}, for all t ∈ GF(q).

Conversely, given σ, D, λ and a permutation π : t ↦ t′, satisfying the above conditions, the following automorphism Θ of G induces a collineation of S, fixing (∞),[A(∞)] and (0, 0, 0).

Θ = Θ(π, λ, σ, D) : G → G :

(α, c, β) ↦ (λ^{−1}α^{σ}D^{−T} , λ^{− \frac{1}{2}} c^{σ} + λ^{−1}\sqrt{α^{σ}D^{−T} A_{0′}D^{−1}(α^{σ})^{T}} , β^{σ}P DP + λ^{−1}y_{0′}α^{σ}D^{−T}).

Theorem 3.4.3 Let C = {A_{t} : t ∈ GF(q)}, q even, be a q-clan normalised as follows:

A_{0} = ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} ); A_{1} = ( \begin{matrix}1 & 1 \\ 0 & δ \end{matrix} ) (with trace(δ) = 1); and A_{t} = ( \begin{matrix}x_{t} & t \\ 0 & y_{t} \end{matrix} ) , t ∈ GF(q).

Let S be the GQ of order (q^{2}, q) defined by C and let S_{α} be the subquadrangle of S defined by the subgroup \overline{G}_{α} of \overline{G}, where α = (α_{1}, α_{2}). Then S_{α} defines an SPG, as in Theorem 3.1.7, if and only if

α_{1}^{2}x_{t} + α_{2}^{2}y_{t} + (α_{1}^{2} + δα_{2}^{2})t = 0 for all t ∈ GF(q) \ {0, 1}.

Proof: By Lemma 3.1.5, it suffices to find conditions under which S admits a non-identity involution Θ that fixes the subGQ S_{α} pointwise.

Now (∞), [A(∞)] and (0, 0, 0) are all contained in the subGQ S_{α} and so Θ must fix these. Thus Θ has the form of the automorphisms of S given by Theorem 3.4.2. Using the notation of Theorem 3.4.2 we have that, A_{0′} = A_{0} (since [A(0)] ∈ S_{α} and so if fixed) and A_{t′} ≡ λD^{T} A_{t}^{σ}D, for some λ ∈ GF(q), λ ≠ 0, σ ∈ Aut(GF(q)) and D ∈ GL(2, q). Since S_{α} is fixed by θ it follows that A_{t′} ≡ A_{t} and so

A_{t} ≡ λD^{T} A_{t}^{σ}D  (3.4.2)

The subgroup G_{α} = {(aα, c, bα) : a, b, c ∈ GF(q)} is fixed element-wise by Θ. But

Θ : (0, c, 0) ↦ (0, λ^{− \frac{1}{2}} c^{σ}, 0),

thus λ^{− \frac{1}{2}} c^{σ} = c for all c ∈ GF(q). Letting c = 1 implies that λ = 1 and so c^{σ} = c, that is σ fixes GF(q) and hence σ is the identity automorphism on GF(q).

Now we know that Θ : (α, c, β) ↦ (αD^{−T} , c, βP DP) so D^{2} is the identity matrix, I_{2} (since

Θ is an involution). We also know that A_{t} ≡ D^{T} A_{t}D. If D = ( \begin{matrix}a & b \\ c & d \end{matrix} ), then D^{2} = I_{2} gives us

the following equations

a^{2} + bc = 1  (3.4.3)
ab + bd = 0  (3.4.4)
ac + cd = 0  (3.4.5)
bc + d^{2} = 1  (3.4.6)

As D is non-singular, equations 3.4.3 and 3.4.4 imply that a = d, in which case equations 3.4.4 and 3.4.5 are satisfied.

Now

D^{T} A_{1}D = ( \begin{matrix}a & c \\ b & d \end{matrix} ) ( \begin{matrix}1 & 1 \\ 0 & δ \end{matrix} ) ( \begin{matrix}a & b \\ c & d \end{matrix} )

= ( \begin{matrix}a & a + cδ \\ b & b + dδ \end{matrix} ) ( \begin{matrix}a & b \\ c & d \end{matrix} )

= ( \begin{matrix}a^{2} + ac + c^{2}δ & ab + ad + cdδ \\ ab + bc + cdδ & b^{2} + bd + d^{2}δ \end{matrix} )

and so by 3.4.2 we have

a^{2} + ac + c^{2}δ = 1  (3.4.7)
b^{2} + bd + d^{2}δ = δ  (3.4.8)
(ab + ad + cdδ) + (ab + bc + cdδ) = 1
⇒ ad + bc = |D| = 1  (3.4.9)

Note that since a = d 3.4.3, 3.4.6 and 3.4.9 are equivalent.

Since Θ fixes S_{α} we have that

Θ : (aα, c, bα) ↦ (aαD^{−T} , c, bαP DP) = (aα, c, bα),

which gives us the following equations:

αD^{−T} = α
αP DP = α.

Now

D^{−1} = \frac{1}{ad + bc} ( \begin{matrix}d & b \\ c & a \end{matrix} ) = ( \begin{matrix}d & b \\ c & a \end{matrix} ) by 3.4.9

and so D^{−T} = ( \begin{matrix}d & c \\ b & a \end{matrix} ).

Now P DP = D^{−T}, and so the two equations in D and α are identical, giving us the equations

dα_{1} + bα_{2} = α_{1}  (3.4.10)
cα_{1} + aα_{2} = α_{2},  (3.4.11)

where α = (α_{1}, α_{2}). Now if α_{1} = 0 then α_{2} ≠ 0 and it follows that b = 0, a = d = 1 and c = 0

or 1/δ. If c = 0, then D = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) and Θ is the identity automorphism of S.

If α_{1} ≠ 0 and α_{2} = 0, then it follows that c = 0, a = d = 1 and b = 0 or 1. In the case

where b = 0, D = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) and Θ is the identity automorphism of S.

Now if α_{1}, α_{2} ≠ 0, then we may rewrite 3.4.10 and 3.4.11 as follows

a + b\frac{α_{2}}{α_{1}} = 1  (3.4.12)
c\frac{α_{1}}{α_{2}} + a = 1,  (3.4.13)

and so b = cα_{1}^{2}/α_{2}^{2}. Substituting this value for b into equation 3.4.3 yields a = d = 1 + cα_{1}/α_{2}. Given these formulae for a, b, d in terms of c, we may rewrite 3.4.7 as

1 + c^{2} \frac{α_{1}^{2}}{α_{2}^{2}} + c + c^{2} \frac{α_{1}}{α_{2}} + c^{2}δ = 1
⟺ c (1 + c (\frac{α_{1}^{2}}{α_{2}^{2}} + \frac{α_{1}}{α_{2}} + δ)) = 0.  (3.4.14)

Note that since trace(δ) = 1, it follows that x^{2}+x+δ is irreducible, and so α_{1}^{2}/α_{2}^{2}+α_{1}/α_{2}+δ ≠ 0. Equation 3.4.14 is satisfied if and only if c = 0 or

c = \frac{1}{\frac{α_{1}^{2}}{α_{2}^{2}} + \frac{α_{1}}{α_{2}} + δ} = \frac{α_{2}^{2}}{α_{1}^{2} + α_{1}α_{2} + δα_{2}^{2}} .

Rewriting 3.4.8 yields

c^{2} \frac{α_{1}^{4}}{α_{2}^{4}} + c\frac{α_{1}^{2}}{α_{2}} (1 + c\frac{α_{1}}{α_{2}} ) + (1 + c^{2} \frac{α_{1}^{2}}{α_{2}^{2}} ) δ = δ

⟺  c^{2} \frac{α_{1}^{4}}{α_{2}^{4}} + c\frac{α_{1}^{2}}{α_{2}^{2}} + c^{2} \frac{α_{1}^{3}}{α_{2}^{3}} + c^{2}δ \frac{α_{1}^{2}}{α_{1}^{2}}  = 0

⟺  c (1 + c (\frac{α_{1}^{2}}{α_{2}^{2}} + \frac{α_{1}}{α_{2}} + δ))  = 0,

which is precisely 3.4.14.

If c = 0, then b = 0 and a = d = a, that is D is the identity matrix and θ the identity automorphism of S. If c = α_{2}^{2}/(α_{1}^{2} + α_{1}α_{2} + δα_{2}^{2}), then

b = \frac{α_{1}^{2}}{α_{1}^{2} + α_{1}α_{2} + δα_{2}^{2}} , a = d = 1 + \frac{α_{1}α_{2}}{α_{1}^{2} + α_{1}α_{2} + δα_{2}^{2}} = \frac{α_{1}^{2} + δα_{2}^{2}}{α_{1}^{2} + α_{1}α_{2} + δα_{2}^{2}} .

Now notice that these formulae also give the previously determined values for a, b, c and d in the case where α_{1} = 0, α_{2} ≠ 0 and D is not the identity matrix and the case where α_{1} ≠ 0, α_{2} = 0 and D is not the identity matrix.

We now need to check whether when the possibilities for D that we have calculated satisfy the property 3.4.2.

D^{T} A_{t}D
= \frac{1}{(α_{1}^{2} + α_{1}α_{2} + δα_{2}^{2})^{2}} ( \begin{matrix}α_{1} + δα_{2}^{2} & α_{2}^{2} \\ α_{1}^{2} & α_{1}^{2} + δα_{2}^{2} \end{matrix} ) ( \begin{matrix}x_{t} & t \\ 0 & y_{t} \end{matrix} ) ( \begin{matrix}α_{1} + δα_{2}^{2} & α_{1}^{2} \\ α_{2}^{2} & α_{1}^{2} + δα_{2}^{2} \end{matrix} )

= \frac{1}{(α_{1}^{2} + α_{1}α_{2} + δα_{2}^{2})^{2}} ( \begin{matrix}(α_{1}^{2} + δα_{2}^{2})x_{t} & (α_{1}^{2} + δα_{2}^{2})t + α_{2}^{2}y_{t} \\ α_{1}^{2}x_{t} & α_{1}^{2}t + (α_{1}^{2} + δα_{2}^{2})y_{t} \end{matrix} ) ( \begin{matrix}α_{1}^{2} + δα_{2}^{2} & α_{1}^{2} \\ α_{2}^{2} & α_{1}^{2} + δα_{2}^{2} \end{matrix} )

= \frac{1}{(α_{1}^{2} + α_{1}α_{2} + δα_{2}^{2})^{2}} ×

( \begin{matrix}(α_{1}^{4} + δ^{2}α_{2}^{4})x_{t} + α_{2}^{4}y_{t} + (α_{1}^{2} + δα_{2}^{2})α_{2}^{2}t & (α_{1}^{2} + δα_{2}^{2})α_{1}^{2}x_{t} + (α_{1}^{4} + δ^{2}α_{2}^{4})t + (α_{1}^{2} + δα_{2}^{2})α_{2}^{2}y_{t} \\ (α_{1}^{2} + δα_{2}^{2})α_{1}^{2}x_{t} + α_{1}^{2}α_{2}^{2}t + (α_{1}^{2} + δα_{2}^{2})α_{2}^{2}y_{t} & α_{1}^{4}x_{t} + (α_{1}^{4} + δ^{2}α_{2}^{4})y_{t} + (α_{1}^{2} + δα_{2}^{2})α_{1}^{2}t \end{matrix} )

Now D^{T} A_{t}D ≡ A_{T} if and only if the following three equations are satisfied

x_{t} = \frac{(α_{1}^{4} + δ^{2}α_{2}^{4})x_{t} + α_{2}^{4}y_{t} + (α_{1}^{2} + δα_{2}^{2})tα_{2}^{2}}{(α_{1}^{2} + α_{1}α_{2} + δα_{2}^{2})^{2}}

y_{t} = \frac{α_{1}^{4}x_{t} + (α_{1}^{4} + δ^{2}α_{2}^{4})y_{t} + (α_{1}^{2} + δα_{2}^{2})α_{1}^{2}t}{(α_{1}^{2} + α_{1}α_{2} + δα_{2}^{2})^{2}}

t = \frac{(α_{1}^{4} + α_{1}^{2}α_{2}^{4} + δ^{2}α_{2}^{4})t}{(α_{1}^{2} + α_{1}α_{2} + δα_{2}^{2})^{2}} = t

Note that the third of these equations always holds. Rewriting the first of the three equations yields

x_{t}(α_{1}^{4} + α_{1}^{2}α_{2}^{2} + δ^{2}α_{2}^{4}) + (α_{1}^{4} + δ^{2}α_{2}^{4})x_{t} + α_{2}^{4}y_{t} + α_{2}^{2}(α_{1}^{2} + δα_{2}^{2})t = 0
⟺  α_{1}^{2}α_{2}^{2}x_{t} + α_{2}^{4}y_{t} + α_{2}^{2}(α_{1}^{2} + δα_{2}^{2})t = 0
⟺  α_{2}^{2}(α_{1}^{2}x_{t} + α_{2}^{2}y_{t} + (α_{1}^{2} + δα_{2}^{2})t) = 0

and similarly the second of the three equations may be rewritten as

α_{1}^{2}(α_{1}^{2}x_{t} + α_{2}^{2}y_{t} + (α_{1}^{2} + δα_{2}^{2})t) = 0.

Now if α_{1} ≠ 0, then the second equation yields

α_{1}^{2}x_{t} + α_{2}^{2}y_{t} + (α_{1}^{2} + δα_{2}^{2})t = 0

and if α_{2} ≠ 0, then the first equation also gives this. Thus the conditions that we require are

α_{1}^{2}x_{t} + α_{2}^{2}y_{t} + (α_{1}^{2} + δα_{2}^{2})t = 0 for all t ∈ GF(q) \ {0, 1}.  (3.4.15)

Note that the normalised form of the q-clan ensures that the equation in 3.4.15 is always satisfied for t = 0 and t = 1.

□

3.4.2 Examples

In this section we consider the examples of q-clan GQs, q even. We show that if S is a known example that has a doubly subtended subquadrangle S_{α}, for some α ∈ GF(q)^{2} \ {(0, 0)}, then S is the classical GQ, that is S ≅ H(3, q^{2}) ≅ Q(5, q)^{∧}. In this case, S_{α} is doubly subtended for all α ∈ GF(q)^{2} \ {(0, 0)}.

Classical (see [44] for example): The GQs exist for all q = 2^{e}. The q-clan is

\{( \begin{matrix}t & t \\ 0 & at \end{matrix} ) : t ∈ GF(q)\} where trace(a) = 1,

and so the condition on α in Theorem 3.4.3 is equivalent to the polynomial in t given by

α_{1}^{2}t + α_{2}^{2}at + (α_{1}^{2} + aα_{2}^{2})t

begin identically zero for (α_{1}, α_{2}) ≠ (0, 0). This is clearly always the case.

Fisher-Thas-Walker (see [63]): The GQs exist for q ≡ −1 (mod 3), so for the q even case this is q = 2^{e}, e odd and e > 1. The q-clan is

\{( \begin{matrix}\sqrt{t} & t \\ 0 & \sqrt{t^{3}} \end{matrix} ) : t ∈ GF(q)\} ,

that is x_{t} = \sqrt{t}, y_{t} = \sqrt{t^{3}} and δ = 1, using the notation of Theorem 3.4.3. The condition for the existence of an SPG is that the polynomial in t given by

α_{1}^{2}\sqrt{t} + α_{2}^{2}\sqrt{t^{3}} + (α_{1}^{2} + α_{2}^{2})t

is identically zero for fixed α_{1}, α_{2}, (α_{1}, α_{2}) ≠ (0, 0). Clearly, this is never the case.

Payne [45]: The GQs exist for q = 2^{e}, e odd and e > 1. The q-clan is

\{( \begin{matrix}t^{1/3} & t \\ 0 & t^{1/5} \end{matrix} ) : t ∈ GF(q)\} ,

and so the condition from Theorem 3.4.3 is equivalent to the polynomial in t given by

α_{1}^{2}t + α_{2}^{2}t^{5} + (α_{1}^{2} + α_{2}^{2})t^{3}

being identically zero for (α_{1}, α_{2}) ≠ 0, which is clearly never the case.

Subiaco [13]: The GQs exist for q = 2^{e} and q ≥ 8. The q-clan is given by

\{( \begin{matrix}f(t)^{2} & t \\ 0 & a^{2}g(t)^{2} \end{matrix} ) : t ∈ GF(q)\} ,

where

f(t) = \frac{d^{2}(t^{4} + t) + d^{2}(1 + d + d^{2})(t^{3} + t^{2})}{(t^{2} + dt + 1)^{2}} + t^{1/2},
g(t) = \frac{d^{4}t^{4} + d^{3}(1 + d^{2} + d^{4})t^{3} + d^{3}(1 + d^{2})t}{(d^{2} + d^{5} + d^{1/2})(t^{2} + dt + 1)^{2}} + \frac{d^{1/2}}{d^{2} + d^{5} + d^{1/2}} t^{1/2}, and
a = \frac{d^{2} + d^{5} + d^{1/2}}{d(1 + d + d^{2})} ,

with d ∈ GF(q) such that d^{2} + d + 1 ≠ 0 and trace(1/d) = 1.

The condition from Theorem 3.4.3 becomes

α_{1}^{2}f(t)^{2} + α_{2}^{2}a^{2}g(t)^{2} + (α_{1}^{2} + δα_{2}^{2})t = 0 for all t ∈ GF(q),

which is equivalent to the polynomial in t given by

[(\frac{dt^{4} + d^{3}(1 + d^{2} + d^{4})t^{3} + d^{3}(1 + d^{2})t}{d(1 + d + d^{2})})^{2} + \frac{td(t^{2} + dt + 1)^{4}}{d^{2}(1 + d + d^{2})^{2}} ] α_{2}^{2}

+ [(d^{2}(t^{4} + t) + d^{2}(1 + d + d^{2})(t^{3} + t^{2}))^{2} + t(t^{2} + dt + 1)^{4}] α_{1}^{2} + (α_{1}^{2} + δα_{2}^{2})t(t^{2} + dt + 1)^{4}

being identically zero. If we equate the coefficient of t^{2} to zero we obtain

\begin{aligned}& α_{1}^{2}(d^{4}) + α_{2}^{2} \frac{d^{6}(1 + d^{4})}{d^{2}(1 + d^{2} + d^{4})} & = & 0 \\ ⟺ & α_{1}^{2}(1 + d^{2} + d^{4}) + α_{2}^{2}(1 + d^{4}) & = & 0 \end{aligned}

and by equating the coefficient of t^{6} to zero we obtain

α_{1}^{2}d^{4}(1 + d^{2} + d^{4}) + α_{2}^{2} \frac{d^{6}(1 + d^{4} + d^{8})}{d^{2}(1 + d + d^{2})} = 0
⟺  (α_{1}^{2} + α_{2}^{2})(1 + d^{2} + d^{4}) = 0
⟺  α_{1} = α_{2}.

Substituting α_{1} = α_{2} into the t^{2} coefficient equation yields α_{1}^{2}d^{2} = 0, which means that α_{1} = α_{2} = 0.

Characterisations of GQs of order (r, r^{2})

Let S = (P, B, I) be a GQ of order (r, r^{2}) and S′ = (P′, B′, I) a subquadrangle of S of order r. In Chapter 3 (Lemma 3.1.1) we observed that every point of P \ P′ is external to S′ and so subtends an ovoid of S′, while every line of B \ B′ is tangent to S′ and so subtends a rosette of ovoids of S′ (see Section 1.4.1 for definitions). Two questions naturally arise at this point. The first is: given S′, a GQ of order r what ovoid/rosette structures of S′ may be subtended by a GQ S of order (r, r^{2}) that contains σ′ as a subquadrangle? This is particularly interesting in the light of Theorem 3.1.10 which says that in the case where S′ is doubly subtended in S (that is, each ovoid of S′ that is subtended by S is subtended by exactly two points of S), that S can be reconstructed from S′, the subtended ovoid/rosette structure (an SPG in this case) and an algebraic 2-fold cover of the subtended ovoid/rosette structure. If we choose the ‘right’ ovoid/rosette structure of S′ can we ‘naturally’ construct a GQ S of order (r, r^{2}) from S′ and this ovoid/rosette structure?

The second question is: given S, a GQ of order (r, r^{2}) containing a subquadrangle S′ of order r, does the subtended ovoid/rosette structure of S′ characterise the GQ S, or is it possible to find another GQ \overline{S} of order (r, r^{2}), not isomorphic to S, but with S′ as a subquadrangle and subtending the same ovoid/rosette structure in S′ as S? A result in this direction, which served as the inspiration for the work in this chapter, is the following theorem due to Thas and Payne [65, VII.1].

Theorem 4.0.4 Let S = (P, B, I) be a GQ of order (q, q^{2}), q even, having a subquadrangle S′ isomorphic to Q(4, q). If in S′ each ovoid O_{x} consisting of all of the points collinear with a given point x of S \ S′ is an elliptic quadric, then S is isomorphic to Q(5, q).

In the first section of this chapter we consider the question: which ovoid/rosette structures of W (q), q even, (recall W (q) is isomorphic to Q(4, q) for q even) may be subtended by a GQ of order (q, q^{2})? As a result we are able to strengthen the characterisation of Theorem 4.0.4.

In the second section of this chapter we consider a GQ S of order (r, r^{2}) containing a doubly subtended subquadrangle S′ of order r. We formulate the question of whether S can be characterised by S′ and the associated SPG (a generalisation of the case Q(4, q) ⊂ Q(5, q) in Theorem 4.0.4) as a cohomological problem, the solution of which allows us to prove the equivalent result to Theorem 4.0.4 for q odd.

4.1 Characterisations of Q(5, q), q even

Recall that for q even the GQ W (q) is isomorphic to Q(4, q) ([49, 3.2.1]). If θ is an ovoid of W (q), then θ is also an ovoid of the ambient PG(3, q) of W (q), and if θ′ is an ovoid of PG(3, q), then there is an automorphism T of PG(3, q) such that T (θ′) is an ovoid of W (q) ([55]). Note that it is not true in general that any ovoid of PG(3, q) is an ovoid of W (q).

For q = 2^{e}, e even, the only known ovoid of W (q) is the elliptic quadric ovoid of PG(3, q) while for q = 2^{e}, e odd, there is also the Tits ovoid (see Section 1.4.6 for the explicit form of the Tits ovoid).

Now if Q(4, q) is the non-singular quadric of PG(4, q), P a point of Q(4, q) and Σ a hyperplane of PG(4, q) not containing P, then the projection of points and lines of Q(4, q) from P onto Σ is the geometry W (q) and Σ is the ambient PG(3, q) of this W (q) (see the proof of [49, 3.2.1]). From this isomorphism it follows that an elliptic quadric ovoid of Q(4, q) is isomorphic to an elliptic quadric ovoid of W (q). Thus Theorem 4.0.4 is equivalent to the following.

Theorem 4.1.1 Let S = (P, B, I) be a GQ of order (q, q^{2}), q even, having a subquadrangle S′ = (P′, B′, I) isomorphic to W (q). If in S′ each ovoid O_{X} subtended by a point X ∈ P \ P′ is an elliptic quadric ovoid, then S is isomorphic to Q(5, q).

4.1.1 The intersection and subtending of ovoids of W (q), q = 2^{e}, e odd

In this section we look at the size of the intersection of a given Tits ovoid of W (q) with elliptic quadrics of W (q). We show that if W (q) is a subquadrangle of a GQ S of order (q, q^{2}) then the set of ovoids of W (q) subtended by S cannot contain both a Tits ovoid and an elliptic quadric and cannot be all Tits ovoids.

Theorem 4.1.2 [3, Theorem 1(a)] The intersection of an elliptic quadric of W (q) with a Tits ovoid of W (q) consists of either q + \sqrt{2q} + 1 points or q − \sqrt{2q} + 1 points.

For the proof of the following theorem recall the preliminary section on graphs, Section 1.1.

Theorem 4.1.3 [50] Let W (q) be a subquadrangle of a GQ S of order (q, q^{2}). Let N be the graph with vertices the set of ovoids of W (q) subtended by S and with two vertices adjacent if their corresponding ovoids meet in exactly one point. Then N is connected.

Proof: Consider the graph M with vertices the points of S not in W (q) and two vertices adjacent if their corresponding points are collinear in S. Let P and Q be two vertices of M. If P and Q are collinear in S, then they are adjacent and therefore connected. Otherwise there exist two lines of S, l and m, on P and Q respectively, such that l and m meet W (q) in P′ and Q′ respectively. We can assume that P′ ≠ Q′. By the third GQ axiom we may choose R ∈ m such that R is not collinear with P′ in S. Also by the third GQ axiom there exists a point T ∈ l, T ≠ P′ such that T is collinear with R in S. Thus P, T, R, Q is a path of M and hence P and Q are connected. We have shown, therefore, that the graph M is connected.

Now we consider the graph N. Let θ and ϕ be two ovoids of W (q) subtended by points P and Q, respectively. If P and Q in S are collinear, then by Lemma 3.1.1 the line z joining them is a tangent to W (q) at some point U. The line z subtends a rosette based at U containing θ and ϕ. Thus θ and ϕ intersect in exactly one point and are adjacent as vertices of N. If P and Q are not collinear in S, then they are connected as vertices of M, since M is connected. Let P, C_{1}, ..., C_{n}, Q be the path of M connecting P and Q and θ, ψ_{1}, ..., ψ_{n}, ϕ the ovoids subtended by the elements of the path. Since P and C_{1} are collinear in S, it follows that θ and ψ_{1} intersect in exactly one point and are hence connected in N. Similarly, C_{i} is connected in N to C_{i+1} for 1 ≤ i ≤ n − 1 and also C_{n} is connected to ϕ in N. Thus θ, ψ_{1}, ..., ψ_{n}, ϕ is a path connecting θ and ϕ. As these were arbitrary points of the graph N, it follows that N is connected.

□

Recall from Section 1.4.1 that a rosette R of a GQ of order r is homogeneous if any two ovoids of R are isomorphic, and inhomogeneous otherwise. We extend this definition to the following.

Definition 4.1.4 Let S be a GQ of order (r, r^{2}) and S′ a subquadrangle of S of order r. S′ is homogeneous in S if the ovoids of S′ subtended by S are all isomorphic. If this is not the case S′ is said to be inhomogeneous in S.

Theorem 4.1.5 [50] Let W (q) be a subquadrangle of a GQ S of order (q, q^{2}) and suppose that W (q) is inhomogeneous in S. Then the set of ovoids of W (q) subtended by S cannot be contained in the union of the set of Tits ovoids of W (q) and the set of elliptic quadrics of W (q).

Proof: Suppose that the set of subtended ovoids is contained in union of the set of Tits ovoids of W (q) and the set of elliptic quadrics of W (q). In other words, that each subtended ovoid is either a Tits ovoid or an elliptic quadric. Then since S′ is inhomogeneous in S, it follows that there occurs at least one ovoid of each type. Now from Theorem 4.1.3 the graph N is connected, which implies that there exists a Tits ovoid θ and an elliptic quadric ϕ of W (q) that are connected vertices of N. Thus there exists a path of N of the form θ = C_{0}, C_{1}, C_{2}, ..., C_{n−1}, C_{n} = ϕ, where C_{i} is either a Tits ovoid or an elliptic quadric. If j is the smallest number such that C_{j} is an elliptic quadric then C_{j−1} is a Tits ovoid, and so by Theorem 4.1.2 they intersect in either exactly q + \sqrt{2q} + 1 or q − \sqrt{2q} + 1 points. But C_{j} and C_{j−1} are adjacent in N and so the corresponding ovoids of W (q) intersect in exactly one point; a contradiction.

□

We now consider the case where S is a GQ of order (q, q^{2}) containing W (q) as a subquadrangle, such that each subtended ovoid of W (q) is a Tits ovoid.

Let G be the graph with vertices the Tits ovoids of W (q) and with two vertices adjacent if they intersect in exactly one point of W (q). We denote the automorphism group of G by aut (G).

Note that an automorphism of W (q) maps a Tits ovoid to a Tits ovoid and preserves collinearity, thus inducing an automorphism of G. Suppose that T is an arbitrary, non-trivial, automorphism of W (q) and let P be a point such that Q = T (P) ≠ P. Now consider a Tits ovoid θ of W (q) such that P ∈ θ and Q ∉ θ. Since Q ∈ T (θ) but Q ∉ θ, it follows that θ ≠ T (θ). Thus the induced action of any group fixing W (q) is faithful (that is, the only element of the group that induces the identity automorphism on G is the identity element of the group). In particular, the induced action of the group Sp(4, q) on G is faithful and so Sp(4, q) is isomorphic to a group of automorphisms of G. Now the Suzuki group Sz(q) is the stabiliser of the Tits ovoid in Sp(4, q) ([25, Corollary 16.4.7]) and an element of the conjugacy class of Sz(q) in Sp(4, q) induces a group of automorphisms on G which is isomorphic to Sz(q). This induced group fixes a vertex in G corresponding to the ovoid fixed by the particular conjugate of Sz(q) in Sp(4, q). For simplicity we will use Sp(4, q) to denote the group of automorphisms of G induced by Sp(4, q) and Sz(q) to denote the group of automorphisms of G induced by a conjugate of Sz(q) fixing a particular Tits ovoid θ. We introduce a slight abuse of notation by using θ to refer to a particular ovoid P G(3, q) and also the corresponding vertex of G.

Lemma 4.1.6 Let G be the graph with vertices the Tits ovoids of W (q) and with two vertices adjacent if they intersect in exactly one point of W (q). Then G is a connected graph.

Proof: Let aut (G)_{θ} denote the stabiliser of θ in aut (G); then we have Sz(q) ⊆ aut (G)_{θ} and Sp(4, q) ⊆ aut (G). By Lemma 1.4.11 we know that θ lies in an elation rosette of W (q), and so is in a non-trivial connected component C of G, with stabiliser denoted by aut (G)_{C}.

Thus Sz(q) ⊆ aut (G)_{C} ∩Sp(4, q) ⊆ Sp(4, q). Now since C is non-trivial it contains a vertex ϕ such that ϕ ≠ θ. The subgroup aut (G)_{ϕ} of aut (G) that fixes ϕ also preserves adjacency in G and so fixes C. Furthermore, it contains a subgroup isomorphic to Sz(q), induced by a conjugate of Sz(q) in Sp(4, q). Thus since Sz(q) and its conjugate that fixes the ovoid ϕ are both in aut (G)_{C} and they are not identical in Sp(4, q) we have that aut (G)_{C} ∩ Sp(4, q) ≠ Sz(q). Finally Sz(q) is maximal in Sp(4, q), so aut (G)_{C} ∩ Sp(4, q) = Sp(4, q), and as Sp(4, q) is transitive on G we have C = G. Thus G is connected.

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Lemma 4.1.7 Let θ and ϕ be two Tits ovoids of W (q) such that θ ∩ϕ = {P }. Then θ and ϕ lie in exactly one common rosette of Tits ovoids based at P, namely the elation rosette with respect to P and θ (or equivalently P and ϕ). Furthermore, for every point Q ∈ θ there is exactly one rosette of Tits ovoids based at Q which contains θ, namely the elation rosette with respect to Q and θ.

Proof: The number of Tits ovoids that intersect θ in exactly one point is (q − 1)(q^{2} + 1) ([3] Theorem 5(b)). By considering the elation rosette with respect to θ and P for all points P ∈ θ, we obtain (q − 1)(q^{2} + 1) Tits ovoids intersecting θ in exactly one point. Thus the ovoids intersecting θ in exactly one point are precisely those in elation rosettes containing θ, and there are only q −1 Tits ovoids intersecting θ in a given point P. Thus since |θ ∩ϕ| = 1, θ and ϕ lie in a common elation rosette: the unique rosette of Tits ovoids containing θ and ϕ. It also follows that the elation rosette with respect to θ and Q is the only rosette of Tits ovoids containing θ and based at Q.

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Theorem 4.1.8 There does not exist a GQ of order (q, q^{2}) such that W (q) is homogeneous in the GQ and all subtended ovoids are Tits ovoids.

Proof: Suppose such a GQ S = (P, B, I) exists. Let θ be a vertex of the graph G subtended by X ∈ P. Let ϕ be a vertex of G adjacent to θ, so that |ϕ ∩ θ| = 1, and let ϕ ∩ θ = {P }. Now by Lemma 4.1.7 there is a unique elation rosette R of Tits ovoids based at P and containing the ovoids θ and ϕ. The tangent joining P and X in S subtends a rosette R′ in W (q). By assumption, all subtended ovoids are Tits ovoids, so R′ is a rosette of Tits ovoids based at P and containing θ. Thus by Lemma 4.1.7 R = R′ and ϕ is a subtended ovoid of W (q).

We have shown that if one vertex G corresponds to a subtended ovoid of W (q) then so does each adjacent vertex. Since G is connected by Lemma 4.1.6 it follows that if one vertex of G corresponds to a subtended ovoid then all of the vertices do. Thus all Tits ovoids of W (q) are subtended. W (q) contains q^{2}(q + 1)(q^{2} − 1) Tits ovoids (= |[Sp(4, q) : Sz(q)]|), but there are only q^{4} − q^{2} points of S subtending ovoids, so we have a contradiction and the theorem is proved.

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4.1.2 Applications to Q(5, q), q even

We now apply the work of Section 4.1.1 to extend the characterisation of Q(5, q) in Theorem 4.1.1 to include Tits ovoids of W (q).

Theorem 4.1.9 [65, VII.1] If a GQ S of order (q, q^{2}), q even, has a subquadrangle isomorphic to W (q) and subtends only ovoids isomorphic to the elliptic quadric, then S is isomorphic to the classical GQ Q(5, q).

Now using the work in this chapter we can extend this characterisation to include Tits ovoids.

Theorem 4.1.10 If a GQ S of order (q, q^{2}), q even, has a subquadrangle isomorphic to W (q) and each subtended ovoid is either an elliptic quadric or a Tits ovoid then S is isomorphic to the classical GQ Q(5, q).

Proof: By Theorem 4.1.5 if W (q) is a subquadrangle of a GQ S of order (q, q^{2}) there cannot be a mixture of Tits ovoids and elliptic quadric ovoids subtended. By Theorem 4.1.8 the subtended ovoids are not all Tits ovoids and thus all the subtended ovoids are elliptic quadric ovoids. Hence by Theorem 4.1.9 S is isomorphic to the classical GQ Q(5, q).

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Corollary 4.1.11 Suppose q is even and that the only ovoids of W (q) are Tits ovoids and the elliptic quadric ovoids. If S is a GQ of order (q, q^{2}) containing a subquadrangle isomorphic to W (q), then S is isomorphic to the classical GQ Q(5, q).

Corollary 4.1.12 If there exists a GQ S of order (q, q^{2}) such that S is not isomorphic to Q(5, q) and S contains a subquadrangle isomorphic to W (q), then there is a new ovoid of P G(3, q).

Corollary 4.1.13 For q ≤ 32, q even, if S is a GQ of order (q, q^{2}) containing a subquadrangle isomorphic to W (q), then S is isomorphic to the classical GQ Q(5, q).

Proof: For q = 2 the GQs of order 2 and (2, 4) are unique ([49], 5.2.3 and 5.3.2 (ii)). By [4] (q = 4), [20] (q = 8), [36] (q = 16) and [37] (q = 32), the only ovoids in W (q) for 4 ≤ q ≤ 32 and q even are Tits ovoids and elliptic quadric ovoids.

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4.2 GQs of order (r, r^{2}) with a doubly subtended subquadrangle of order r

Let S = (P, B, I) be a GQ of order (r, r^{2}) and S′ = (P′, B′, I) a subquadrangle of order r. Recall from Chapter 3 that S′ is said to be doubly subtended in S if each ovoid of S′ subtended by a point of P \ P′ is subtended by exactly two points of P \ P′. Recall from Theorem 3.1.7 that there is an associated SPG T . In this section we will investigate the number of embeddings of S′ in a GQ of order (r, r^{2}) such that S′ is doubly subtended in the GQ, with associated SPG T .

Suppose that T_{W′} and T_{S′} are SPGs constructed by the subtending of the GQs W′ and S′, of order r, in the GQs W and S, of order (r, r^{2}) (note that in Chapter 3 we denoted these SPGs by T_{W} and T_{S} respectively) . Let (T_{W′}^{c_{W}} , p_{W}) and (T_{S′}^{c_{S}}, p_{S}) be the algebraic 2-fold covers of T_{W′} and T_{S′}, given by W and S respectively, as in Theorem 3.1.9. Recall from Section 1.1 that if G is a graph of diameter d and G_{d} is the graph with the same vertex set as G and two vertices adjacent if they are at distance d in G, then G is antipodal if G_{d} is the disjoint union of cliques. Now if S = (P, B, I) and S′ = (P′, B′, I), then T_{S′}^{c_{S}} = (P \ P′, B \ B′, I) (this is the definition in Theorem 3.1.9), from which we may easily show that the point graph of T_{S′}^{c_{S}} has diameter 3. We can also show that the point graph of T_{S′}^{c_{S}} is antipodal, with two points antipodal if they subtend the same ovoid of S′, or (equivalently) if they cover the same point of T_{S′}.

If there exists an isomorphism from T_{W′}^{c_{W}} to T_{S′}^{c_{S}}, then the isomorphism must induce a graph isomorphism on the corresponding point graphs. Since the antipodal nature of the graphs must be preserved under an isomorphism, it follows that a set of antipodal vertices of the point graph of T_{W′}^{c_{W}} maps onto a set of antipodal vertices of the point graph of T_{S′}^{c_{S}}. Thus we have an induced bijection from the pointset of T_{W′} onto the pointset of T_{S′}. Further, we can easily show that this bijection induces an isomorphism from the geometry T_{W′} to the geometry T_{S′}. We say that this isomorphism is induced by the isomorphism from T_{W′}^{c_{W}} to T_{S′}^{c_{S}}. Note that Section 5.3.2 contains a more detailed exposition of the fact that an isomorphism of covering geometries induces an isomorphism of the geometries being covered. Although Section 5.3.2 deals with different geometries to those considered here, the general arguments are still applicable.

The following result is a consequence of the proof of Theorem 3.2.1.

Theorem 4.2.1 Let W and S be two GQs of order (r, r^{2}) and let W′ and S′ be subGQs of W and S respectively, of order r. Let W′ and S′ be doubly subtended in W and S and let the SPGs constructed as in Theorem 3.1.7 be T_{W′} and T_{S′}. Let the algebraic 2-fold covers of T_{W′} and T_{S′} determined by W and S as in Theorem 3.1.9 be (T_{W′}^{c_{W}} , p_{W}) and (T_{S′}^{c_{S}}, p_{S}), defined by the 1-cochains c_{W} and c_{S} respectively. If i : T_{W′}^{c_{W}} → T_{S′}^{c_{S}} is an isomorphism, then i extends uniquely to an an isomorphism \overline{i} : W → S. Further, \overline{i}(W′) = S′ and i and \overline{i} induce the same isomorphism from T_{W′} to T_{S′}.

Proof: From the discussion above it follows that the isomorphism i : T_{W′}^{c_{W}} → T_{S′}^{c_{S}} induces a unique isomorphism from T_{W′} to T_{S′}. In the proof of Theorem 3.2.1 it was shown that given i and the induced isomorphism from T_{W′} to T_{S′} there is a unique isomorphism \overline{i} from W to S, taking W′ to S′ and inducing the same isomorphism from T_{W′} to T_{S′} as i.

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Theorem 4.2.1 means that, in order to characterise a GQ of order (r, r^{2}) in terms of a doubly subtended subGQ of order r, we need to consider the isomorphism classes of covering geometries of algebraic 2-fold covers of the associated SPG. Note that Theorem 4.2.1 does not preclude the possibility that given an SPG T_{W′}, two covers (T_{W′}^{c}, p) and (T_{W′}^{c′}, p′) which give rise to the GQs S and W of order (r, r^{2}), as in Theorem 3.1.10, and T_{W′}^{c} not isomorphic to T_{W′}^{c′} that S ≅ W. In this case S and W are isomorphic but there is no isomorphism from S to W that maps W′ to W′ and T_{W′} and T_{W′}.

Suppose that T_{W′} is an SPG constructed by the GQ W′ of order r being doubly subtended in a GQ W of order (r, r^{2}). We now consider isomorphisms between algebraic 2-fold covers of T_{W′} such that the defining 1-cochain satisfies the GQ condition (3.1.1); that is, covers that give rise to a GQ of order (r, r^{2}), as in Theorem 3.1.10. Let (T_{W′}^{c}, p) and (T_{W′}^{c′}, p′) be two algebraic 2-fold covers of T_{W′}, defined by c and c′ respectively. Let \overline{U} : T_{W′}^{c′} → T_{W′}^{c} be an isomorphism. If (P, α) is a point of T_{W′}^{c′}, then \overline{U} acts by (P, α) ↦ (U(P), u_{P} (α)), where U is the automorphism of T_{W′} induced by \overline{U} and u_{P} is a permutation of ℤ_{2}, such that

c′(P, Q) = α + β ⟺ c(U(P), U(Q)) = u_{P} (α) + u_{Q}(β).  (4.2.1)

Now there are only two permutations on ℤ_{2}, the identity permutation and the permutation that swaps 0 and 1. Let b be the 0-cochain defined by b(P) = 0 if u_{P} is the identity and b(P) = 1 if u_{P} swaps 0 and 1. Thus u_{P} (α) = α + b(P). So, rewriting the condition 4.2.1 U and the permutations u_{P}, we have that

c′(P, Q) = α + β ⟺ c(U(P), U(Q)) = α + b(P) + β + b(Q).

That is, c′(P, Q) = c(U(P), U(Q)) + δ^{0}b(P, Q). Conversely, if b is a 0-cochain and U an automorphism of T_{W}, we define c′(P, Q) = c(U(P), U(Q)) + δ^{0}b(P, Q). Then the map (P, α) ↦ (U(P), b(P) + α) is an isomorphism from T_{W′}^{c′} to T_{W′}^{c}.

The above work gives us the following theorem.

Theorem 4.2.2 Let W be a GQ of order (r, r^{2}) and let W′ be a subGQ of W of order r. Let W′ be doubly subtended in W and the SPG constructed as in Theorem 3.1.7 be T_{W′}. Let (T_{W′}^{c}, p) and (T_{W′}^{c′}, p′) be two algebraic 2-fold covers of T_{W′}, defined by c and c′ respectively, such that c and c′ satisfy the GQ condition 3.1.1. Then T_{W′}^{c} and T_{W′}^{c′} are isomorphic if and only if c′(P, Q) = c(U(P), U(Q)) + δ^{0}b(P, Q), where U is an automorphism of T_{W′} and b a 0-cochain. In this case the map i : T_{W′}^{c′} → T_{W′}^{c} that acts by (P, α) ↦ (U(P), α + b(P)) is an isomorphism.

Proof: Let (P, α) and (Q, β) be points of T_{W′}. Then i(P, α) = (U(P), b(P)+α), and i(Q, β) = (U(Q), b(Q) + β) and i(P, α) and i(Q, β) are collinear if and only if

c(U(P), U(Q)) = α + β + b(P) + b(Q).

This is the case if and only if c(U(P), U(Q)) + δ^{0}(P, Q) = c′(P, Q) = α + β, which is true if and only if (P, α) and (Q, β) are collinear.

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Recall from Section 1.6 that two 1-cochains c and c′ are said to be equivalent if T_{W′}^{c} and T_{W′}^{c′} are isomorphic, that is, if and only if c′(P, Q) = c(U(P), U(Q)) + δ^{0}(P, Q), for some automorphism U of T_{W′} and 1-coboundary δ^{0}b. Note that if b is a 0-cochain, then the 1-cochains c and c + δ^{0}b are equivalent.

Having established the form of the 1-cochains which give isomorphic covers of a given SPG, we now consider automorphisms of a cover. From the above, we know that an automorphism of T_{W′}^{c} has the form (P, α) ↦ (U(P), α + b(P)) where c(U(P), U(Q)) = c(P, Q) + δ^{0}b(P, Q), for U an automorphism of T_{W′} and b a 0-cochain. Such an automorphism U is said to be admitted by T_{W′}^{c}. The automorphisms of T_{W′} admitted by T_{W′}^{c} form a subgroup of the automorphism group of T_{W′}, which we denote by aut(T_{W′})_{c}.

4.2.1 Introducing the cohomology

Suppose that T_{W′} is an SPG constructed by the GQ W′ of order r being doubly subtended in a GQ W of order (r, r^{2}). Recall that if G is the point graph of T_{W′} and Γ_{G} the simplicial complex of G, then c ∈ C^{1}(Γ_{G}, ℤ_{2}) (that is, c is a 1-cochain) satisfies the GQ condition 3.1.1 if and only if the following condition holds

δ^{1}c(P, Q, R) = 0 ⟺ P, Q, R are distinct collinear points of T_{W′}.

Since we are working over ℤ_{2}, if δ^{1}c(P, Q, R) is not 0 then it must be 1, and hence every 1-cochain that satisfies the GQ condition has the same coboundary. Thus if c is a particular 1-cochain satisfying the GQ condition, then the set of all 1-cochains satisfying the GQ condition is

c + Z^{1}(Γ_{G}, ℤ_{2}),

where Z^{1}(Γ_{G}, ℤ_{2}) is the set of 1-cocycles (that is, 1-cochains with a zero coboundary). So, given one algebraic 2-fold cover of T_{W′} satisfying the GQ condition, we can find all others by adding on 1-cocycles. The question we now answer is: what is the relationship between the set of covers equivalent to c and the first cohomology group H^{1}(Γ_{G}, ℤ_{2})?

Recall from Section 1.5.2 that the first cohomology group is given by H^{1}(Γ_{G}, ℤ_{2}) = Z^{1}(Γ_{G}, ℤ_{2})/B^{1}(Γ_{G}, ℤ_{2}); that is it’s the group that results from factoring the 1-coboundaries out of the 1-cocycles. Thus an element of H^{1}(Γ_{G}, ℤ_{2}) is a set of 1-cocycles of the form z + B^{1}(Γ_{G}, ℤ_{2}) = {z + δ^{0}b : δ^{0}b ∈ B^{1}(Γ_{G}, ℤ_{2})}, where z is a fixed 1-cocycle. Since two 1-cochains whose difference is a 1-coboundary are equivalent, it follows that any two elements of c + (z + B^{1}(Γ_{G}, ℤ_{2})) are equivalent.

We now consider the conditions under which two 1-cochains equivalent to c are in a common element of c + H^{1}(Γ_{G}, ℤ_{2}). Let c′ and c′′ be two 1-cochains equivalent to c, such that

c′(P, Q) = c(S(P), S(Q)) + δ^{0}b′(P, Q) and
c′′(P, Q) = c(T (P), T (Q)) + δ^{0}b′′(P, Q),

where S and T are automorphisms of T_{W′} and δ^{0}b′ and δ^{0}b′′ are 1-coboundaries. The 1-cochains c′ and c′′ are in the same element of c + H^{1}(Γ_{G}, ℤ_{2}) if and only if their sum is a 1-coboundary. Now

c′(P, Q) + c′′(P, Q) = c(S(P), S(Q)) + c(T (P), T (Q)) + δ^{0}b′(P, Q) + δ^{0}b′′(P, Q), (4.2.2)

but δ^{0}b′(P, Q) + δ^{0}b′′(P, Q) is a 1-coboundary. So 4.2.2 holds if and only if

c(S(P), S(Q)) + c(T (P), T (Q)) = δ^{0}b(P, Q)

for some 1-coboundary δ^{0}b. This is the case if and only if

c(P, Q) + c(S^{−1} ∘ T (P), S^{−1} ∘ T (Q)) = δ^{0}b(S^{−1} ∘ T (P), S^{−1} ∘ T (Q)),  (4.2.3)

but since δ^{0}b(S^{−1} ∘ T (P), S^{−1} ∘ T (Q)) is a 1-coboundary, it follows that 4.2.3 holds if and only if S^{−1} ∘ T ∈ aut(T_{W′})_{c}. This is the case if and only if S and T are in the same left coset of aut(T_{W′})_{c} in aut(T_{W′}). Thus, we have a one-to-one correspondence from the set of left cosets of aut(T_{W′})_{c} in aut(T_{W′}) to the set of elements of H^{1}(Γ_{G}, ℤ_{2}) (note that the correspondence need not be onto). Note that if h ∈ H^{1}(Γ_{G}, ℤ_{2}) corresponds to the left coset H of aut(T_{W′})_{c} in aut(T_{W′}), then every element of h acts by

c(S(P), S(Q)) + δb(P, Q),

for some S ∈ H and 1-coboundary δb.

From the above we have the following theorem.

Theorem 4.2.3 Let W and S be two GQs of order (r, r^{2}) and let W′ be a subGQ of order r of both W and S. Suppose that W′ is doubly subtended in both W and S and that T_{W′} is the SPG constructed from both W and T , as in Theorem 3.1.7. Let the 2-fold algebraic cover of T_{W′} constructed from W as in Theorem 3.1.9 be (T_{W′}^{c}, p) and let the 1-cochain c define (T_{W′}^{c}, p). Let Γ_{G} be the simplicial complex of the point graph of T_{W′}^{c}. If |aut(T_{W′}) : aut(T_{W′})_{c}| = |H^{1}(Γ_{G}, ℤ_{2})|, then there exists an isomorphism i : W → S, such that i(W′) = W′.

Proof: Let h be an element of H^{1}(Γ_{G}, ℤ_{2}). Since |aut(T_{W′}) : aut(T_{W′})_{c}| = |H^{1}(Γ_{G}, ℤ_{2})|, by the correspondence above it follows that if c′ is any 1-cochain in the set c + h, then c′ acts by c′(P, Q) = c(S(P), S(Q)) + δ^{0}b(P, Q) for some 1-coboundary δ^{0}b and some automorphism S of T_{W′} contained in the coset of aut(T_{W′})_{c} corresponding to h. Thus c is equivalent to c′. Since this holds for any h ∈ H^{1}(Γ_{G}, ℤ_{2}) it follows that any 1-cochain of the form c+ 1-cocycle is equivalent to c. These are precisely the 1-cochains of Γ_{G} over ℤ_{2} that satisfy the GQ condition. Thus if (T_{W′}^{c′}, p_{S}) is the algebraic 2-fold cover of T_{W′} constructed from S, then it follows that c and c′ are equivalent and by Theorem 4.2.1 we have the existence of the isomorphism i.

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If we let aut(W′)_{W} denote the subgroup of aut(W′) that is induced by aut(W), then Theorem 4.2.3 has the following corollary.

Corollary 4.2.4 Let W and S be two GQs of order (r, r^{2}) and let W′ be a subGQ of order r of both W and S. Suppose that W′ is doubly subtended in both W and S and that T_{W′} is the SPG constructed from both W and T , as in Theorem 3.1.7. Let the 2-fold algebraic cover of T_{W′}, constructed from W as in Theorem 3.1.9, be (T_{W′}^{c}, p) and let the 1-cochain c define (T_{W′}^{c}, p). Let Γ_{G} be the simplicial complex of the point graph of (T_{W′}^{c}, c). If |aut(T_{W′}) : aut(W′)_{W}| = |H^{1}(Γ_{G}, ℤ_{2})|, then there exists an isomorphism i : W → S, such that i(W′) = W′.

Proof: We show that aut(T_{W′}) ≅ aut(W′)_{W}, and so the result follows from Theorem 4.2.3. Let T be an element of aut(W′)_{W} and let \overline{T} be an automorphism of W that induces T on W′. Thus \overline{T} induces an element of aut(T_{W′})_{c}. Let i : aut(W′)_{W} → aut(T_{W′})_{c} be the map taking an element of aut(W′)_{W} to the (unique) element of aut(T_{W′}) it induces. We can easily show that i is a group homomorphism, so now we show that it is also both one-to-one and onto and hence an isomorphism. By Corollary 3.2.2 any element S of aut(T_{W′})_{c} is induced by a unique element of aut(W′), and so i is one-to-one. Now any element S of aut(T_{W′})_{c} is induced by an automorphism of T_{W′}^{c} (by definition), and so by Theorem 4.2.1 there is an element of aut(W′)_{W} inducing S. Hence i is onto and so is an isomorphism, that is, aut(T_{W′}) ≅ aut(W′)_{W}.

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4.2.2 Calculating the homology

This section depends heavily on the definitions and results of Section 1.5 and Section 1.1. In Section 4.2.1 we established that characterising a GQ of order (r, r^{2}) by a doubly subtended subGQ of order r depends on the order of the first cohomology group of the simplicial complex of the point graph of the subtended SPG. By Theorem 1.5.4, we know that the first cohomology group, over a field, is isomorphic to the first homology group. In this section we will work towards calculating the first homology group of the simplicial complex of the point graph of the subtended SPG.

From Section 1.5.5, the fact that we are calculating homology over ℤ_{2} means that we may represent a 1-chain as the set of 1-simplexes with a coefficient of 1 in the 1-chain. The sum of two 1-chains is the symmetric difference of the corresponding sets of 1-simplexes.

Let W be a GQ of order (r, r^{2}) with a subGQ W′ of order r, and suppose that W′ is doubly subtended in W, we will denote the associated SPG by T . Let G be the point graph of T and Γ_{G} the simplicial complex of G.

Recall from the discussion after Theorem 1.5.8 that the 1-cycles of Γ_{G} are the edge sets of circuits of G. The elementary 1-cycles of Γ_{G} are the edge sets of elementary circuits of G and the induced 1-cycles of Γ_{G} are the edge sets of the induced circuits of G. We use this representation of 1-cycles of Γ_{G} as it is more intuitive, and leads us to results that will allow us to calculate the homology directly from G.

A 1-boundary of Γ_{G} is a triangle in the graph G. In much of the work that follows in this section we will be selectively replacing an edge e_{1} of the edge set of a circuit of G with two edges e_{2} and e_{3} of G, where {e_{1}, e_{2}, e_{3}} is a triangle of G. This is equivalent to adding the boundary of a 2-simplex to a 1-cycle of Γ_{G}, which produces a homologous 1-cycle. We use this technique to manipulate 1-cycles into more convenient forms.

In the following theorem we show that any induced 1-cycle of Γ_{G} is homologous to a sum of induced 1-cycles, each of which consists of four 1-simplexes. A circuit of G with edge set that is a 1-cycle of Γ_{G} consisting of four 1-simplexes will be called a four-circuit of G. If in addition the 1-cycle is induced, then we call the corresponding circuit and induced four-circuit. In other words, an induced four-circuit is a circuit of length four such that “opposite” vertices of the circuit are not adjacent. Recall from Section 1.1 that if X_{1}, X_{2}, …, X_{n}, X_{n+1} is a circuit of G of length n (and so X_{n+1} = X_{1}) then we represent it by (X_{1}, X_{2}, …, X_{n}).

Recall from Section 1.5 that two 1-cycles of Γ_{G} are homologous if their difference is a 1boundary and that any 1-cycle that is homologous to the zero 1-cycle (that is, any 1-boundary) is null homologous.

Theorem 4.2.5 Let σ be an induced 1-cycle of Γ_{G} consisting of at least four 1-simplexes. Then there exist induced 1-cycles σ_{1}, σ_{2}, …, σ_{r} such that each σ_{i} consists of four 1-simplexes and σ is homologous to the sum of the σ_{i}.

Proof: Let σ consist of n 1-simplexes; we proceed by induction on n. If n = 4, then the result is immediate. If n ≥ 5, then let σ be the edge set of the circuit (X_{1}, X_{2}, …, X_{n}). Recall from the construction of T (Theorem 3.1.7) that the X_{i} are ovoids of W′, and that X_{i} and X_{j} are adjacent (in G) when the ovoids they represent intersect in exactly one point. Now suppose that X_{1} ∩ X_{2} = {P } and that R is the rosette, with basepoint P, containing both X_{1} and X_{2}. By the proof of Theorem 3.1.7, if X_{4} does not contain P then X_{4} intersects exactly two ovoids of R in one point, and if X_{4} does contain P then X_{4} does not intersect any ovoid of R in exactly one point. Suppose first that P is not a point of X_{4}. Since σ is an induced circuit and we have assumed that n ≥ 5, it follows that X_{4} is adjacent to neither X_{1} nor X_{2}. Thus there exists an ovoid X, X ≠ X_{1}, X_{2}, such that X is in R and X intersects X_{4} in exactly one point. Hence the induced 1-cycle σ may be expressed as the sum of the 1-cycles that are the edge sets of the circuits (X_{1}, X, X_{4}, X_{5}, …, X_{n}), (X_{1}, X_{2}, X) and (X_{2}, X_{3}, X_{4}, X). If X_{3} is adjacent to X, then (X_{2}, X_{3}, X) and (X_{4}, X_{3}, X) are triangles in G and so the 1-cycle corresponding to (X_{2}, X_{3}, X_{4}, X) is null homologous. If X_{3} is not adjacent to X, then (X_{2}, X_{3}, X_{4}, X) is an induced four-circuit. Now the induced 1-cycle σ contains n 1-simplexes, and the 1-cycles corresponding to the circuits (X_{1}, X, X_{4}, X_{5}, …, X_{n}) and (X_{2}, X_{3}, X_{4}, X) contain n − 1 and four 1-simplexes, respectively. Thus the induced 1-cycle σ is homologous to the sum of an elementary 1-cycle containing n−1 1-simplexes and a 1-cycle containing four 1-simplexes (both of which may or may not be induced). By the proof of Lemma 1.5.6 we can write both of these 1-cycles as the sum of induced 1-cycles, each consisting of fewer than n 1-simplexes.

Now suppose that P is a point of the ovoid X_{4}. Let Q be a point contained in the ovoid X_{1} but in neither X_{3} nor X_{4}. By Corollary 3.1.6, there is a (unique) subtended rosette R′, containing X_{1} and having basepoint Q. Since X_{3} does not contain the basepoint of R′, it follows, from the proof of Theorem 3.1.7, that X_{3} is adjacent to two ovoids in R′. If we let one such ovoid be X′, then σ can be expressed as the sum of the 1-cycles corresponding to the circuits (X_{1}, X_{2}, X_{3}, X′) and (X_{1}, X′, X_{3}, X_{4}, …, X_{n}). Now, since X_{4} does not contain the basepoint of the rosette R′, and X_{4} is not adjacent to X_{1}, there is a second ovoid X′′, contained in R′, such that X_{4} is adjacent to X′′. The 1-cycle corresponding to the circuit (X_{1}, X′, X_{3}, X_{4}, …, X_{n}) can be expressed as the sum of the 1-cycles corresponding to the circuits (X′, X_{3}, X_{4}, X′′), (X_{1}, X′, X^{′,′}) and (X_{1}, X′′, X_{4}, …, X_{n}). The circuit c = (X_{1}, X′′, X_{4}, …, X_{n}) has length n−1, but is not necessarily induced. If σ_{c} is the 1-cycle corresponding to the circuit c, then σ_{c} contains n − 1 1-simplexes and is not necessarily induced. If σ_{c} is not induced, then by the proof of Lemma 1.5.6, we may write σ_{c} as the sum of induced 1-cycles, each consisting of fewer than n 1-simplexes. Thus the 1-cycle σ may be written as the sum of induced circuits each of which contains fewer than n 1-simplexes. Thus, σ is homologous to the sum of induced circuits each of which has at least four and at most n − 1 1-simplexes. The result follows by induction.

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By using Theorem 1.5.8 and Theorem 4.2.5 we have the following.

Theorem 4.2.6 Let W be a GQ of order (r, r^{2}) with a subGQ W′ of order r. Suppose that W′ is doubly subtended in W, with associated SPG T . Let G be the point graph of T and Γ_{G} the simplicial complex of G. Then H_{1}(Γ_{G}, ℤ_{2}) is trivial if and only if each induced 1-cycle of Γ_{G}, consisting of four 1-simplexes, is a 1-boundary (that is, null homologous).

Proof: By Theorem 1.5.8 H_{1}(Γ_{G}, ℤ_{2}) is trivial if and only if every induced 1-cycle is null homologous and by Theorem 4.2.5 every induced 1-cycle of Γ_{G} is null homologous if and only if every induced 1-cycle consisting of four 1-simplexes is null homologous.

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Given Theorem 4.2.6 we now need a method for showing that a given induced four-circuit is null homologous. The following lemma provides this.

Lemma 4.2.7 Let σ be an induced 1-cycle of Γ_{G} and let c = (X, Y, Z, W) be the corresponding induced four-circuit of G. If G_{{X,Z}} is connected, then σ is null homologous.

Proof: Suppose that G_{{X,Z}} is connected and let Y, v_{0}, v_{1}, …, v_{n}, W be a path connecting Y and W in G_{{X,Z}}. Then σ is equal to the sum of the 1-cycles corresponding to the circuits (X, Y, v_{0}), (Z, Y, v_{0}), (X, v_{0}, v_{1}), (Z, v_{0}, v_{1}), … (X, v_{n−1}, v_{n}), (Z, v_{n−1}, v_{n}), (X, v_{n}, W) and (Z, v_{n}, W). Since each of these 1-cycles is a 1-boundary, it follows that c is null homologous.

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4.3 Application to Q(5, q)

In this section we apply the work of the previous section to the classical GQ Q(5, q) where q is odd and show that Q(5, q) is characterised by a Q(4, q) subquadrangle and the subtended ovoid/rosette structure. Recall from Section 3.1 that Q(5, q) contains subGQs, of order q, isomorphic to Q(4, q). Each such subGQ is doubly subtended in Q(5, q) and the set of such subGQs form a single orbit under the action of the group of Q(5, q) [27, Theorem 22.6.6]. The set of subtended ovoids of Q(4, q) is exactly the set of elliptic quadric ovoids of Q(4, q). If we use the notation of the previous section, that is, W = Q(5, q), W′ = Q(4, q) with SPG T , then every automorphism of Q(4, q) fixes T and so by Corollary 3.2.2 aut(T_{W′}) ≅ aut(W′) ≅ aut(T_{W′})_{W′}. Since every automorphism of Q(4, q) is induced by an automorphism of Q(5, q) we have aut(W′)_{W} = aut(W′). Thus by Theorem 4.2.3 Q(5, q) is characterised by having Q(4, q) as a doubly subtended subGQ with associated SPG T , if and only if H^{1}(Γ_{G}, ℤ_{2}) is trivial, which is the case if and only if H_{1}(Γ_{G}, ℤ_{2}) is trivial, by Theorem 1.5.4.

As in Section 4.2.2 let G be the point graph of the SPG T and Γ_{G} the simplicial complex of G. To show H_{1}(Γ_{G}, ℤ_{2}) is trivial we will employ Lemma 4.2.7 and so need a “user-friendly” representation of the graph G. For q odd under the polarity of Q(4, q) the ambient threedimensional subspace of an elliptic quadric ovoid maps to a non-singular point of PG(4, q) and two elliptic quadric ovoids are adjacent in G if and only if the corresponding polar points span a tangent to Q(4, q). In this polar representation, a rosette is the set of non-singular points on a tangent to Q(4, q) (note that if one non-singular point on a line tangent to Q(4, q) has a polar hyperplane that intersects Q(4, q) in an elliptic quadric ovoid, then so do all the other non-singular points on the line). In the following section we show that H_{1}(Γ_{G}, ℤ_{2}) is trivial for q odd.

For q even the theory of Section 4.2.2 also applies and the most obvious representation to use seems to be that given in [26]. The calculation, however, would seem to be more involved than that for the q odd case, and certainly longer than the proof of the same result by Thas and Payne in [65].

4.3.1 Explicit homology calculation for Q(4, q), q odd

Let T be the SPG with pointset the elliptic quadric ovoids of Q(4, q) and lineset the rosettes of elliptic quadric ovoids. Let G be the point graph of T and Γ_{G} the simplicial complex of G. In this section we show that H_{1}(Γ_{G}, ℤ_{2}) is trivial. We show that for each induced four-circuit (X_{1}, X_{2}, X_{3}, X_{4}) of G either the subgraph G_{{X_{1},X_{3}}} or the subgraph G_{{X_{2},X_{4}}} is connected. By Lemma 4.2.7 this implies that each induced 1-cycle of Γ_{G} over ℤ_{2} that consists of four 1-simplexes is null homologous. By Theorem 4.2.6 this implies that H_{1}(Γ_{G}, ℤ_{2}) is trivial.

Let Q = Q(4, q) be the non-singular (parabolic) quadric of PG(4, q) defined by Q(X) = x_{0}^{2} + x_{1}x_{2} + x_{3}x_{4} = 0, where X has coordinates (x_{0}, x_{1}, x_{2}, x_{3}, x_{4}). The bilinear form associated with Q is given by β(x, y) = 2x_{0}y_{0} + x_{1}y_{2} + x_{2}y_{1} + x_{3}y_{4} + x_{4}y_{3} and if ⊥ denotes the polarity associated with Q, then (x_{0}, x_{1}, x_{2}, x_{3}, x_{4})^{⊥} = [2x_{0}, x_{2}, x_{1}, x_{4}, x_{3}] and [a_{0}, a_{1}, a_{2}, a_{3}, a_{4}]^{⊥} = (a_{0}/2, a_{2}, a_{1}, a_{4}, a_{3}). If X and Y are two points of Q such that the line ⟨X, Y ⟩ is singular (that is, β(X, Y) = 0), then we write X ∼ Y. If X is a point of PG(4, q) not on the quadric Q, and Y is a second point of PG(4, q) (which may be either singular or non-singular) such that the line ⟨X, Y ⟩ is tangent to Q, then we say that X and Y are cotangent and write X ⌢ Y.

It will be useful in what follows to have an algebraic condition that determines when two points of PG(3, q) are cotangent with respect to Q. Let X and Y be two points of PG(4, q) such that X is not contained in Q. Then the set of points incident with the line ⟨X, Y ⟩ but not equal to X is {λX + Y : λ ∈ GF(q)}. The points X and Y are cotangent if and only if the line ⟨X, Y ⟩ intersects Q in precisely one point; that is, the equation Q(λX + Y) = 0 has exactly one solution in λ. Now

Q(λX + Y) = β(λX, Y) + Q(λX) + Q(Y)
= Q(X)λ^{2} + β(X, Y)λ + Q(Y).

This quadratic in λ has a unique solution, λ = −β(X, Y)/(2Q(X)), if and only if the discriminant of the quadratic is zero. That is,

β(X, Y)^{2} − 4Q(X)Q(Y) = 0.

We define Δ(X, Y) = β(X, Y)^{2} − 4Q(X)Q(Y).

If X is a non-singular point of PG(4, q), then the lines tangent to Q containing X form a quadric T_{X}. The quadric T_{X} is a cone with vertex X and base X^{⊥} ∩ Q. The cone T_{X} is called the tangency cone of X.

If X is a non-singular point of PG(3, q), then the hyperplane X^{⊥} intersects Q in a nonsingular hyperbolic quadric if and only if Q(X) is a non-zero square of GF(q), and in a nonsingular elliptic quadric if and only if Q(X) is a non-square of GF(q). If Σ_{1} and Σ_{2} are two hyperplanes of PG(4, q), such that Σ_{1}∩Q and Σ_{2}∩Q are both non-singular elliptic quadrics, then the two elliptic quadrics intersect in exactly one point if and only if the line ⟨Σ_{1}^{⊥}, Σ_{2}^{⊥}⟩ is tangent to Q. That is, Σ_{1}^{⊥} and Σ_{2}^{⊥} are cotangent. Given this, we have the following representation of G:

Vertex set: {X ∈ PG(4, q) : Q(X) is a non-square of GF(q)}.

Adjacency: Cotangency.

In this representation of the points of T the lines of T are the lines of PG(4, q) that are tangent to Q and have a polar plane intersecting Q in exactly one point (the base point of the rosette corresponding to the line of T ). Recall that T is an SPG with parameter α = 2 and so if (P, ℓ) is a non-incident point/line pair of T , then P is collinear to exactly 0 or 2 points of ℓ. Recall from the proof of Theorem 3.1.7 that P is collinear to 0 points of ℓ when it is an ovoid containing the base point of the rosette corresponding to ℓ, and collinear to two points of ℓ when it does not. We state the equivalent result in the current setting, which will prove useful for the calculations that follow.

Lemma 4.3.1 Let ℓ be a line of PG(4, q) tangent to Q at the point Q, such that the polar space of each non-singular point of ℓ intersects Q in a non-singular elliptic quadric. Let P be a point of PG(4, q) such that P^{⊥} intersects Q in a non-singular elliptic quadric and P ∉ ℓ. Then one of the following must be the case

(i) There are exactly two points incident with ℓ and cotangent to P, both of which are not on Q; or

(ii) The point Q ∈ Q is the only point incident with ℓ that is cotangent to P.

Proof: Consider the plane π = ⟨ℓ, P ⟩. If π intersects Q in a conic, then P is an exterior point to the conic and there are exactly two points of ℓ cotangent to P, both of which are non-singular with respect to Q. The other possibilities for π are that it intersects Q in a line pair (where the lines intersect in Q) or that it intersects Q in the unique point Q. In either case, π contains a unique tangent on P and Q is the unique point of ℓ cotangent to P.

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We now consider four-circuits of G. If c = (X_{1}, X_{2}, X_{3}, X_{4}) and c′ = (X′_{1}, X′_{2}, X′_{3}, X′_{4}) are two four-circuits of G and S is a collineation of PG(4, q) that fixes Q, such that S : (X_{1}, X_{2}, X_{3}, X_{4}) ↦ (X′_{1}, X′_{2}, X′_{3}, X′_{4}), then we say that S maps c to c′ and write S(c) = c′. Now, any collineation of PG(4, q) that fixes Q also fixes both the vertex set of G and the set of tangents to Q and so preserves cotangency amongst the vertices of G. Thus S induces an automorphism of the graph G. From this we have the following result.

Lemma 4.3.2 Let c and c′ be two four-circuits of G and let S be a collineation of PG(4, q) such that S fixes Q and S(c) = c′. Let the edge sets of c and c′ be the 1-cycles σ and σ′ of Γ_{G}, respectively. Then σ is null homologous if and only if σ′ is null homologous.

Proof: Suppose that σ is null homologous, so that S is the boundary of some 2-chain {Δ_{1}, Δ_{2}, …, Δ_{n}}, where each Δ_{i} is a 2-simplex of Γ_{G}. Each Δ_{i} is a triangle in G and since σ = ∂({Δ_{1}, Δ_{2}, …, Δ_{n}}) it follows that the edge set of c is the symmetric difference of the edge sets of the triangles Δ_{1}, Δ_{2}, …, Δ_{n}. This property is preserved by any automorphism of G and, since the above discussion shows that S induces an automorphism of G, it follows that σ′ is null homologous.

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By Theorem 4.2.6, to prove that H_{1}(Γ_{G}, ℤ_{2}) is trivial we need to show that the edge set of every induced four-circuit is a null homologous 1-cycle. By Lemma 4.3.2, we can do this by directly showing that the edge set of a given induced four-circuit of G is null homologous or by showing it is isomorphic in PG(4, q) to an induced four-circuit with a null homologous edge set. We proceed by using the elements of the group of Q in PG(4, q) to choose canonical forms for the induced four-circuits of G, and then apply Lemma 4.2.7 to the canonical induced four-circuits.

The following result gives a canonical form for the induced four-circuits of G.

Lemma 4.3.3 Let c be an induced four-circuit of G. Then c is isomorphic to an induced fourcircuit of the form (O, A, B, C(λ)) where O = (0, 1, η, 0, 0) for η a fixed non-square of GF(q), A = (0, 1, η, 0, 1), C(λ) = (0, 1, η, λ, 0), for some λ ∈ GF(q) \ {0, −4η} and B is a non-singular point of PG(4, q) such that Δ(O, B) ≠ 0.

Proof: Let c = (X_{1}, X_{2}, X_{3}, X_{4}) and let the plane ⟨X_{1}, X_{2}, X_{4}⟩ = π. Since c is an induced four-circuit X_{2} and X_{4} are not adjacent in G. Thus the lines ⟨X_{1}, X_{2}⟩ and ⟨X_{1}, X_{4}⟩ are tangents at distinct points of Q and the line ⟨X_{2}, X_{4}⟩ is not a tangent to Q. From this it follows that π intersects Q in a conic and that π^{⊥} is a line exterior to Q. By [27, Theorem 22.6.6] we know that the group of Q is transitive on such planes, so we may assume that π = [0, η, −1, 0, 0] ∩ [1, 0, 0, 0, 0], where η is a fixed non-square of GF(q). Now let ⟨X_{1}, X_{2}⟩ ∩ Q = P_{1}, ⟨X_{1}, X_{4}⟩ ∩ Q = P_{2} and X_{2}^{⊥} ∩ π ∩ Q = {P_{1}, P_{3}}. By [24, Lemma 7.2.3 Corollary 8] we know that the group of Q is three transitive on the singular points incident with π. So we may assume that P_{1} = (0, 0, 0, 0, 1), P_{2} = (0, 0, 0, 1, 0) and P_{3} = (0, −1/(2η), −1/2, 1, −1/(4η)). This means that X_{1} is the intersection of the tangent on (0, 0, 0, 0, 1) incident with π and the tangent on (0, 0, 0, 1, 0) incident with π. That is, X_{1} = O = (0, 1, η, 0, 0). Similarly X_{2} = A and X_{4} = C(λ) for some λ ∈ GF(q) \ {0}. Now the circuit c is induced if and only if X_{3} is a non-singular point of PG(4, q) such that Δ(O, X_{3}) ≠ 0 and Δ(A, C(λ)) ≠ 0. Since

Δ(A, C(λ)) = Δ((0, 1, η, 0, 1), (0, 1, η, λ, 0))
= (2η + λ)^{2} − 4(η)(η)
= λ(4η + λ),

we have the required result.

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We now consider induced four-circuits of the form (O, A, B, C(λ)) (using the notation of Lemma 4.3.3) where λ ≠ 0, −4η. First we need to establish for a given λ, λ ≠ 0, −4η, the possible points B. Since (O A B C(λ)) is a four-circuit if and only if B ⌢ A, B ⌢ C(λ) and B is non-singular, then, for a fixed λ, λ ≠ 0, −4η, the set of possible points B is (T_{A} ∩ T_{C(λ)}) \ Q.

It is straight-forward to show that the quadric T_{A} has equation

x_{0}^{2} − \frac{ηx_{1}^{2}}{4} − \frac{x_{2}^{2}}{4η} − \frac{x_{3}^{2}}{4η} + \frac{x_{1}x_{2}}{2} − \frac{x_{1}x_{3}}{2} − \frac{x_{2}x_{3}}{2η} + x_{3}x_{4} = 0,

and the quadric T_{C(λ)} has equation

x_{0}^{2} − \frac{ηx_{1}^{2}}{4} − \frac{x_{2}^{2}}{4η} − \frac{λ^{2}x_{4}^{2}}{4η} + \frac{x_{1}x_{2}}{2} − \frac{λx_{1}x_{4}}{2} − \frac{λx_{2}x_{4}}{2η} + x_{3}x_{4} = 0.

We will now show that for λ ≠ 0, −4η, there exist two hyperplanes of PG(4, q), Σ_{1}(λ) and Σ_{2}(λ), such that Σ_{1}(λ) ∩ Q = E_{1}(λ) and Σ_{2}(λ) ∩ Q = E_{2}(λ) are both non-singular elliptic quadrics and T_{A} ∩ T_{C(λ)} = E_{1}(λ) ∪ E_{2}(λ).

Let Σ_{1}(λ) be the hyperplane of PG(4, q) with coordinates [0, 0, 0, 1, −λ]. Since Σ_{1}(λ) contains neither A nor C(λ) it meets T_{A} in a non-singular elliptic quadric, and similarly for T_{C(λ)}. The equation for the section of T_{A} in Σ_{1}(λ) is that of T_{A} with the substitution x_{3} = λx_{4}:

x_{0}^{2} − \frac{ηx_{1}^{2}}{4} − \frac{x_{2}^{2}}{4η} − \frac{λ^{2}x_{4}^{2}}{4η} + \frac{x_{1}x_{2}}{2} − \frac{λx_{1}x_{4}}{2} − \frac{λx_{2}x_{4}}{2η} + λx_{4}^{2} = 0,

which is the same as the equation for the section of T_{C(λ)} in Σ_{1}(λ), that is Σ_{1}(λ) ∩ T_{A} = Σ_{1}(λ) ∩ T_{C(λ)}. Let E_{1}(λ) be the non-singular elliptic quadric Σ_{1}(λ) ∩ T_{A} = Σ_{1}(λ) ∩ T_{C(λ)}.

Similarly, let Σ_{2}(λ) be the hyperplane of PG(4, q) with coordinates [0, 2η, 2, 1, λ]. The hyperplane Σ_{2}(λ) contains neither A nor C(λ) and the intersection of T_{A} with Σ_{2}(λ) has equation:

x_{0}^{2} − \frac{ηx_{1}^{2}}{4} − \frac{x_{2}^{2}}{4η} − \frac{(−2ηx_{1} − 2x_{2} − λx_{4})^{2}}{4η} + \frac{x_{1}x_{2}}{2}

−\frac{x_{1}(−2ηx_{1} − 2x_{2} − λx_{4})}{2} − \frac{x_{2}(−2ηx_{1} − 2x_{2} − λx_{4})}{2η} + x_{3}x_{4} = 0

⟺ x_{0}^{2} + (−\frac{η}{4} − η + η)x_{1}^{2} + (−\frac{1}{4η} − \frac{1}{η} + \frac{1}{η})x_{2}^{2} − \frac{λ^{2}}{4η} x_{4}^{2} + (\frac{1}{2} − 2 + 1 + 1)x_{1}x_{2}

+(−λ + \frac{λ}{2})x_{1}x_{4} + (−\frac{λ}{η} + \frac{λ}{2η})x_{2}x_{4} + x_{3}x_{4} = 0

⟺ x_{0}^{2} − \frac{ηx_{1}^{2}}{4} − \frac{x_{2}^{2}}{4η} − \frac{λ^{2}x_{4}^{2}}{4η} + \frac{x_{1}x_{2}}{2} − \frac{λx_{1}x_{4}}{2} − \frac{λx_{2}x_{4}}{2η} + x_{3}x_{4} = 0.

Thus the intersection of T_{A} with Σ_{2}(λ) is the same as the intersection of T_{C(λ)} with Σ_{2}(λ). Let this non-singular elliptic quadric section be E_{2}(λ).

The intersection of the quadrics E_{1}(λ) and E_{2}(λ) is contained in the plane π_{λ} = Σ_{1}(λ)∩Σ_{2}(λ), which has equations x_{3} = λx_{4} and 2ηx_{1} + 2x_{2} + x_{3} + λx_{4} = 0. Substituting the first equation of π_{λ} into the second yields ηx_{1} + x_{2} + λx_{4} = 0, or ηx_{1} + x_{2} + x_{3} = 0, which are the equations of C(λ)^{⊥} and A^{⊥} respectively. So π_{λ} ⊂ C(λ)^{⊥}, A^{⊥} and in fact π_{λ} = C(λ)^{⊥} ∩ A^{⊥}. Thus (T_{A} ∩ T_{C(λ)}) ∩ Q = π_{λ} ∩ Q, and so E_{1}(λ) ∩ E_{2}(λ) = E_{1}(λ) ∩ Q = E_{2}(λ) ∩ Q.

Using the above we now prove the following result.

Lemma 4.3.4 Let λ ∈ GF(q)\{0, −4η} and let Σ_{1}(λ) and Σ_{2}(λ) be the hyperplanes of PG(4, q) with the equations x_{3}−λx_{4} = 0 and 2ηx_{1}+2x_{2}+x_{3}+λx_{4} = 0, respectively. Let E_{1}(λ) and E_{2}(λ) be the non-singular elliptic quadrics, Σ_{1}(λ)∩Q and Σ_{2}(λ)∩Q, respectively. If A = (0, 1, η, 0, 1) and C(λ) = (0, 1, η, λ, 0), then T_{A} ∩T_{C(λ)} = E_{1}(λ)∪E_{2}(λ) and E_{1}(λ)∩Q = E_{2}(λ)∩Q = E_{1}(λ)∩E_{2}(λ).

Proof: We have already established that E_{1}(λ) ∩ Q = E_{2}(λ) ∩ Q = E_{1}(λ) ∩ E_{2}(λ) and that E_{1}(λ) ∪ E_{2}(λ) ⊆ T_{A} ∩ T_{C(λ)}, so we must now show that T_{A} ∩ T_{C(λ)} ⊆ E_{1}(λ) ∪ E_{2}(λ).

Let P be a point of T_{A} ∩ T_{C(λ)}. If P ∈ Q, then by the above T_{A} ∩ T_{C(λ)} ∩ Q = π_{λ} ∩ Q = E_{1}(λ) ∩ E_{2}(λ) and so P ∈ E_{1}(λ) ∪ E_{2}(λ). If P ∉ Q, then consider the line ℓ = ⟨P, A⟩ tangent to Q. By Lemma 4.3.1, since C(λ) is cotangent to P and P ∉ Q, it follows that C(λ) is cotangent to exactly two points of ℓ, both of which are non-singular with respect to Q and one of which is P. We also know that C(λ) cotangent to the points P_{1} = ℓ ∩ E_{1}(λ) and P_{2} = ℓ ∩ E_{1}(λ). Since E_{1}(λ) ∩ E_{2}(λ) ⊂ Q, it follows that P_{1} and P_{2} are distinct and so either P = P_{1} or P_{2}. Thus P ∈ E_{1}(λ) ∪ E_{2}(λ).

□

From Lemma 4.2.7, if c is the induced four-circuit (O, A, B, C(λ)) of G, for B ∈ (E_{1}(λ) ∪ E_{2}(λ)) \ Q, and the subgraph G_{{A,C(λ)}} of G is connected, then the edge set of c is null homologous. Thus if G_{{A,C(λ)}} is connected, then all the canonical induced four-circuits of G have null homologous edge sets. In the next section we show that G_{{A,C(λ)}} is connected.

4.3.2 The graph G_{{A,C(λ)}}, λ ≠ 0, −4η

The subgraph G_{{A,C(λ)}} of G has vertex set the set of points (E_{1}(λ) ∪ E_{2}(λ)) \ Q = (E_{1}(λ) ∪ E_{1}(λ)) \ (E_{1}(λ) ∩ E_{2}(λ)) of PG(4, q). Two vertices of this set are adjacent if they are cotangent. We now prove a result which simplifies the task of showing that G_{{A,C(λ)}} is connected.

Lemma 4.3.5 Let the graph Γ_{1}(λ) be the subgraph of G with vertex set E_{1}(λ) \ Q. If Γ_{1}(λ) is connected then G_{{A,C(λ)}} is connected.

Proof: Suppose that Γ_{1}(λ) is connected and that X and Y are two vertices of G_{{A,C(λ)}}. If X and Y are both vertices of Γ_{1}(λ), then there is a path connecting them in Γ_{1}(λ) and hence in G_{{A,C(λ)}}. Now suppose that X is a vertex of Γ_{1}(λ) and that Y is not. That is, X is a point of E_{1}(λ) \ Q and Y is a point of E_{2}(λ) \ Q. Consider the line ⟨Y, A⟩ that is tangent to Q. Since ⟨Y, A⟩ contains a point of E_{2} \Q it also contains a (distinct) point of E_{1}(λ), Y′ say. As X and Y′ are connected in Γ_{1}(λ), it follows that X and Y are connected in G_{{A,C(λ)}}. A similar argument shows that if neither X nor Y are vertices of Γ_{1}(λ), then they are connected in G_{{A,C(λ)}}.

□

We now prove that Γ_{1}(λ) is connected. The first step is to prove some results on the group of Γ_{1}(λ).

Lemma 4.3.6 The stabiliser of E_{1}(λ) in the group of Q is transitive on the points of E_{1}(λ) \ Q.

Proof: Let X and Y be two points of E_{1}(λ) \ Q. Since E_{1}(λ), ⟨X, Y ⟩ and the plane π_{λ} are contained in the hyperplane Σ_{1}(λ) of PG(4, q), it follows that the tangent ⟨X, Y ⟩ meets the plane π_{λ} in a point P. Recall from Section 4.3.1 that π_{λ} was the plane such that E_{1}(λ)∩Q = Q∩π_{λ} and so X, Y ∉ π_{λ} and by Lemma 4.3.1 P ∉ Q. Since P is not on Q, we consider the automorphism μ_{P} of Q (see Section 1.2). Now from Section 4.3.1 we have that π_{λ} ⊂ A^{⊥}, C(λ)^{⊥} and so since P ∈ π_{λ}, it follows that μ_{P} fixes both A^{⊥} and C(λ)^{⊥}. Thus μ_{P} fixes T_{A} ∩ T_{C(λ)}. Also since P ∈ Σ_{1}(λ) it follows that μ_{P} fixes Σ_{1}(λ) and so μ_{P} fixes E_{1}(λ). Since μ_{P} fixes the lien ⟨X, Y ⟩ and E_{1}(λ) it also fixes ⟨X, Y ⟩ ∩ E_{1}(λ) = {X, Y }. If μ_{P} fixes X, then it must also fix Y. Now μ_{P} fixes X if and only if

μ_{P} (X) = X − \frac{β(X, P)}{Q(P)}P = X,

which is the case if and only if β(X, P) = 0. That is, X ∈ P^{⊥}. So if μ_{P} fixes both X and Y, then the line ⟨X, Y ⟩ is contained in P^{⊥}, which implies that P ∈ P^{⊥}. But this is a contradiction as P ∉ Q. Thus μ_{P} (X) = Y.

□

Corollary 4.3.7 The automorphism group of the graph Γ_{1}(λ) is transitive on the vertices of Γ_{1}(λ).

Proof: Any automorphism of Q fixing E_{1}(λ) induces an automorphism of Γ_{1}(λ).

□

Corollary 4.3.8 The automorphism group of Γ_{1}(λ) is transitive on the connected components of Γ_{1}(λ).

We now proceed to show that Γ_{1}(λ) has a connected component containing more than half the vertices of Γ_{1}(λ), since by Corollary 4.3.8 this means that Γ_{1}(λ) is connected.

Now if X and Y are two points of E_{1}(λ) \ Q, then they are cotangent if and only if the line ⟨X, Y ⟩ is tangent to the quadric Σ_{1}(λ) ∩ Q. That is, the graph Γ_{1}(λ) can be defined from E_{1}(λ) \ Q and Σ_{1}(λ) ∩ Q. We first define an isomorphism from Σ_{1}(λ) to PG(3, q) which will make calculating the connected components of Γ_{1}(λ) a little easier.

Let U : Σ_{1}(λ) → PG(3, q) (where PG(3, q) has coordinates x_{0}, x_{1}, x_{2}, x_{3}) be such that U(x_{0}, x_{1}, x_{2}, λx_{4}, x_{4}) = (x_{0}, x_{1}, x_{2}, x_{4}). If E_{1}(λ)′ = U(E_{1}(λ)), Q′ = U(Σ_{1}(λ) ∩ Q) and π_{λ} = U(π_{λ}), then E_{1}(λ)′, Q′ and π′_{λ} have equations

\begin{aligned}E_{1}(λ)′ & : & x_{0}^{2} − \frac{ηx_{1}^{2}}{4} − \frac{x_{2}^{2}}{4η} − \frac{λ^{2}x_{3}^{2}}{4η} + \frac{x_{1}x_{2}}{2} − \frac{λx_{1}x_{3}}{2} − \frac{λx_{2}x_{3}}{2η} + λx_{3}^{2} & = & 0 \\ Q′ & : & x_{0}^{2} + x_{1}x_{2} + λx_{3}^{2} & = & 0 \\ π′_{λ} & : & ηx_{1} + x_{2} + λx_{3} & = & 0 \end{aligned}

If X is the point of PG(3, q) with coordinates (x_{0}, x_{1}, x_{2}, x_{3}), then let Q′(X) = x_{0}^{2} + x_{1}x_{2} + λx_{3}^{2}. The bilinear form of PG(3, q) associated with Q′ is β′(X, Y) = 2x_{0}y_{0}+x_{1}y_{2}+x_{2}y_{1}+2λx_{3}y_{3}, and we will denote the polarity associated with Q′ by ⊥′. If X and Y are two distinct, non-singular points of PG(3, q), then X and Y are cotangent with respect to Q′ if and only if Δ′(X, Y) = 0 where Δ′(X, Y) = (β′(X, Y))^{2} − 4Q′(X)Q′(Y). Using the above, an alternative representation of Γ_{1}(λ) is

Vertex set: E_{1}(λ)′ \ Q′.

Adjacency: Cotangency with respect to Q′.

At this point we note that, as in Theorem 4.3.6, if X and Y are two points of E_{1}(λ)′ \ Q′ and P = ⟨X, Y ⟩ ∩ π′_{λ}, then μ_{P} fixes both E_{1}(λ)′ and Q′, as well as interchanging X and Y.

Now O ∈ E_{1}(λ) \ Q and so O′ = (0, 1, η, 0) is the image of O under U, O′ = (0, 1, η, 0) and O′ ∈ E_{1}(λ)′ \ Q′. We now determine the points of E_{1}(λ)′ that are cotangent to O′. Let π_{O′} be the plane of PG(3, q) with equation 2ηx_{1} + 2x_{2} + λx_{3} = 0 and note that O′ ∉ π_{O′}. The points of π_{O′} that lie in E_{1}(λ)′ have equation

x_{0}^{2} − \frac{ηx_{1}^{2}}{4} − \frac{x_{2}^{2}}{4η} − \frac{(2ηx_{1} + 2x_{2})^{2}}{4η} + \frac{x_{1}x_{2}}{2} − \frac{(−2ηx_{1} − 2x_{2})x_{1}}{2}

−\frac{(−2ηx_{1} − 2x_{2})x_{2}}{2η} + \frac{(2ηx_{1} + 2x_{2})^{2}}{λ}  = 0
⟺ x_{0}^{2} − \frac{ηx_{1}^{2}}{4} − \frac{x_{2}^{2}}{4η} − ηx_{1}^{2} − \frac{x_{2}^{2}}{η} − 2x_{1}x_{2} + \frac{x_{1}x_{2}}{2} + ηx_{1}^{2} + x_{1}x_{2} + x_{1}x_{2}
+\frac{x_{2}^{2}}{η} + \frac{(2ηx_{1} + 2x_{2})^{2}}{λ}  = 0
⟺  x_{0}^{2} − \frac{ηx_{1}^{2}}{4} − \frac{x_{2}^{2}}{4η} + \frac{x_{1}x_{2}}{2} + \frac{(2ηx_{1} + 2x_{2})^{2}}{λ}  = 0.

The points of π_{O′} that are cotangent to O′ are those that satisfy the equation

Δ′ (O′, (x_{0}, x_{1}, x_{2}, −\frac{2ηx_{1}}{λ} − \frac{2x_{2}}{λ} ))  = 0
⟺ (ηx_{1} + x_{2})^{2} − 4η (x_{0}^{2} + x_{1}x_{2} + \frac{(2ηx_{1} + 2x_{2})^{2}}{λ} ) = 0
⟺  x_{0}^{2} − \frac{ηx_{1}^{2}}{4} − \frac{x_{2}^{2}}{4η} + \frac{x_{1}x_{2}}{2} + \frac{(2ηx_{1} + 2x_{2})^{2}}{λ}  = 0.  (4.3.4)

Thus the set of points of π_{O′} that are cotangent to O′ is exactly the intersection of E_{1}(λ)′ with π_{O′}. Now the set of lines incident with O′ that are tangent to Q′ form a quadratic cone T′_{O′} (regardless of whether Q′ is a hyperbolic or elliptic quadric), and so it follows that the set of points of π_{O′} cotangent to O′ is the conic T′_{O′} ∩ π_{O′} = C_{O′}. Also since the point O′ is on exactly q + 1 tangents to Q′, it follows that the set of points of E_{1}(λ), that are cotangent to O′, is exactly C_{O′}.

Note that the discussion preceding Lemma 4.3.4 shows that E_{1}(λ)′ ∩ Q′ = π′_{λ} ∩ Q′. Thus the points of C_{O′} that are also on Q′ are precisely the points of E_{1}(λ)′ on the line π_{O′} ∩ π′_{λ}. Now any point of E_{1}(λ)′ on the line π_{O′} ∩ π′_{λ} must satisfy the equations, x_{2} = −ηx_{1} and x_{3} = 0. Since η is a non-square of GF(q) there is no point that satisfies these equations. Hence C_{O′} contains no points of Q′.

By the proof of Lemma 4.3.6 the stabiliser of E_{1}(λ)′ in the group of Q′ is transitive on the points of E_{1}(λ)′ \ Q′ and so the above results for O′ apply for all X ∈ E_{1}(λ)′ \ Q′.

Lemma 4.3.9 Let X be a point of E_{1}(λ)′ \Q′ (for λ ≠ 0, −4η). Then the set of points of E_{1}(λ)′ that are cotangent to X form a conic, C_{X} in the plane π_{X}. Furthermore, C_{X} contains no points of Q′.

We now calculate the conditions under which π_{X} = π_{Y} for X, Y ∈ E_{1}(λ)′ \ Q′.

Lemma 4.3.10 Let X ∈ E_{1}(λ)′ \ Q′. If λ ≠ −8η, then for each Y ∈ E_{1}(λ)′ \ Q′ and X ≠ Y it follows that π_{X} ≠ π_{Y}. If λ = −8η, then Y = (π_{X})^{⊥′} is the unique point of E_{1}(λ)′ \ Q′ such that π_{X} = π_{Y} and X ≠ Y.

Proof: We show that the result holds for the case X = O′ and so the general result holds by the transitivity on the points of E_{1}(λ)′ (Lemma 4.3.6).

Now it follows from the proof of Lemma 4.3.6 that if Y ∈ E_{1}(λ)′ \ Q′ and π′_{λ} ∩ ⟨Y, O′⟩ = P, then μ_{P} fixes Q′ and E_{1}(λ)′ \ Q′, and interchanges O′ and Y. Hence π_{O′} = π_{Y} if and only if μ_{P} (π_{O′}) = π_{O′}. We also know from the proof of Lemma 4.3.6 that any point P ∈ π′_{λ} \ Q′ has the property that μ_{P} fixes Q′ and E_{1}(λ)′ \ Q′ and either fixes O′ of maps it to a different point of E_{1}(λ)′ \ Q′. Thus we look for points P of π′_{λ} \ Q′ such that μ_{P} fixes π_{O′} but not O′.

Now μ_{P} fixes π_{O′} if and only if μ_{P} fixes the point (π_{O′})^{⊥′} and this is the case if and only if P ∈ π_{O′} or P = (π_{O′})^{⊥′}. In the first case P is any point of π_{O′} ∩ π′_{λ} which is a line with equations 2ηx_{1} + 2x_{2} + λx_{3} = 0 and ηx_{1} + x_{2} + λx_{3} = 0, that is ηx_{1} + x_{2} = x_{3} = 0. However, the polar plane of O′ with respect to Q′ has equation ηx_{1} + x_{2} = 0 and so contains the line π_{O′} ∩ π′_{λ}. Thus if P ∈ π_{O′} ∩ π′_{λ}, then it is also contained in the polar plane of O′, and so μ_{P} fixes O′.

For the second case we note that P = π_{O′}^{⊥′} = (0, 4, 4η, 1), which lies on π′_{λ} if and only if λ = −8η. In this case, μ_{P} (π_{O′}) = π_{O′} and μ_{P} (O′) ≠ O′.

□

Given Lemma 4.3.9 and Lemma 4.3.10, we are now equipped to show that Γ_{1}(λ) is connected.

Theorem 4.3.11 The graph Γ_{1}(λ) is connected.

Proof: We show that the connected component of Γ_{1}(λ) containing O′ has size greater than |Γ_{1}(λ)|/2 = (q^{2} − q)/2. Recall from Lemma 4.3.9 that the vertices at distance 1 to X form the conic C_{X} in the plane π_{X} of PG(3, q). We now find a lower bound for the number of vertices at distance 0, 1 or 2 to O′.

We separate the proof into two cases, λ = −8η and λ ≠ −8η (note that we still have the restriction that λ ≠ 0, −4η).

Suppose that λ = −8η. To obtain a lower bound for the number of points at distance 2 to O′ we consider the points at distance 1 to a point X ∈ C_{O′}, that is, the points of the conic C_{X}. Since λ = −8η we have that O′, (π_{O′})^{⊥′} ∈ C_{X}. Since there are at most two lines contained in π_{O′} that are incident with X and tangent to Q′, it follows that |C_{O′} ∩ C_{X}| ≤ 2. Thus there at least q −3 of the points in the set C_{X} \{(π_{O′})^{⊥′}} that are at distance 2 to O′. Now suppose that Y ∈ C_{O′} \ {X}, then either π_{Y} ≠ π_{X} and C_{X} ∩ C_{Y} = {O′, (π_{O′})^{⊥′}}, or π_{X} = π_{Y} and C_{X} = C_{Y}. Thus if Z ∈ C_{X} and is at distance 2 to O′, and not equal to (π_{O′})^{⊥′}, then Z is at distance 1 to at most one other point of C_{O′}. Hence a lower bound on the number of points at distance 2 to O′ is (q + 1)(q − 3)/2 + 1 and so a lower bound for the number of points at distance 0, 1 or 2 to O′ is:

\frac{(q + 1)(q − 3)}{2} + 1 + (q + 1) + 1 = \frac{(q + 1)(q − 1)}{2} + 2 > \frac{q^{2} − q}{2} = \frac{|Γ_{1}(λ)|}{2}.

This shows that Γ_{1}(λ) is connected for λ = −8η.

Now suppose that λ ≠ −8η. As above we consider a point X ∈ C_{O′}. By an argument above there are at most two points at distance 1 to X in C_{O′}. Hence at least q − 2 of the points of C_{X} are at distance 2 to O′. Now let Z be a point at distance 2 to O′. Since π_{Z} ≠ π_{O′}, it follows that π_{Z} ∩ π_{O′} is a line and |C_{Z} ∩ C_{O′}| ≤ 2, that is, any point at distance 2 to O′ is at distance 1 to at most two points of C_{O′}. Hence a lower bound for the number of points at distance 2 to O′ is (q + 1)(q − 2)/2. Thus a lower bound for the number of points at distance 0, 1 or 2 to O′ is

\frac{(q + 1)(q − 2)}{2} + (q + 1) + 1 = \frac{(q + 1)q}{2} + 1 > \frac{q^{2} − q}{2} = \frac{|Γ_{1}(λ)|}{2}.

This shows that Γ_{1}(λ) is connected for λ ≠ −8η.

□

We now have the following string of consequences of Theorem 4.3.11.

Corollary 4.3.12 The graph G_{{A,C(λ)}} is connected.

Proof: By Lemma 4.3.5.

□

Theorem 4.3.13 H_{1}(Γ_{G}, ℤ_{2}) is trivial.

Proof: By Theorem 4.2.6 H_{1}(Γ_{G}, ℤ_{2}) is trivial if and only if the edge set of each induced four-circuit of G is null homologous. By Lemma 4.3.2 and Lemma 4.3.3 the edge set of every induced four-circuit of G is null homologous if all the canonical induced four-circuits have null homologous edge sets (where the canonical induced four-circuit are the induced four-circuits (O, A, B, C(λ)) for some λ ∈ GF(q)\{0, −4η}). By Lemma 4.2.7 if G_{{A,C(λ)}} is connected, then the edge set of (O, A, B, C(λ)) is null homologous. Thus Corollary 4.3.12 proves the result.

□

Theorem 4.3.13 has the following corollary:

Theorem 4.3.14 Let S be a GQ of order (q, q^{2}), q odd, and S′ a subquadrangle of S of order q isomorphic to Q(4, q). If each ovoid of S′ subtended by S is isomorphic to an elliptic quadric ovoid, then S is isomorphic to Q(5, q).

Proof: Since H_{1}(Γ_{G}, ℤ_{2}) = H^{1}(Γ_{G}, ℤ_{2}) is trivial, the result follows from Theorem 4.2.3.

□

Combining Theorem 4.3.14 with the corresponding result for q even given in [65] we have the following result for general q.

Theorem 4.3.15 Let S be a GQ of order (q, q^{2}) and S′ a subquadrangle of S of order q isomorphic to Q(4, q). If each ovoid of S′ subtended by S is isomorphic to an elliptic quadric ovoid, then S is isomorphic to Q(5, q).

4.4 Remarks

Theorem 4.3.14 now makes it possible to prove results similar to those in Section 4.1 for the case where q is odd. In the case where q = 9 Penttila and Royle have shown that the only ovoids of Q(4, q) are K1(σ) ovoids, where σ : x ↦ x^{3}, and the elliptic quadric ovoid ([53]). Further (still for q = 9) Penttila has shown that there is no GQ of order (q, q^{2}) with a subquadrangle isomorphic to Q(4, q) that subtends both an elliptic quadric ovoid and a K1(σ) ovoid ([51]). Thus, by Theorem 4.3.14, a new GQ S of order (9, 81) with a subquadrangle isomorphic to Q(4, 9) must subtend an ovoid-rosette configuration in Q(4, 9) where every subtended ovoid is a K1(σ) ovoid. Using the work in Section 4.2 it may be possible to rule out the existence of a new GQ subtending the SPG found in Chapter 3 by characterising S(C_{σ})^{∧} (as in Chapter 3) by its doubly subtended Q(4, 9) subquadrangle. Of course it may also be possible to prove this characterisation for q = p^{h}, where p is an odd prime and h > 1, using the work of Section 4.2, although it is likely to be a more difficult problem than the classical case solved in Section 4.3.

Affine planes and GQs of order s

In this section we study the connections between affine planes and GQs of order s containing a regular point. In particular, it is known that if S is a GQ of order s, with a regular point, then there is an associated affine plane of order s ([49, 1.3.1]). The GQ defines an s-fold cover of the affine plane and we investigate the relationships between the GQ, the affine plane and the associated cover of the affine plane.

To begin we consider the classical case, for q even. Let Q = Q(4, q), the non-singular (parabolic) quadric in PG(4, q) and let I denote incidence between points and lines of Q. For such a quadric there is a unique point N ∉ Q, called the nucleus of Q. The point N has the property that every tangent space to Q contains N. In addition, every line of PG(4, q) incident with N is either tangent to Q or is external to Q [27, Lemma 22.3.3(ii) and Lemma 22.3.1(i)].

Let V be a fixed point of Q. By [49, 3.3.1.(i)] every point of Q(4, q), q even, is regular, and so it follows that V is regular. Let V^{⊥} denote the tangent space to Q at V, then V^{⊥} ∩ Q is a quadratic cone K with vertex V [27, Lemma 22.7.3]. If X is any point of Q \ V^{⊥}, that is, a point of Q not collinear to V in Q, then X^{⊥} ∩ V^{⊥} is a plane of PG(4, q) that intersects K in a conic, C_{X}. We say that X subtends the conic C_{X} in K.

Let ℓ be a line of Q such that V is not incident with ℓ. Then there exists a unique point/line pair (Y, m) such that V I m I Y I ℓ. If Z is a point of the set ℓ \ {Y } (where ℓ is considered as the set of points incident with it), then by the above, Z subtends a conic C_{Z} in K. If Z_{1} and Z_{2} are two distinct point of ℓ \ {Y }, then the tangent spaces Z_{1}^{⊥} and Z_{2}^{⊥} intersect in the plane ⟨ℓ, N⟩ of PG(4, q). Thus, C_{Z_{1}} ≠ C_{Z_{2}}, and in fact C_{Z_{1}} ∩ C_{Z_{2}} = {Y }. Hence each point of ℓ \ {Y } subtends a distinct conic and the line ⟨N, Y ⟩ is a tangent to each of these conics. We say that the tangent ⟨N, Y ⟩ is subtended by ℓ.

Let π be the incidence structure with pointset the set of conics of K whose plane contains N, lineset the tangents to Q contained in V^{⊥}, except the tangent ⟨V, N⟩, and incidence inherited from PG(4, q). Then π is an affine plane, and in fact π is the affine plane obtained by removing

the line (\frac{⟨N, V ⟩}{N})^{D} from the projective plane (\frac{V^{⊥}}{N} )^{D}, and so π ≅ AG(2, q).

If the line m of V^{⊥} is a line of π, then it is determined by the unique point of Q incident with it, since all such lines are incident with N. Thus an equivalent description of π is to redefine the lineset as the set K \ {V }, keep the pointset as the set of conics of V^{⊥} whose plane contains N, and keep incidence as incidence in PG(4, q). This is a specific case of the following theorem.

Theorem 5.0.1 [49, 1.3.1] Let X be a regular point of the GQ S = (P, B, I) of order (s, t). Then the incidence structure with pointset X^{⊥} \ {X}, lineset the set of spans {Y, Z}^{⊥⊥}, where Y, Z ∈ X^{⊥} \ {X}, Y ≁ Z and with natural incidence, is the dual of a net of order s and degree t + 1. If s = t ≥ 2 it is a dual affine plane of order s.

5.1 Covering affine planes

Let S = (P, B, I) be a GQ of order s and (∞) a point of S that is regular. Let π = π_{(∞)} be the affine plane constructed from S and (∞) as in Theorem 5.0.1. We will define a map p from S \ (∞)^{⊥} to π that defines an s-fold cover of π.

Let X be a point of P \ (∞)^{⊥}. Then {X, (∞)}^{⊥} is a point of π. Since (∞) is regular it follows that |{X, (∞)}^{⊥⊥}| = s + 1, and if Y ∈ P \ (∞)^{⊥} then {Y, (∞)}^{⊥} = {X, (∞)}^{⊥} if and only if Y ∈ {X, (∞)}^{⊥⊥}. So if we define the map p from P \ (∞)^{⊥} into the pointset of π by p(X) = {X, (∞)}^{⊥}, then p has the property that for any point {Y, (∞)}^{⊥} of π, the set p^{−1}({Y, (∞)}^{⊥}) = {Y, (∞)}^{⊥⊥} consists of s pairwise non-adjacent points of P \ (∞)^{⊥}.

Now suppose that {X, (∞)}^{⊥} and {Y, (∞)}^{⊥} are two distinct points of π. Then {X, (∞)}^{⊥}∩ {Y, (∞)}^{⊥} = Z for some point Z ∈ (∞)^{⊥}. Now each of the s lines of S incident with Z, but not incident with (∞), is incident with exactly one point of {X, (∞)}^{⊥⊥} \ {(∞)} and exactly one point of {Y, (∞)}^{⊥⊥} \ {(∞)}. Also, any line of S that contains one point of {X, (∞)}^{⊥⊥} \ {(∞)} and one point of {Y, (∞)}^{⊥⊥} \ {(∞)} must contain Z. Thus the set p^{−1}({{X, (∞)}^{⊥}, {Y, (∞)}^{⊥}}) consists of s disjoint, incident pairs of points of P \ (∞)^{⊥}.

If Γ is the subgraph of the point graph of S defined on the vertex set P \ (∞)^{⊥}, then thus far we have shown that (Γ, p) satisfies requirements (i) and (ii) of an s-fold cover of the point graph of π (see Section 1.6). Since any two points of π are collinear, requirement (iii) is satisfied vacuously; so (Γ, p) is an s-fold cover of the point graph of π. We now show that this gives rise to an s-fold cover of π.

Let P be a fixed line of π: that is, a point of (∞) \ (∞)^{⊥}. Then the set {p^{−1}({X, (∞)})^{⊥} : P is incident with {X, (∞)}^{⊥} in π} = {X ∈ P \ (∞)^{⊥} : X I P }. Now if Γ_{p} is the subgraph of Γ defined on the vertex set {X ∈ P \ (∞)^{⊥} : X I P }, then the s lines of S incident with P, but not (∞), divide Γ_{p} into s disjoint complete graphs, each of size s. Thus the map p satisfies condition (iv) in Section 1.6, and so we have the following theorem.

Theorem 5.1.1 Let S = (P, B, I) be a GQ of order s with a regular point (∞). Let π be the affine plane with lineset (∞)^{⊥} \ (∞), pointset the set of spans {X, Y }^{⊥⊥}, where X, Y ∈ (∞)^{⊥} \ (∞), X ≁ Y and natural incidence. Let \overline{π} be the geometry with pointset P \ (∞)^{⊥}, lineset the set of lines of S not incident with (∞), and incidence as in S. Let p : \overline{π} → π be such that: p(X) = {X, (∞)}^{⊥} for X ∈ P \ (∞)^{⊥}, and for ℓ a line of S not incident with (∞), p(ℓ) = Y, where Y is the unique point such that Y I ℓ and Y ∼ (∞). Then (\overline{π}, p) is an s-fold cover of π.

Note that in Theorem 5.1.1 p denotes the map induced on \overline{π} by the map p in the discussion preceding Theorem 5.1.1.

At this point it is natural to consider under what conditions we can “reverse” the covering process in Theorem 5.1.1 and construct a GQ of order s, with a regular point, from an s-fold cover of an affine plane of order s.

Theorem 5.1.2 Let π be an affine plane of order s and (\overline{π}, p) an s-fold cover of π. Let S be the following incidence structure

Points:  (i) Lines of π
(ii) Points of \overline{π}
(iii) (∞)
Lines:  (a) Parallel classes of π
(b) Lines of \overline{π}
Incidence I: (i) (a) Containment

(b) The s covers of the point of type (i)

(ii) (a) None

(b) Incidence in \overline{π}

(iii) (a) All

(b) None

The geometry S is a GQ of order s if and only if for each non-incident point-line pair (P, ℓ), where P is of type (ii) and ℓ is of type (b) and p(P) is not incident with p(ℓ) in π, there is a unique incident point-line pair (Q, m) such that P I m I Q I ℓ. Further, if S is a GQ, then the point (∞) is regular.

Proof: In Section 1.6 it was noted that if ℓ is a line of π and is incident with the set of points {P_{1}, P_{2}, …, P_{s+1}}, then each element of p^{−1}(ℓ), as a set of points, has the form {P′_{1}, P′_{2}, …, P′_{s+1}}, where p(P′_{i}) = P_{i} for i = 1, 2, …, s + 1. We will be using this fact, implicitly, throughout this proof.

Let ℓ be a line of type (a) (that is, a parallel class of π). Then, in S, ℓ is incident with the s lines of π it contains as a parallel class of π and the point (∞), s + 1 in total. Let ℓ be a line of type (b) (that is, a line of \overline{π}). Then, in S, ℓ is incident with the s points incident with it in \overline{π} and the point of type (i) p(ℓ), s + 1 in total. We now show that two distinct lines of S intersect in at most one point. Let ℓ and m be two lines of S, with ℓ ≠ m. If ℓ and m are both of type (a), then they are concurrent at the unique point (∞). If ℓ is of type (a) and m is of type (b), then they are concurrent if and only if the line m (as a line of \overline{π}) is a cover of the line n of π, which is contained in the parallel class m. In this case n is the unique point incident with both ℓ and m. If ℓ and m are both of type (b) and cover the same line n of π, then they are concurrent at the point n and no other point of S. If ℓ and m cover different lines of π and X is a point of S such that X I ℓ, m, then X is a point of \overline{π}. Since \overline{π} is a geometry there is at most one such point.

Let m be a point of type (i) (that is, a line of π). Then, in S, m is incident with the s lines of type (b), p^{−1}(m), and the parallel class of π containing it, s + 1 in total. Let P be a point of type (ii) (that is, a point of \overline{π}). Then, in S, P is incident with the s + 1 lines of \overline{π} incident with it in \overline{π}. The point (∞) is incident with the s + 1 lines of type (a). Now let X be a point of type (i) and Y a point of type (ii). Any line of S incident with both X and Y must be a line of \overline{π}, covering X (as a line of π) and incident with Y in \overline{π}. Since it is a property of the cover (\overline{π}, p) that the s covers of m are pairwise non-concurrent, it follows that m is incident with at most one such line. It is straight-forward to show for the other cases that two points of S are incident with at most one common line.

It now remains to check the GQ incidence axiom. Let (P, ℓ) be a non-incident point-line pair of S. Suppose P is of type (i) and ℓ is of type (a). Let L be the parallel class of π containing P. Then ((∞), L) is the unique point-line pair such that P I L I (∞) I ℓ. Now suppose that P is of type (i) and ℓ is of type (b). Then p(ℓ) is a line of π such that P ≠ p(ℓ). If P and p(ℓ) belong to the same parallel class of π, L say, then (p(ℓ), L) is the unique point-line pair such that P I L I p(ℓ) I ℓ. If P and p(ℓ) intersect in the point Q, say, then there is a unique point Q′ of π such that Q′ I ℓ and p(Q′) = Q. There is also a unique line m of \overline{π} such that p(m) = P and Q′ I m. Now suppose that P is of type (ii) and ℓ is of type (a). Let m be the unique line of π such that p(P) ∈ m and m ∈ ℓ (as a parallel class). Then there is a unique element m′ of p^{−1}(m), such that P is incident with m in \overline{π} and (m, m′) is the unique point-line pair such that P I m′ I m I ℓ. Now suppose that P is (∞) and ℓ is of type (b). Let L be the parallel class of π containing p(ℓ), then (L, p(ℓ)) is the unique point-line pair such that P I L I p(ℓ) I ℓ.

The only cases that we haven’t yet considered are where P is (∞) and ℓ is of type (a), in which case P and ℓ are always incident, and where P is of type (ii) and ℓ is of type (b). In this second case, if P and ℓ are non-incident in S and p(P) is incident with p(ℓ) in π, then there exists a unique element of p^{−1}(p(ℓ)), m say, such that P I m. Then (p(ℓ), m) is the unique point-line pair such that P I m I p(ℓ) = p(m) I ℓ.

This leaves the case where P is of type (ii), ℓ is of type (b) and p(P) is not incident with p(ℓ) in π. Hence S is a GQ if and only if pairs of this type satisfy the GQ incidence axiom.

If S is a GQ, then consider the non-collinear pair of points of S, {(∞), X}. It follows that X must be a point of type (ii). Now {X, (∞)}^{⊥⊥} = p^{−1}(p(X))∪(∞) and so |{X, (∞)}^{⊥⊥}| = s+1. Thus the pair {(∞), X} is regular and therefore (∞) is also regular.

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Note that if π is an affine plane of order s and (\overline{π}, p) is an s-fold cover of π such that the incidence structure S defined in Theorem 5.1.2 is a GQ with regular point (∞), then the s-fold cover of π constructed as in Theorem 5.1.1 is (\overline{π}, p).

Let π be an affine plane of order s and (\overline{π}, p) an s-fold cover of π giving rise to a GQ of order s as in Theorem 5.1.2. If Γ is the point graph of π (a complete graph on s^{2} vertices) and \overline{Γ} the point graph of \overline{π}, then (\overline{Γ}, p) is an s-fold cover of Γ. We now prove that \overline{Γ} is antipodal and distance-regular. (See Section 1.1 for the definition of an antipodal graph and a distance regular graph.)

Theorem 5.1.3 Let S be a GQ of order s, with a regular point (∞), and let π be the affine plane constructed from S and (∞) as in Theorem 5.0.1. Let (\overline{π}, p) be the s-fold cover of π constructed in Theorem 5.1.1. If Γ is the point graph of \overline{π}, then Γ is an antipodal distance-regular graph, with intersection set {s^{2} − 1, s(s − 1), 1; 1, s^{2}, s^{2} − 1}.

Proof: For the purposes of this proof we will use ∼ to denote adjacency in the graph \overline{Γ} and d(X, Y) to denote the distance between two vertices X and Y of \overline{Γ}.

First observe that Γ is a regular graph with valence s^{2} − 1. Now let X and Y be two non-collinear points of \overline{π}. If p(X) ≠ p(Y) then, in S, \{(∞), X\}^{⊥} ≠ \{(∞), Y \}^{⊥}. Let Z be a point of the set {(∞), X}^{⊥} \ {(∞), Y }^{⊥} and ℓ the line ⟨Z, X⟩. Now Y ∉ ℓ and Y is not collinear to Z since Z ∉ {(∞), Y }^{⊥}. So it follows from the GQ incidence axiom that there exists a point Z′ ∈ ℓ \ {Z} collinear to both Y and X. Hence d(X, Y) = 2. If p(X) = p(Y), then {(∞), X}^{⊥} = {(∞), Y }^{⊥} in S and so in S we have that {X, Y }^{⊥} = {(∞), X}^{⊥} ⊂ (∞)^{⊥}. Thus there are no points of \overline{π} collinear to both X and Y and so d(X, Y) > 2. If Z is a point of \overline{π} collinear to X, then Z is not collinear to Y and p(Z) ≠ p(Y). Hence d(Y, Z) = 2 and therefore d(X, Y) = 3. Thus, the diameter of \overline{Γ} is 3. We now consider the graph \overline{Γ}_{3}, which is defined on the same vertex set as \overline{Γ} and has two vertices adjacent if they are at distance 3 in \overline{Γ}. If P is a point of π, then the set p^{−1}(P), as vertices of Γ_{3}, is a clique and so \overline{Γ} is an antipodal graph.

We now show that \overline{Γ} is distance-regular by calculating its intersection array. Let X and Y be two distinct vertices of \overline{Γ}. The parameter b_{0} is the valence of \overline{Γ}, s^{2} − 1, and c_{0} = 0. Now suppose that d(X, Y) = 1 (so X ∼ Y), Z is a vertex of \overline{Γ} and X ∼ Z. If, as a point of S, Z is not incident with the line spanned by X and Y (considered as points of S), then Y ≁ Z, since otherwise \{X, Y, Z\} is a triangle in S; and so d(Y, Z) = 2. Hence, the vertices of \overline{Γ} at distance 1 to X and distance 2 to Y are precisely the points of S \ (∞)^{⊥} that are collinear to X but not incident with the line ⟨X, Y ⟩. Thus b_{1} = s(s − 1). The parameter c_{1} = 1. Now suppose that d(X, Y) = 2. Since p(X) and p(Y) are adjacent in the point graph of π, then it follows that Y is adjacent to a unique element of the set p^{−1}(p(X)), that is b_{2} = 1. The set of vertices at distance 1 to both X and Y is the set of points {X, Y }^{⊥} \ (∞)^{⊥} of S. Since |{X, Y }^{⊥} ∩ (∞)^{⊥}| = 1, it follows that c_{2} = s. Finally, suppose that d(X, Y) = 3. If Z is a vertex adjacent to X, then d(Y, Z) ≠ 1 and d(Y, Z) ≠ 3, since in the case d(Y, Z) = 3 the point Z is antipodal to Y and so must also be antipodal to X. Thus every point adjacent to X in \overline{Γ} is at distance 2 to Y, and so c_{3} = s^{2} − 1.

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5.2 Algebraic covers and GQs

In Theorem 5.1.2 we constructed a geometry S from an affine plane π of order s and (\overline{π}, p), an s-fold cover of π. The geometry S was a GQ of order s if and only if \overline{π} satisfied certain conditions. In this section we consider the case where (\overline{π}, p) is an algebraic s-fold cover of π over the abelian group A, defined by the 1-cochain c. We establish the conditions on the 1-cochain c under which S is a GQ of order s and rewrite Theorem 5.1.2 for the case where (\overline{π}, p) is an algebraic s fold cover of π over A. We will then show that every known GQ of order s, with a regular point, can be constructed via an algebraic cover of AG(2, q).

Let π be an affine plane of order s and (\overline{π}, p) and algebraic s-fold cover of π over the abelian group A, defined by the 1-cochain c. The 1-cochain c is said to satisfy the GQ condition if

δc(P, Q, R) = 0 ⟺ P, Q, R are distinct, collinear points of π.  (5.2.1)

In the following theorem we shall prove that this condition on c is equivalent to the conditions on (\overline{π}, p) under which the geometry S in Theorem 5.1.2 is a GQ of order s.

Note that in 5.2.1 and for the rest of the chapter we have adopted the convention of representing any coboundary operator by δ and allowing the context to determine which coboundary operator it is. For example, in 5.2.1 δ is the coboundary operator δ^{1}.

Theorem 5.2.1 Let π be an affine plane of order s and let (\overline{π}, p) be an algebraic s-fold cover of π over the abelian group A, defined by the 1-cochain c. Let S be the incidence structure defined from π and (\overline{π}, p) as in Theorem 5.1.2. Then, S is a GQ if and only if the 1-cochain c satisfies the GQ condition 5.2.1. Further, if S is a GQ, then the point (∞) is regular.

Proof: First we observe that since (\overline{π}, p) is an algebraic s-fold cover of π and is defined by c, it follows from Section 1.6 that if P, Q, R are distinct collinear points of π, then δc(P, Q, R) = 0. Thus we are required to show that S is a GQ if and only if δc(P, Q, R) = 0 implies that P, Q, R are distinct collinear points of π: equivalently, if P, Q, R is a triple of distinct non-collinear points of π, then δc(P, Q, R) ≠ 0.

Now recall from Theorem 5.1.2 that S is a GQ if and only if each non-incident point-line pair (\overline{P} , ℓ), where \overline{P} is a point of type (ii) and ℓ is a line of type (b) and p(\overline{P}) is not incident with p(ℓ) in π, satisfies the GQ incidence axiom (axiom (iii)).

Let (\overline{P} , ℓ) be such a point-line pair, then \overline{P} = (P, α) for some P ∈ π and α ∈ A (that is, P = p(\overline{P})) such that P ∉ p(ℓ). If p^{−1}(p(ℓ)) = \{ℓ_{1}, ℓ_{2}, …, ℓ_{s}\} and (without loss of generality) ℓ = ℓ_{1}, then for each i = 1, 2, …, s the point-line pair ((P, α), ℓ_{i}) is a non-incident pointline pair, where \overline{P} = (P, α) is a point of type (ii), ℓ_{i} a line of type (b) and P ∉ p(ℓ_{i}). If p(ℓ_{1}) = {Q_{1}, Q_{2}, …, Q_{s}}, then (P, α) is collinear to exactly one cover of each of the Q_{i}. In particular, (P, α) is collinear to (Q_{i}, α − c(P, Q_{i})) for each i = 1, 2, …, s. Thus the s nonincident point-line pairs ((P, α), ℓ_{1}), ((P, α), ℓ_{2}), …, ((P, α), ℓ_{s}) each satisfy the GQ incidence axiom if and only if each line ℓ_{i} is incident with a unique point of the form (Q_{j}, α − c(P, Q_{j})). Now each point (Q_{j}, α − c(P, Q_{j})) is incident with exactly one line of p^{−1}(p(ℓ_{1})) and so each line ℓ_{i} is incident with a unique point of the form (Q_{j}, α − c(P, Q_{j})) if and only if no line of p^{−1}(p(ℓ_{1})) is incident with two points of (Q_{1}, α − c(P, Q_{1})), …, (Q_{s}, α − c(P, Q_{s})). Now if the points (Q_{i}, α − c(P, Q_{i})) and (Q_{j}, α − c(P, Q_{j})) are collinear, then they span a line of p^{−1}(p(ℓ)), and so no line of p^{−1}(p(ℓ_{1})) is incident with two points of (Q_{1}, α−c(P, Q_{1})), …, (Q_{s}, α−c(P, Q_{s})) if and only if no two points of (Q_{1}, α − c(P, Q_{1})), …, (Q_{s}, α − c(P, Q_{s})) are collinear. That is

(Q_{i}, α − c(P, Q_{i})) ≁ (Q_{j}, α − c(P, Q_{j})) for all Q_{i}, Q_{j} such that i, j = 1, …, s with i ≠ j.

Thus the point-line pairs ((P, α), ℓ_{1}), …, ((P, α), ℓ_{s}) all satisfy the GQ incidence axiom if and only if

δc(P, Q_{i}, Q_{j}) ≠ 0 for all Q_{i}, Q_{j} such that i, j = 1, …, s with i ≠ j.

Hence every non-incident point-line pair (\overline{P} , ℓ), where \overline{P} is a point of type (ii) and ℓ is a line of type (b) and p(\overline{P}) is not incident with p(ℓ) in π, satisfies the GQ incidence axiom if and only if

δc(P, Q, R) ≠ 0 for all distinct non-collinear triples P, Q, R of points of π .

From this and our observation at the start of the proof, it follows S is a GQ if and only if the 1-cochain c satisfies the GQ condition 5.2.1.

By Theorem 5.1.2 if S is a GQ then the point (∞) is regular.

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Theorem 5.2.1 demonstrates that if c is a 1-cochain on the simplicial complex of the point graph of an affine plane π of order s and c satisfies the GQ condition, then c defines an algebraic s-fold cover of π. Further, this algebraic s-fold cover gives rise to a GQ S of order s by the construction in Theorem 5.1.2. In this case the GQ S is said to be defined by c and we say that c defines S.

The only known examples of GQs of order s containing a regular point are: W (q) which has all points regular by [49, 3.2.1 and 3.3.1], T_{2}(O) where O is an oval of PG(2, q), q even, which has regular point (∞) by [49, 3.3.2 (i)] (for the construction of T_{2}(O) see Section 1.4.3) and the dual T_{2}(O)^{∧} of T_{2}(O), where O is not a translation oval of PG(2, q), which has regular points corresponding to the lines of type (b) of T_{2}(O) by [49, 3.2.2]. It should be noted that for q even W (q) ≅ Q(4, q) ([49, 3.2.2]) and that Q(4, q) ≅ T_{2}(O) where O is a non-degenerate conic ([49, 3.2.2]. Also, by [49, 12.5.2], the GQ T_{2}(O)^{∧} is isomorphic to the GQ T_{2}(O′), for some O′ if and only if O is a translation oval. In this case, T_{2}(O)^{∧} ≅ T_{2}(O). If O is a translation oval, then T_{2}(O) has q + 2 regular points, while if O is not a translation oval, then T_{2}(O) has a unique regular point (∞), by [49, 12.4.5 (ii)].

In the following two sections we will derive algebraic q-fold covers of AG(2, q) which give rise to W (q) and T_{2}(O), respectively, by the construction in Theorem 5.2.1. In the second case the regular point is (∞). In Section 5.6 we will consider the regular points of T_{2}(O)^{∧} dual to the lines of type (b) of T_{2}(O) and derive corresponding algebraic covers.

5.2.1 A cover associated with the GQ W (q)

Consider W (q) defined as the set of absolute points and absolute lines of a symplectic polarity of PG(3, q). If P is a point of W (q), then P is a regular point of W (q). The q + 1 lines of W (q) that contain P form a pencil of lines, through P, on the polar plane π_{P} of P ([49, 3.1.1(iii)]).

Now let P be a fixed point of W (q). We will determine the affine plane π constructed from W (q) and the regular point P via Theorem 5.0.1, and also calculate the cover of π constructed from W (q), as in Theorem 5.1.1. The lines of π are the points of P^{⊥} \ {P }, which is the set of points π_{P} \ {P } of PG(3, q). The points of π are the sets of points {P, X}^{⊥}, where X ∈ W (q) \ P^{⊥}. If π_{X} is the polar plane of X, then {P, X}^{⊥} is equal to the set of points on the line π_{P} ∩ π_{X} (note that P ∉ π_{X}). Hence, π = (π_{P} \ {P })^{D} ≅ AG(2, q).

If ℓ is a point of π (and so ℓ is a line of π_{P}), then the point X of W (q) \ P^{⊥} is a cover of ℓ if and only if ℓ ⊂ π_{X}. This is the case if and only if X is incident with the image of ℓ under the symplectic polarity. Thus if ℓ′ is the image of ℓ under the symplectic polarity then P ∈ ℓ′ and the q covers of ℓ are the points of ℓ′ \ {P }. If Q is a line of π, and so a point of π_{P}, then it is covered by the set of lines of W (q), not ⟨P, Q⟩, that are contained in the plane π_{Q}.

Now we introduce coordinates. Let W (q) have the canonical bilinear form x_{0}y_{1} − x_{1}y_{0} + x_{2}y_{3} − x_{3}y_{2} = 0. If X = (x_{0}, x_{1}, x_{2}, x_{3}), then π_{X} has plane coordinates [−x_{1}, x_{0}, −x_{3}, x_{2}]. If P = (0, 1, 0, 0), then π_{P} = [1, 0, 0, 0], π has pointset {[0, 1, x_{1}, x_{2}]∩π_{P} : x_{1}, x_{2} ∈ GF(q)} and the covers of the point [0, 1, x_{1}, x_{2}]∩π_{P} of π are the elements of the set {(1, λ, x_{2}, −x_{1}) : λ ∈ GF(q)}. If we denote [0, 1, x_{1}, x_{2}] ∩ π_{P} by (x_{1}, x_{2}), then π assumes the canonical form of AG(2, q). In addition, we denote (1, λ, x_{2}, −x_{1}) by the pair ((x_{1}, x_{2}), λ), where (x_{1}, x_{2}) ∈ AG(2, q) and λ ∈ GF(q). With this notation the covering map (defined as in Theorem 5.1.1) (1, λ, x_{2}, −x_{1}) ↦ [0, 1, x_{1}, x_{2}]∩π_{P} becomes ((x_{1}, x_{2}), λ) ↦ (x_{1}, x_{2}). If we can find a 1-cochain c (on the simplicial complex of the point graph of AG(2, q) and over the abelian group (GF(q), +)) such that ((x_{1}, x_{2}), α) ∼ ((y_{1}, y_{2}), β) if and only if (x_{1}, x_{2}) ∼ (y_{1}, y_{2}) and c((x_{1}, x_{2}), (y_{1}, y_{2})) = α−β, then the q-fold cover of π, defined by W (q) as in Theorem 5.1.1, is an algebraic q-fold cover.

Now (1, x_{1}, x_{2}, x_{3}) and (1, y_{1}, y_{2}, y_{3}) are collinear points of W (q) if and only if y_{1} − x_{1} + x_{2}y_{3} − x_{3}y_{2} = 0. So (1, x_{1}, x_{2}, x_{3}) ∼ (1, y_{1}, y_{2}, y_{3}) if and only if β − α − x_{2}y_{1} + x_{1}y_{2} = 0 and hence ((x_{1}, x_{2}), α) ∼ ((y_{1}, y_{2}), β) if and only if α − β = x_{1}y_{2} − x_{2}y_{1}. Now the map c((x_{1}, x_{2}), (y_{1}, y_{2})) = x_{1}y_{2} − x_{2}y_{1} is symmetric in (x_{1}, x_{2}) and (y_{1}, y_{2}), so it follows that c is a 1-cochain and is such that the q-fold cover of π constructed from W (q) in Theorem 5.1.1 is an algebraic cover defined by c. Note that from Theorem 5.2.1 c satisfies the GQ condition 5.2.1. The GQ constructed from AG(2, q) and c in Theorem 5.2.1 is W (q).

5.2.2 A cover associated with the GQ T_{2}(O), q even

Let Σ = PG(3, q), q even, and let H = PG(2, q) be a plane of Σ. Let O be an oval of H with nucleus N. Let T_{2}(O) be the GQ, of order q, constructed from Σ, H and O, as in Section 1.4.3.

First we calculate the affine plane π constructed from T_{2}(O) and its regular point (∞), and the q-fold cover of π defined by T_{2}(O), as in Theorem 5.1.1. Consider a point X of T_{2}(O) that is not an element of (∞)^{⊥}. Then X is a point of type (ii) of T_{2}(O), that is, a point of Σ \ H. The set {(∞), X}^{⊥} consists of the points of type (ii) of T_{2}(O) that are collinear with X. That is, planes of Σ containing X and meeting H in a tangent to O. Thus {(∞), X}^{⊥} is the pencil of planes of Σ on the line ⟨N, X⟩, and this is the point of π that is covered by X. Let ℓ be a line of T_{2}(O) not incident with (∞). Then ℓ is a line of type (a) of T_{2}(O): that is, a line of Σ not contained in H and incident with a point of O. The line ℓ is incident with a unique point of (∞)^{⊥} and this point is the point of type (ii) ⟨ℓ, N⟩. Hence ⟨ℓ, N⟩ is the line of π covered by ℓ.

Thus π is the incidence structure with pointset the pencils of planes of Σ about lines of the form ⟨N, X⟩, where X ∈ Σ \ H; lineset the set of planes of Σ, not including H, that contain N; and incidence inherited from Σ. If we let the point of π that is the pencil of planes about the line ⟨N, X⟩, where X ∈ Σ \ H, be represented by the line ⟨N, X⟩, then π is isomorphic to the incidence structure

Points : Lines ⟨N, X⟩ with X ∈ Σ \ H
Lines : Planes of Σ, not H, containing N
Incidence : Inherited from Σ.

For this description of π, the point X ∈ Σ \ H of T_{2}(O) covers the point ⟨N, X⟩ of π, and the line ℓ of type (a) of T_{2}(O)) covers the line ⟨ℓ, N⟩ of π. Given this form of π we have

π ≅ \frac{Σ}{N} \ \frac{H}{N} ≅ AG(2, q).

We now introduce coordinates. Let H be the plane of Σ = PG(3, q) with equation x_{0} = 0. We may assume that

O = {(0, 1, t, F (t)) : t ∈ GF(q)} ∪ {(0, 0, 1, 0)},

where F is a polynomial satisfying the conditions of [24, 8.4.2 Theorem]. In this case the nucleus N is the point (0, 0, 0, 1). The affine plane π has pointset {⟨N, (1, x_{1}, x_{2}, x_{3})⟩ : (1, x_{1}, x_{2}, x_{3}) ∈ Σ \ H} and the covers of ⟨N, (1, x_{1}, x_{2}, x_{3})⟩ are the points ⟨N, (1, x_{1}, x_{2}, x_{3})⟩ \ {N}. That is, the elements of the set {(1, x_{1}, x_{2}, λ) : λ ∈ GF(q)}. If we denote the point ⟨N, (1, x_{1}, x_{2}, x_{3})⟩ of π by (x_{1}, x_{2}), then π assumes the canonical form of AG(2, q). In addition, we denote the point (1, x_{1}, x_{2}, λ) of Σ\H by the pair ((x_{1}, x_{2}), λ), where (x_{1}, x_{2}) is a point of AG(2, q) and λ ∈ GF(q). With this notation the covering map onto π given by ⟨N, (1, x_{1}, x_{2}, x_{3})⟩ ↦ (1, x_{2}, x_{2}, λ) becomes ((x_{1}, x_{2}), λ) ↦ (x_{1}, x_{2}). If we can find a 1-cochain c (on the simplicial complex of the point graph of AG(2, q) over the abelian group (GF(q), +)) such that ((x_{1}, x_{2}), α) ∼ ((y_{1}, y_{2}), β) if and only if c((x_{1}, x_{2}), (y_{1}, y_{2})) = α + β, then the q-fold cover of π defined by T_{2}(O), as in Theorem 5.1.1, is an algebraic q-fold cover.

Now (1, x_{1}, x_{2}, x_{3}) and (1, y_{1}, y_{2}, y_{3}) are collinear points of T_{2}(O) if and only if ⟨(1, x_{1}, x_{2}, x_{3}), (1, y_{1}, y_{2}, y_{3})⟩ intersects H in a point of O. The point of intersection of the line ⟨(1, x_{1}, x_{2}, x_{3}), (1, y_{1}, y_{2}, y_{3})⟩ with H is the point (0, x_{1} + y_{1}, x_{2} + y_{2}, x_{3} + y_{3}), and this is a point of O if and only if

x_{3} + y_{3} = \{ \begin{matrix}0 & if x_{1} = y_{1} \\ (x_{1} + y_{1})F (\frac{x_{2}+y_{2}}{x_{1}+y_{1}} ) & if x_{1} ≠ y_{1} \end{matrix} .

Now ((x_{1}, x_{2}), α) ∼ ((y_{1}, y_{2}), β) if and only if (1, x_{1}, x_{2}, x_{3}) ∼ (1, y_{1}, y_{2}, y_{3}), which is the case if and only if

α + β = \{ \begin{matrix}0 & if x_{1} = y_{1} \\ (x_{1} + y_{1})F (\frac{x_{2}+y_{2}}{x_{1}+y_{1}} ) & if x_{1} ≠ y_{1} \end{matrix} .

If we define the map

c((x_{1}, x_{2}), (y_{1}, y_{2})) = \{ \begin{matrix}0 & if x_{1} = y_{1} \\ (x_{1} + y_{1})F (\frac{x_{2}+y_{2}}{x_{1}+y_{1}} ) & if x_{1} ≠ y_{1} \end{matrix} ,

then since c is symmetric in (x_{1}, x_{2}) and (y_{1}, y_{2}) it is a 1-cochain. Further, ((x_{1}, x_{2}), α) ∼ ((y_{1}, y_{2}), β) if and only if c((x_{1}, x_{2}), (y_{1}, y_{2})) = α + β. Thus the q-fold cover of π constructed from T_{2}(O) in Theorem 5.1.1 is an algebraic cover defined by c. Note that Theorem 5.2.1 implies that c satisfies the GQ condition 5.2.1. The GQ constructed from AG(2, q) and c in Theorem 5.2.1 is T_{2}(O).

5.2.3 A “geometric” construction of a cover of T_{2}(O), with regular point (∞)

Let O be an oval of PG(2, q), q even, with canonical form

O = {(1, t, F (t)) : t ∈ GF(q)} ∪ {(0, 1, 0)},

where F is a polynomial satisfying the conditions of [24, 8.4.2 Theorem]. In this case, the nucleus of O is the point (0, 0, 1).

Let AG(2, q) be the subgeometry of PG(2, q) with pointset {(x_{1}, x_{2}, 1) : x_{1}, x_{2} ∈ GF(q)}, and denote the point (0, 0, 1) by O. Note that the line at infinity of AG(2, q) contains the points (1, 0, 0) and (0, 1, 0) of O, which correspond to the parallel classes {x_{2} = d : d ∈ GF(q)} and {x_{1} = d : d ∈ GF(q)} of AG(2, q) respectively. If X and Y are two distinct points of AG(2, q), then consider the line ℓ_{XY} = ⟨O, X − Y ⟩ of AG(2, q). If ℓ_{XY} is neither the line given by the equation x_{1} = 0, nor that given by x_{2} = 0, then ℓ_{XY} contains a point of O, say Φ_{XY}. Let X = (x_{1}, x_{2}, 1), Y = (y_{1}, y_{2}, 1) and define the 1-cochain c as follows:

c(X, Y) = \{ \begin{matrix}0 & if x_{1} = y_{1} or x_{2} = y_{2} \\ r & where x_{1} ≠ y_{1}, x_{2} ≠ y_{2} and X − Y = r.Φ_{XY} \end{matrix} .

Now, let X, Y and Z = (z_{1}, z_{2}, 1) be three distinct points of AG(2, q) and suppose that the sets {x_{1}, y_{1}, z_{1}} and {x_{2}, y_{2}, z_{2}} both consist of three distinct elements. In this case X, Y and Z are collinear if and only if Φ_{XY} = Φ_{XZ} = Φ_{Y Z}. Now Φ_{XY} = Φ_{XZ} = Φ_{Y Z} if and only if

O = (X − Y) − (X − Z) + (Y − Z)
= c(X, Y)Φ_{XY} − c(X, Z)Φ_{XZ} + c(Y, Z)Φ_{Y Z}
= δc(X, Y, Z)Φ_{XY} ,

which holds if and only if δc(X, Y, Z) = 0.

Now suppose that x_{1} = y_{1}. Then X, Y, Z are collinear if and only if x_{1} = y_{1} = z_{1}, which holds if and only if δc(X, Y, Z) = 0. Similarly, in the case where x_{2} = y_{2}, X, Y, Z are collinear if and only if δc(X, Y, Z) = 0.

Thus for X, Y, Z any three distinct points of AG(2, q), X, Y, Z are collinear if and only if δc(X, Y, Z) = 0.

Now suppose that x_{1} ≠ y_{1}, x_{2} ≠ y_{2} and that Φ_{XY} = (1/F (t), t/F (t), 1), for some t ∈ GF(q). Then it follows that X − Y = r.Φ_{XY}, where r and t satisfy the equations

x_{1} + y_{1} = \frac{r}{F (t)} and
x_{2} + y_{2} = \frac{rt}{F (t)}.

Then it follows that t = \frac{x_{2} + y_{2}}{x_{1} + y_{1}} , r = (x_{1} + y_{1})F (\frac{x_{2} + y_{2}}{x_{1} + y_{1}} ) and so

c(X, Y) = (x_{1} + y_{1})F (\frac{x_{2} + y_{2}}{x_{1} + y_{1}} ).

If x_{1} = y_{1}, or x_{1} = y_{2}, then c(X, Y) = 0. Thus, c is precisely the 1-cochain that was derived in Section 5.2.2.

5.3 Equivalence of covers of π

5.3.1 Normalised algebraic covers

Let π be an affine plane of order s and (\overline{π}, p) an algebraic s-fold cover of π, defined by the 1-cochain c. It was noted in Section 1.6 that if c′ = c+δb for any 1-coboundary δb and if (\overline{π}′, p′) is the algebraic s-fold cover of π defined by c′, then the geometries \overline{π} and \overline{π}′ are isomorphic. Thus, by judiciously choosing the 1-coboundary δb, we may assume a normalised form for the 1-cochain c.

Let O be a fixed point of π. The 1-cochain c is normalised with respect to O if c(O, P) = 0 for all P ∈ π \ \{0\}. If (\overline{π}, c) is any algebraic s-fold cover of \overline{π}, then define c′ = c + δb, where b(P) = c(O, P) if P ≠ O and b(O) = 0. The 1-cochain c′ is normalised with respect to O and is called the form of c normalised with respect to O. Suppose that c + δb and c + δb′ are both normalised with respect to the point O. Then δb(O, P) = δb′(O, P) = 0 for all points P of π such that P ≠ O. Let b(O) − b′(O) = k, then it follows that b(P) − b′(P) = k for all points P of π such that P ≠ O. Thus δb = δb′ and so the normalised form of a 1-cochain with respect to a fixed point of π is unique.

Now, let L be a fixed parallel class of π and let ℓ be an element of L. Then the simplicial complex Γ_{ℓ} of the point graph of ℓ is a subcomplex of the simplicial complex Γ of the point graph of π. Thus the 1-cochain c of Γ over the abelian group A induces a 1-cochain on Γ_{ℓ} over A, c_{ℓ} say. Since c defines a cover of the geometry π it follows that δc(P, Q, R) = 0 for P, Q, R any three distinct points of ℓ. Thus c_{ℓ} is a 1-cocycle and since the point graph of ℓ is complete, it follows from [12, Proposition 2.2] that H^{1}(Γ_{ℓ}, A) is trivial and so c_{ℓ} is a 1-coboundary, δb_{ℓ} say. Now let c′ = c + δb be a 1-cochain on Γ over A such that if P ∈ ℓ, where ℓ ∈ L, then b(P) = b_{ℓ}(P). Thus, c′ has the property that c′(P, Q) = 0 if ⟨P, Q⟩ ∈ L. In this case we say that c′ is normalised with respect to the parallel class L.

5.3.2 Equivalent algebraic covers of π

Let π be an affine plane of order s and let A be an abelian group of order s. We will represent A as an additive group with identity element 0. Let (\overline{π}, p) and (\overline{π}′, p′) be two algebraic s-fold covers of π defined by c and c′ respectively, such that c and c′ both satisfy the GQ condition. Let S = (P, B, I) and S′ = (P′, B′, I′) be the GQs of order s constructed from (\overline{π}, p) and (\overline{π}′, p′), respectively, as in Theorem 5.1.2. Let the point of type (iii) of S be denoted by (∞) and the point of type (iii) of S′ be denoted by (∞)′. We are naturally interested in the conditions under which S and S′ are isomorphic. A first step is to determine the conditions under which there is an isomorphism from S to S′ that maps (∞) to (∞)′. This is the objective of the current section.

Theorem 5.3.1 Let i : S → S′ be an isomorphism such that i : (∞) ↦ (∞)′. If \overline{i} is the map obtained by restricting i to the subgeometry \overline{π}, then \overline{i} : \overline{π} → \overline{π}′ is an isomorphism.

Proof: The pointset of \overline{π} is P \ (∞)^{⊥} and the lineset is B \ \{ℓ : (∞) I ℓ\}. Since i maps (∞) to (∞)′, it also maps \overline{π} to \overline{π}′. As \overline{π} and \overline{π}′ are subgeometries of S and S′ respectively, it follows that i induces an isomorphism from \overline{π} to \overline{π}′.

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Recall from Section 1.6 that if there exists an isomorphism from \overline{π} to \overline{π}′, then we say c and c′ are equivalent. If there exists an isomorphism i : S → S′, such that i : (∞) ↦ (∞)′, we say that the 1-cochains c and c′ are GQ equivalent.

Our eventual aim is to determine the conditions under which two 1-cochains that satisfy the GQ condition are GQ equivalent. First we show that any isomorphism from \overline{π} to \overline{π}′ may be “completed” to an isomorphism from S to S′.

Lemma 5.3.2 Let \overline{i} : \overline{π} → \overline{π}′ be an isomorphism. Then for each point P of π, \overline{i} : \{(P, α) : α ∈ A} ↦ {(P′, α) : α ∈ A}, for some point P′ of π. Furthermore, the map T : P ↦ P′ defines an automorphism of π.

Proof: Since \overline{i} is an isomorphism, it induces an isomorphism from the point graph of \overline{π} to the point graph of \overline{π}′. By Theorem 5.1.3 the point graph of \overline{π} and the point graph of \overline{π}′ are both antipodal graphs, with the sets of antipodal points being {(P, α) : α ∈ A} for both graphs. Thus, for P a point of π, \overline{i} : \{(P, α) : α ∈ A\} ↦ \{(P′, α) : α ∈ A\}, for some point P′ of π. Now let ℓ be a line of \overline{π} represented by the set \{(P_{1}, α_{1}), (P_{2}, α_{2}), …, (P_{s}, α_{s})\} of points of \overline{π} incident with ℓ. Then the image of ℓ under \overline{i} is the set of points \{(T (P_{1}), α′_{1}), (T (P_{2}), α′_{2}), …, (T (P_{s}), α′_{s})\} for some α′_{i} ∈ A, where i = 1, 2, …, s. Since {(T (P_{1}), α′_{1}), (T (P_{2}), α′_{2}), …, (T (P_{s}), α′_{s})} is a line of \overline{π} it follows that \{P′_{1}, P′_{2}, …, P′_{s}\} is a line of π, ℓ′ say. Thus, \overline{i} extends to a well defined map on the lines of π, mapping ℓ to ℓ′.

Now from Section 1.6 we know that if the point (P, α) of \overline{π} (or \overline{π}′ respectively) is incident with the line ℓ of \overline{π} (or \overline{π}′ respectively), then the point P of π is incident with the line p(ℓ). Further if (P, m) is an incident point-line pair of π, then there exists an incident point-line pair ((P, α), ℓ) of \overline{π} (or \overline{π}′ respectively) such that m = p(ℓ) (or m = p′(ℓ) respectively). In fact for each α ∈ A there is a unique such element l of p^{−1}(m). So suppose that (P, m) is a incident point-line pair of π and ((P, α), ℓ) is an incident point-line pair of \overline{π} such that m = p(ℓ). Now \overline{i}((P, α), ℓ) = ((P′, α′), ℓ′) for some α′ ∈ A and ((P′, α′), ℓ′) is an incident point-line pair of \overline{π}′, from which it follows that (P′, p′(ℓ′)) is an incident point-pair of π. Now p(\overline{i}(ℓ)) = m′ (where m′ is the image of m under the map on π induced by \overline{i}) and so the map on π induced by \overline{i} preserves incidence and is therefore an automorphism of π.

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Theorem 5.3.3 Let \overline{i} : \overline{π} → \overline{π}′ be an isomorphism. Then there exists a unique isomorphism i : S → S′, such that i : (∞) ↦ (∞)′ and i induces \overline{i}, as in Theorem 5.3.1.

Proof: Let \overline{i} induce the collineation T on π, as in Lemma 5.3.2. Consider the map i : S → S′ defined as follows. If ℓ is a point of S of type (i) (that is, a line of π), then i(ℓ) = T (ℓ). If (P, α) is a point of S of type (ii) (that is, a point of \overline{π}), then i((P, α)) = \overline{i}((P, α)). Let i : (∞) ↦ (∞)′.

If L is a line of S of type (a) (that is, a parallel class of π), then i(L) = T (L). Finally, if \overline{ℓ} is a line of S of type (b) (that is, a line of \overline{π}), then i(\overline{ℓ}) = \overline{i}(\overline{ℓ}).

The map i is an isomorphism of the form required. Since T determines the action of i on (∞)^{⊥}, \overline{i} determines the action of i on S \ (∞)^{⊥} and i : (∞) ↦ (∞)′, it follows that i is the unique isomorphism from S to S′, mapping (∞) to (∞)′ that induces \overline{i} for \overline{π} to \overline{π}′. □

By Theorem 5.3.1 and Theorem 5.3.3, we have the following corollary.

Corollary 5.3.4 Let (\overline{π}, p) and (\overline{π}′, p′) be two algebraic s-fold covers of π, defined by c and c′, respectively, such that c and c′ both satisfy the GQ condition. Then c and c′ are GQ equivalent if and only if they are equivalent.

We want to determine when S ≅ S′ by an isomorphism that maps (∞) to (∞)′ and by Corollary 5.3.4 this occurs if and only if c and c′ are equivalent. Our aim now is to determine the conditions under which c and c′ are equivalent. The first step is to determine the conditions under which there exists an isomorphism from \overline{π} to \overline{π}′ that induces a given collineation T of π.

If \overline{i} is an isomorphism from \overline{π} to \overline{π}′ that induces the collineation T of π, then \overline{i} must act on the points of \overline{π} by

\overline{i} : (P, α) ↦ (T (P), t_{P} (α)) for P ∈ π and α ∈ A,

where {t_{P} : P ∈ π} is some set of permutations on A.

Now suppose that \overline{i} is a map from the pointset of \overline{π} to the pointset of \overline{π}′, such that \overline{i} : (P, α) ↦ (T (P), t_{P} (α)), where t_{P} is an automorphism on A. If we determine the conditions under which this map is an isomorphism from \overline{π} to \overline{π}′, then since, by the above, all isomorphisms from \overline{π} to \overline{π}′ have this form, we will have determined all isomorphisms from \overline{π} to \overline{π}′. The map \overline{i} is an isomorphism if and only if

(P, α) ∼ (Q, β) in \overline{π} ⟺ (T (P), t_{P} (α)) ∼ (T (Q), t_{Q}(β)) in \overline{π}′,

for all P, Q ∈ π, P ≠ Q and α, β ∈ A. Now (P, α) ∼ (Q, β) in \overline{π} if and only if β = α − c(P, Q), and so \overline{i} is an isomorphism from \overline{π} to \overline{π}′ if and only if

(T (P), t_{P} (α)) ∼ (T (Q), t_{Q}(α − c(P, Q)) for all P, Q ∈ π, P ≠ Q and α ∈ A,  (5.3.2)

This is the case if and only if

c′(T (P), T (Q)) = t_{P} (α) − t_{Q}(α − c(P, Q)) for all P, Q ∈ π, P ≠ Q and α ∈ A. (5.3.3)

We now consider the case where c is normalised with respect to the point O of π and c′ is normalised with respect to the point T (O) of π.

For k a constant element of A, the map (P, α) ↦ (P, α + k) is an automorphism of \overline{π}′. Thus it follows that there is an isomorphism from \overline{π} to \overline{π}′ that induces T if and only if there is an isomorphism \overline{i} : (P, α) ↦ (T (P), t_{P} (α)), such that t_{O}(0) = 0. By 5.3.3 such an isomorphism exists if and only if

c′(T (P), T (Q)) = t_{P} (α) − t_{Q}(α − c(P, Q)) for all P, Q ∈ π, P ≠ Q and α ∈ A. (5.3.4)

Substituting Q = O we have that t_{P} (α) = t_{O}(α) for all P ∈ π and α ∈ A, and so 5.3.4 is equivalent to

c′(T (P), T (Q)) = t_{O}(α) − t_{O}(α − c(P, Q)) for all P, Q ∈ π, P ≠ Q and α ∈ A. (5.3.5)

By substituting α = 0 into this expression it follows that

c′(T (P), T (Q)) = −t_{O}(−c(P, Q)) for all P, Q ∈ π, P ≠ Q and α ∈ A
⟺ −c′(T (P), T (Q)) = t_{O}(c(Q, P)) for all P, Q ∈ π, P ≠ Q and α ∈ A
⟺ c′(T (Q), T (P)) = t_{O}(c(Q, P)) for all P, Q ∈ π, P ≠ Q and α ∈ A
⟺ c′(T (P), T (Q)) = t_{O}(c(P, Q))  for all P, Q ∈ π, P ≠ Q and α ∈ A.

Thus with c′(T (P), T (Q)) = t_{O}(c(P, Q)) 5.3.5 is equivalent to

t_{O}(α) = t_{O}(α − c(P, Q)) + t_{O}(c(P, Q)) for all P, Q ∈ π, P ≠ Q and α ∈ A.  (5.3.6)

If we can show that t_{O} maps onto A, then t_{O} must be an automorphism of A. Let ℓ be a line of π such that O ∈ ℓ and X a point of π not incident with ℓ. Now if c(X, Y) = c(X, Y′), for Y, Y′ distinct points of ℓ, then

δc(X, Y, Y′) = c(X, Y) − c(X, Y′) + c(Y, Y^{p}) = c(X, Y) − c(X, Y′) = 0.  (5.3.7)

However, X, Y, Z are not collinear in π and so 5.3.7 contradicts the fact that c satisfies the GQ condition 5.2.1. Thus {c(X, Y) : Y ∈ ℓ} = A. Hence t_{O} maps onto A and is an automorphism of A.

Thus we have established that there exists an isomorphism from \overline{π} to \overline{π}′ that induces the collineation T on π if and only if

c′(T (P), T (Q)) = σ(c(P, Q)), where σ is an automorphism of A.  (5.3.8)

In this case \overline{i} : (P, α) ↦ (T (P), σ(α)) is such an isomorphism.

In the following lemma we restate this result and calculate all isomorphisms from \overline{π} to \overline{π}′ that induce the collineation T on π.

Lemma 5.3.5 Let π be an affine plane of order s. Let (\overline{π}, p) and (\overline{π}′, p′) be two algebraic s-fold covers of π over the abelian group A, defined by the 1-cochains c and c′ respectively, such that c and c′ satisfy the GQ condition. Let T be a collineation of π and suppose that c and c′ are normalised with respect to the points O and T (O) respectively. Then there exists an isomorphism from \overline{π} to \overline{π}′ that induces T if and only if there exists a unique automorphism σ of A such that

c′(T (P), T (Q)) = σ(c(P, Q)).

In this case, the full set of isomorphisms from \overline{π} to \overline{π}′ that induce T on π is \{\overline{i}_{k} : k ∈ A\}, where \overline{i}_{k}(P, α) = (T (P), σ(α) + k).

Proof: The condition for the existence of an isomorphism that induces T comes from 5.3.8, and we are also furnished with an example of such an isomorphism, namely \overline{i} where \overline{i} : (P, α) ↦ (T (P), σ(α)). The isomorphisms from \overline{π} to \overline{π}′ that induce T on π are exactly those of the form τ\overline{i}, where τ is an automorphism of \overline{π}′ inducing the identity on π. Thus we now calculate the set of such τ. Since τ induces the identity on π, it is of the form τ : (P, α) ↦ (P, τ_{P} (α)), where τ_{P} is a permutation on A. Now a map of this form is an automorphism of π if and only if

(P, α) ∼ (Q, β) ⟺ (P, t_{P} (α)) ∼ (Q, t_{Q}(β)) for P, Q ∈ π, P ≠ Q and α, β ∈ A.

This is the case if and only if α − β = τ_{P} (α) − τ_{Q}(β) for all P, Q ∈ π, P ≠ Q and α, β ∈ A. Let O be a fixed point of π and let τ_{O}(0) = k. Then τ_{P} (α) = α + k, for all points P, not O and for all α ∈ A. To determine the permutation τ_{O} note that τ_{O}(α) = α − β + τ_{P} (β) = α + k. Thus we have the full set of isomorphisms from \overline{π} to \overline{π}′ that induce T on π.

□

We now generalise to the case where c and c′ are any pair of 1-cochains satisfying the GQ condition. Let T be a collineation of π, c + δb the form of c normalised with respect to O and c′ + δb′ the form of c′ normalised with respect to T (O). Let \overline{π}_{b} be the covering geometry of the algebraic s-fold cover of π defined by c + δb. The map from \overline{π} to \overline{π}_{b} acting by (P, α) ↦ (P, α + b(P)) is an isomorphism. Similarly, let \overline{π}′_{b′} be the covering geometry of the algebraic s-fold cover of π defined by c′ + δb′. Then the map from \overline{π}′ to \overline{π}′_{b′} acting by (P, α) ↦ (P, α + b′(P)) is an isomorphism.

Now there is an isomorphism from \overline{π} to \overline{π}′ that induces T on π if and only if there is an isomorphism from \overline{π}_{b} to \overline{π}′_{b′}, and by Lemma 5.3.5 this is the case if and only if

c′(T (P), T (Q)) = σ(c(P, Q)) + σ(δb(P, Q)) − δb′(T (P), T (Q)) for all P, Q ∈ π, P ≠ Q,

for some σ ∈ aut(A).  (5.3.9)

Note that by Lemma 5.3.5 the automorphism σ of A is unique.

We now show that 5.3.9 holds if and only if

c′(T (P), T (Q)) = σ(c(P, Q)) + δd(P, Q) for all P, Q ∈ π, P ≠ Q,
for some σ ∈ aut(A) and 1-coboundary δd.  (5.3.10)

Note that the map σ(δb(P, Q))−δb′(T (P), T (Q)) is a 1-coboundary. It follows that 5.3.9 satisfies 5.3.10.

Now suppose that δd is a 1-coboundary that acts by δd(P, Q) = c′(T (P), T (Q))−σ(c(P, Q)), for some σ ∈ aut(A). That is, the 1-cochains c and c′ satisfy 5.3.10. Then define δd′ to be the 1-coboundary that acts by

δd′(P, Q) = σ^{−1}(δd(P, Q) + δb′(T (P), T (Q))).

Thus δd(P, Q) = σ(δd′(P, Q)) − δb′(T (P), T (Q)) and so

σ(δd′(P, Q)) − δb′(T (P), T (Q)) = c′(T (P), T (Q)) − σ(c(P, Q)).

From this it follows that

c′(T (P), T (Q)) + δb′(T (P), T (Q)) = σ(c(P, Q) + δd′(P, Q)).

Now c′ + δb′ is normalised with respect to T (O) and so c + δd′ is normalised with respect to O. Hence δd′ = δb. Substituting δd′ = δb into the definition of δd yields

δd(P, Q) = σ(δb(P, Q)) − δb′(T (P), T (Q)),

which when substituted into 5.3.10 yields 5.3.9. Hence we have that 5.3.10 is equivalent 5.3.9 and thus holds if and only if there is an isomorphism from \overline{π} to \overline{π}′ that induces T on π. Note that we observed that σ is the unique automorphism of A such that 5.3.9 holds, and so since δd is determined by c, c′ and σ, it follows that (σ, δd) is the unique pair such that 5.3.10 holds.

Lemma 5.3.5 and the above discussion give the following result.

Lemma 5.3.6 Let π be an affine plane of order s. Let (\overline{π}, p) and (\overline{π}′, p′) be two algebraic s-fold covers of π over the abelian group A, defined by the 1-cochains c and c′ respectively, such that c and c′ satisfy the GQ condition. Let T be a collineation of π. Then there exists an isomorphism from \overline{π} to \overline{π}′ that induces T if and only if there exists a unique automorphism σ of A and unique 1-coboundary δb, such that

c′(T (P), T (Q)) = σ(c(P, Q)) + δb(P, Q).

In this case, the full set of isomorphisms from \overline{π} to \overline{π}′ that induce T on π is \{\overline{i}_{k} : k ∈ A\}, where \overline{i}_{k}(P, α) = (T (P), σ(α) + k + d(P)).

Proof: From the above discussion there is an isomorphism from \overline{π} to \overline{π}′ that induces T on π if and only if there is a unique pair (σ, δd) with σ ∈ aut(A) and δd a 1-coboundary such that

c′(T (P), T (Q)) = σ(c(P, Q)) + δd(P, Q) for all P, Q ∈ π, P ≠ Q.

Now δb normalises c with respect to the point O and δb′ normalises c′ with respect to the point T (O). If \overline{π}_{b} and \overline{π}′_{b′} are the covering geometries defined by c + δb and c′ + δb′, then by Lemma 5.3.5 the full group of isomorphisms from \overline{π}_{b} to \overline{π}′_{b′} that induce T on π is \{(P, α) ↦ (T (P), σ(α) + k) : k ∈ A\}. Since (P, α) ↦ (P, α + b(P)) is an isomorphism from \overline{π} to \overline{π}_{b} and (P, α) ↦ (P, α − b′(P)) is an isomorphism from \overline{π}′_{b′} to \overline{π}′, it follows that the full group of isomorphisms from \overline{π} to \overline{π}′ that induce T on π is

{(P, α) ↦ (T (P), σ(α+b(P))−b′(P)+k) : k ∈ A} = {(P, α) ↦ (T (P), σ(α)+d(P)+k) : k ∈ A}.

□

Lemma 5.3.6 allows to calculate all of the 1-cochains equivalent to a fixed 1-cochain c (such that c satisfies the GQ condition). In the next section we will use this to calculate the stabiliser of the regular point (∞) in the group of the GQ S constructed from c (as in Theorem 5.2.1).

Theorem 5.3.7 Let π be an affine plane of order s and (\overline{π}, p) an algebraic s-fold cover of π, over the abelian group A, defined by the 1-cochain c, such that c satisfies the GQ condition. The set of 1-cochains equivalent to c is

{c′ : c′(P, Q) = σ(c(T (P), T (Q)) + δb(T (P), T (Q)), for all P, Q ∈ π, P ≠ Q
where σ ∈ aut(A) and T ∈ aut(π)}

Proof: This result follows directly from Lemma 5.3.6.

□

If σ is an automorphism of A, then let c_{σ} be the 1-cochain that acts by c_{σ}(P, Q) = σ(c(P, Q)), and if T is a collineation of π, then let c_{T} be the 1-cochain that acts by c_{T} (P, Q) = c(T (P), T (Q)). Thus if c is a 1-cochain and c′ is the 1-cochain such that c′(P, Q) = σ(c(T (P), T (Q))) + δb(P, Q), then c′ = (c_{T})_{σ} + δb.

Now by [12, Proposition 2.2], since the point graph of π is complete, it follows that two 1cochains are cohomologous if and only if their difference is a 1-coboundary. Thus c′ = (c_{T})_{σ} +δb if and only if δc′ = δ(c_{T})_{σ}. Note that in this case δc′(P, Q, R) = σ(δc_{T} (P, Q, R)). Given this we have the following corollary to Theorem 5.3.7.

Corollary 5.3.8 Let π be an affine plane of order s. Let (\overline{π}, p) and (\overline{π}′, p′) be algebraic s-fold covers of π, over the abelian group A, defined by the 1-cochains c and c′ respectively, such that c and c′ satisfy the GQ condition. Then c and c′ are equivalent if and only if there exists an automorphism σ of A, and T collineation of π, such that δc′ = δ(c_{T})_{σ}.

5.4 The automorphism group of \overline{π} and the group of S fixing (∞)

Let π be an affine plane of order s and (\overline{π}, p) an algebraic s-fold cover of π over the abelian group A, defined by the 1-cochain c, such that c satisfies the GQ condition. Let S be the GQ constructed from (\overline{π}, p) as in Theorem 5.1.2. In this section we show that the automorphism group of \overline{π} is isomorphic to the subgroup of the automorphism group of S that fixes (∞). Then we calculate the automorphism group of the geometry \overline{π}.

If we consider Theorem 5.3.1 and Theorem 5.3.3 for the case where c = c′ (so \overline{π} = \overline{π}′ and S = S′) and (∞) = (∞)′, then the automorphisms of S fixing (∞) are “equivalent” to those fixing \overline{π}. More precisely:

Theorem 5.4.1 Let π be an affine plane of order s and (\overline{π}, p) an algebraic s-fold cover over the abelian group A, defined by the 1-cochain c, such that c satisfies the GQ condition. Let S be the GQ of order s constructed from (\overline{π}, p), as in Theorem 5.1.2. If aut(S)_{(∞)} denotes the subgroup of the group of S fixing (∞), and aut(\overline{π}) denotes the group of \overline{π}, then

aut(S)_{(∞)} ≅ aut(\overline{π}).

Proof: Let ϕ : aut(S)_{(∞)} → aut(\overline{π}) be the map that takes an element of aut(S)_{(∞)} to the element of aut(\overline{π}) it induces as in Theorem 5.3.1 (with c = c′). Then ϕ is a homomorphism. By Theorem 5.3.3 ϕ is one-to-one and onto and so is an isomorphism.

□

In the previous section it was shown that if \overline{i} is an automorphism from \overline{π} to \overline{π}′, then \overline{i} induces a collineation of π (Lemma 5.3.2). In the case where c = c′, \overline{i} is an automorphism of \overline{π} and the collineation of π that is induced by \overline{i} is said to be admitted by c. In the following theorem we determine the conditions under which a collineation of π is admitted by c.

Theorem 5.4.2 Let π be an affine plane of order s and (\overline{π}, p) an algebraic s-fold cover of π over the abelian group A, defined by the 1-cochain c, such that c satisfies the GQ condition. Let T be a collineation of π. Then the following statements are equivalent

1. The collineation T is admitted by c.

2. There exists a unique automorphism σ of A such that

δc_{T} = δc_{σ}.

3. There is a unique 1-coboundary, δb and a unique automorphism σ_{T} of A such that

c_{T} = c_{σ} + δb.

Proof: The collineation T is admitted by c if and only if there is an automorphism of \overline{π} that induces T. By setting c = c′ in Lemma 5.3.6, it follows that 1 is equivalent to 3 By [12, Proposition 2.2], it follows that 2 is equivalent to 3.

□

Note that the set of collineations of π that are admitted by c forms a group, which will be denoted by aut(π)_{c} and is called the subgroup of aut(π) that is admitted by c.

We now calculate the automorphism group of the geometry \overline{π}.

Theorem 5.4.3 Let π be an affine plane of order s and (\overline{π}, p) an algebraic s-fold cover of π, over the abelian group A, defined by the 1-cochain c, such that c satisfies the GQ condition. Then the full automorphism group aut(\overline{π}) of the geometry \overline{π} comprises the elements

(P, α) ↦ (T (P), σ_{T} (α) + k + b_{T} (P)),

where k ∈ A, T ∈ aut(π)_{c} and (σ_{T} , δb_{T}) is the unique pair, such that c_{T} = c_{(σ_{T})} + δb_{T}.

Proof: Use c = c′ in Lemma 5.3.6.

□

In the proof of Theorem 5.3.3 the unique element of aut(S)_{(∞)} that induces a given element of aut(\overline{π}) was explicitly constructed from the element of aut(\overline{π}). Combining this construction and Theorem 5.4.3 enables the explicit construction of the subgroup of the automorphism group of S that fixes (∞).

5.4.1 Elations and symmetries about (∞)

Let π be an affine plane of order s and (\overline{π}, p) an algebraic s-fold cover over the abelian group A, defined by c, such that c satisfies the GQ condition. Let S = (P, B, I) be the GQ of order s constructed from π, \overline{π} and p as in Theorem 5.1.2. In this section we consider particular types of elements of the group aut(S)_{(∞)}, called elations and symmetries. The terminology and definitions in this section are taken from [49, Chapter 8].

A whorl of S about a point P is an automorphism of S that fixes each line incident with P. An elation of S about P is a whorl of S about P, that fixes no point of P \ P^{⊥}. If θ is a whorl of S about (∞), then θ induces an automorphism θ′ of π that fixes each parallel class of π. (We can show this by using Theorem 5.3.1 and Lemma 5.3.2, or it is relatively straight-forward to verify directly). If Π is the projective completion of π with line at infinity ℓ_{∞}, then by [28, Theorem 4.9], the extension of θ′ to Π is either an elation of a homology of Π and hence either a translation or a dilatation of π.

A symmetry of S about P is an automorphism of S that fixes each point of P^{⊥}. Any symmetry about P is an elation about P. Now by [49, Chapter 8] the set of symmetries about P forms a group, the order of which must divide s. In the case where the order of the symmetry group is s, the point P is called a centre of symmetry (and must be regular). By Theorem 5.4.3, the group of symmetries about (∞) is {(P, α) ↦ (P, α + k) : k ∈ A} and so (∞) is a centre of symmetry of S. Note that any symmetry about (∞) induces the identity automorphism on π. A symmetry about a line ℓ of S is defined dually (that is, as an automorphism of σ that fixes each line of ℓ^{⊥}). A line whose symmetry group has maximal order s is an axis of symmetry. Suppose that ℓ is a line of S incident with (∞) and S is a symmetry about ℓ. Now S fixes (∞) linewise and ℓ pointwise, and so S induces a collineation of π that fixes each parallel class of π and fixes the parallel class of π corresponding to ℓ linewise. Thus by [28, Theorem 4.9] S induces a translation on π. Note that this implies that each symmetry about a line incident with (∞) is an elation about (∞), although the converse of this does not necessarily hold. The line ℓ is an axis of symmetry if and only if π has a full translation group, of order s, fixing the parallel class of π corresponding to ℓ linewise, and c admits this group.

The GQ S is an Elation Generalized Quadrangle (EGQ), with basepoint P, if G is a group of elations about P acting regularly on the points of P \ P^{⊥}. Since G acts regularly on the points of P \ P^{⊥} it is a maximal group of elations about P.

The GQ S is a Translation Generalized Quadrangle (TGQ), with basepoint P, if S is an EGQ with basepoint P, such that the elation group G contains a full group of s symmetries about each line through P.

Theorem 5.4.4 The GQ S is a TGQ with base point (∞) if and only if π is a translation plane and c admits the translation group of π.

Proof: Suppose that S is a TGQ with base point (∞). Recall from the above discussion that if ℓ is a line of S incident with (∞), then ℓ is an axis of symmetry if and only if π has a full translation group, of order s, fixing the parallel class of π corresponding to ℓ linewise, and c admits this group. Thus if each line incident with (∞) is an axis of symmetry, then π is a translation plane and c admits the full translation group of π.

Now if π is a translation plane and c admits the full translation group G, then by Theorem 5.4.3 the group

{(P, α) ↦ (T (P), σ_{T} (α) + b_{T} (P) + k) : T ∈ G, k ∈ A and c_{T} = c_{(σ_{T})} + δb_{T} }

is a group fixing \overline{π}. By Theorem 5.3.3 this group is isomorphic to a group of S that is an elation group about (∞) acting regularly on the points of P \ (∞)^{⊥}. Thus S is an EGQ with base point (∞).

Let ℓ be a line of S incident to (∞) and G_{ℓ} the full group of translations of π fixing the parallel class of π corresponding to ℓ. By Theorem 5.4.3 the group

{(P, α) ↦ (T (P), σ_{T} (α) + b_{T} (P) + k) : T ∈ G_{ℓ}, k ∈ A and c_{T} = c_{(σ_{T})} + δb_{T} }

is a group fixing \overline{π}, which by Theorem 5.3.3 extends to a group of symmetries of S about ℓ. Thus ℓ is an axis of symmetry. Since this is true for each line incident with (∞), it follows that S is a TGQ with base point (∞).

□

The discussion of TGQ was introduced for GQ of order s by Thas in [60], and technical aspects were developed by Payne in [42], before it assumed the form given in [49, Chapter 8].

5.4.2 Example: The subgroup of AΓL(3, q) admitted by x_{1}y_{2} − x_{2}y_{1}

Consider the classical affine plane AG(2, q) with pointset {(x_{1}, x_{2}) : x_{1}, x_{2} ∈ GF(q)} and lines given by the equations x_{2} = mx_{1} + d and x_{1} = d, where m, d ∈ GF(q). Then

aut(AG(2, q)) = AΓL(2, q)
= {T : T (x_{1}, x_{2}) = (a + bx_{1}^{ρ} + cx_{2}^{ρ}, d + ex_{1}^{ρ} + fx_{2}^{ρ}),
such that a, b, c, d, e, f ∈ GF(q) and bf − ec ≠ 0},

which may be derived from the fundamental theorem of projective geometry (see [24, 2.1(ii)]).

From Section 5.2.1 the algebraic q-fold cover of AG(2, q) giving rise to W (q) is defined by the 1-cochain c, where c((x_{1}, x_{2}), (y_{1}, y_{2})) = x_{1}y_{2} − x_{2}y_{1}. If T ∈ aut(AG(2, q)), then

c(T (x_{1}, x_{2}), T (y_{1}, y_{2})) = (a + bx_{1}^{ρ} + cx_{2}^{ρ})(d + ey_{1}^{ρ} + fy_{2}^{ρ}) − (d + ex_{1}^{ρ} + fx_{2}^{ρ})(a + by_{1}^{ρ} + cy_{2}^{ρ})
= (bf − ce)(x_{1}y_{2} − x_{2}y_{1})^{ρ} + (bd − ae)(x_{1} − y_{1})^{ρ}
+(cd − af)(x_{2} − y_{2})^{ρ}
= (bf − ce)c(x_{1}, x_{2}), (y_{1}, y_{2}))^{ρ} + ((bd − ae)x_{1}^{ρ} + (cd − af)x_{2}^{ρ})
−((bd − ae)y_{1}^{ρ} + (cd − af)y_{2}^{ρ}).

That is, c_{T} = c_{(σ_{T})}+δb_{T}, where σ_{T} (x) = (bf −ec)x^{ρ} and δb_{T} (x_{1}, x_{2}) = (bd−ae)x_{1}^{ρ}+(cd−af)x_{2}^{ρ}. Thus, by Theorem 5.4.2 we have that

aut(AG(2, q))_{c} = AΓL(3, q).

5.4.3 Example: Covers of AG(2, q), q even, of the form (x_{1} + y_{1})F (\frac{x_{2} + y_{2}}{x_{1} + y_{1}} )

Let π be an affine plane of order s and (\overline{π}, p) an algebraic s-fold cover of π over the abelian group A, defined by the general 1-cochain c, (that is, c need not satisfy the GQ condition). Let T be a collineation of π and σ an automorphism of A. Recall from Section 5.4 that c_{T} denotes the 1-cochain which acts by c_{T} (X, Y) = c(T (X), T (Y)) and that c_{σ} denotes the 1-cochain which acts by c_{σ}(X, Y) = σ(c(X, Y)).

The collineation T is said to be admitted by c if there exists an automorphism σ of A and a 1-coboundary δb such that c_{T} = c_{σ} + δb. If T is admitted by c, then it follows that δc_{T} = δc_{σ} and so δc_{T} = 0 if and only if δc = 0. Note that if c satisfies the GQ condition, then by Theorem 5.4.2 this definition of admitted agrees with that given in Section 5.4.

We will now give an example showing how the definition of T being admitted by a general 1-cochain c may be used to help determine if c satisfies the GQ condition. Let q = 2^{e}, π = AG(2, q), A = GF(q) and for X = (x_{1}, x_{2}), Y = (y_{1}, y_{2}) define

c(X, Y) = \{ \begin{matrix}(x_{1} + y_{1})F (\frac{x_{2} + y_{2}}{x_{1} + y_{1}} ) & if x_{1} ≠ y_{1} \\ 0 & if x_{1} = y_{1}, \end{matrix}

for some function F on GF(q). We know from Section 5.2.2 and Section 5.2.3 that if F has the property that O = {(1, t, F (t)) : t ∈ GF(q)} ∪ {(0, 1, 0)} is an oval of PG(2, q) with nucleus (0, 0, 1), then c satisfies the GQ condition. We wish to determine all functions F such that c satisfies the GQ condition.

Let T_{(a,b)} be the collineation of AG(2, q) that acts by translating the points of AG(2, q) by (a, b). That is T_{(a,b)}(x_{1}, x_{2}) = (x_{1} + a, x_{2} + b).

Now given the points X = (x_{1}, x_{2}) and Y = (y_{1}, y_{2}) of AG(2, q) such that x_{1} ≠ y_{1}, we have

c(T_{(a,b)}(X), T_{(a,b)}(Y)) = c((x_{1} + a, x_{2} + b), (y_{1} + a, y_{2} + b))
= (x_{1} + y_{1} + a + a)F (\frac{x_{2} + y_{2} + b + b}{x_{1} + y_{1} + a + a})
= c(X, Y).

If x_{1} = y_{1}, then x_{1} + a = y_{1} + a and so

c(T_{(a,b)}(X), T_{(a,b)}(Y)) = 0 = c(X, Y).

Thus T_{(a,b)} is admitted by c for all a, b ∈ GF(q). Now c satisfies the GQ condition if and only if for X, Y, Z distinct points of AG(2, q)

δc(X, Y, Z) = 0 ⟺ X, Y, Z are collinear.

Now

δc(X, Y, Z) = δc(T_{X}(X), T_{X}(Y), T_{X}(Z))
= δc(O, T_{X}(Y), T_{X}(Z)), where O = (0, 0),

since T_{X} is admitted by c. So in this case, the GQ condition on c is equivalent to

δc(O, X, Y) = 0 ⟺ O, X, Y are collinear.

If O, X, Y are collinear, either x_{1} = y_{1} = 0, in which case δc(O, X, Y) = 0, or \frac{x_{2}}{x_{1}} = \frac{y_{2}}{y_{1}} = \frac{x_{2} + y_{2}}{x_{1} + y_{1}} = m, where the line collinear with O, X, Y has slope m. In this case

δc(O, X, Y) = x_{1}F (\frac{x_{2}}{x_{1}} ) + y_{1}F (\frac{y_{2}}{y_{1}} ) + (x_{1} + y_{1})F (\frac{x_{2} + y_{2}}{x_{1} + y_{1}} ) = 0.

Now suppose that O, X, Y are not collinear and that x_{1} = 0 (so y_{1} ≠ 0). Then

δc(O, X, Y) = y_{1}F (\frac{y_{2}}{y_{1}} ) + y_{1}F (\frac{x_{2} + y_{2}}{y_{1}} )

= y_{1} (F (\frac{y_{2}}{y_{1}} ) + F (\frac{x_{2} + y_{2}}{y_{1}} )).

Thus, δc(O, X, Y) ≠ 0 if and only if F is a permutation of GF(q). A similar result holds for y_{1} = 0 and the case where x_{1} = y_{1} ≠ 0.

Now suppose that x_{1} ≠ y_{1} and x_{1}, y_{1} ≠ 0. Then

δc(O, X, Y) = x_{1}F (\frac{x_{2}}{x_{1}} ) + y_{1}F (\frac{y_{2}}{y_{1}} ) + (x_{1} + y_{1})F (\frac{x_{2} + y_{2}}{x_{1} + y_{1}} ),

so δc(O, X, Y) ≠ 0 if and only if

\frac{F (\frac{x_{2}}{x_{1}} ) + F (\frac{x_{2} + y_{2}}{x_{1} + y_{1}} )}{\frac{x_{1}y_{2} + y_{1}x_{2}}{x_{1}(x_{1} + y_{1})}} ≠ \frac{F (\frac{y_{2}}{y_{1}} ) + F (\frac{x_{2} + y_{2}}{x_{1} + y_{1}} )}{\frac{x_{1}y_{2} + y_{1}x_{2}}{y_{1}(x_{1} + y_{1})}},

where x_{1}y_{2} + x_{2}y_{1} ≠ 0, since O, X, Y are non-collinear (this is the 1-cochain derived from W (q) in Section 5.2.1). Now setting

u = \frac{x_{2}}{x_{1}} , t = \frac{x_{2} + y_{2}}{x_{1} + y_{1}} , s = \frac{y_{2}}{y_{1}} ,

we have that

δc(O, X, Y) ≠ 0 ⟺ \frac{F (u) + F (t)}{u + t} ≠ \frac{F (u) + F (s)}{u + s}.

These conditions that we have determined for the function F are exactly those under which O = {(1, t, F (t)) : t ∈ GF(q)} ∪ {(0, 1, 0)} is an oval of PG(2, q) with nucleus (0, 0, 1). Thus the only

1-cochains of the form (x_{1} + y_{1})F (\frac{x_{2} + y_{2}}{x_{1} + y_{1}} ) satisfying the GQ condition are precisely those

that define T_{2}(O), with regular point (∞) as derived in Section 5.2.2. (Note that in Section 5.2.2 we supposed that F had the form as in [24, 8.4.2 Theorem]. All that the construction required, however, was that F was such that the set of points O formed an oval).

5.5 Automorphisms of A and coboundaries associated with collineations of π

In this section we consider 1-cochains that define an s-fold cover of an affine plane π of order s over an abelian group A, and admit a particular subgroup G of aut(π). We investigate conditions under which such a 1-cochain satisfies the GQ condition.

Now suppose that π is an affine plane of order s, (\overline{π}, p) an algebraic s-fold cover of π, over the abelian group A, defined by the 1-coboundary c such that c satisfies the GQ condition. Recall that if T is a collineation of π, then T is admitted by c if there is a (unique) pair (σ_{T} , δb_{T}), such that

c(T (P), T (Q)) = σ_{T} (c(P, Q)) + δb_{T} (P, Q).

Let c be normalised with respect to a fixed point O of π. If P and Q are any two points of π such that O ∈ ⟨P, Q⟩, then c(P, Q) = 0. Further, if T is a collineation of π admitted by c, then c(T (P), T (Q)) = δb_{T} (P, Q). So c is at least partially determined by the admitted collineations and their associated 1-coboundaries. Now we establish conditions under which we may construct a 1-cochain satisfying the GQ condition from collineations of π and associated 1-coboundaries.

If S and T are collineations of π admitted by c then

c(S ∘ T (P), S ∘ T (Q)) = σ_{S}(c(T (P), T (Q)) + δb_{S}(T (P), T (Q))
= σ_{S}(σ_{T} (c(P, Q)) + δb_{T} (P, Q)) + δb_{S}(T (P), T (Q))
= σ_{S} ∘ σ_{T} (c(P, Q)) + δb_{T} (P, Q) + δb_{S}(T (P), T (Q)),

and hence σ_{S∘T} = σ_{S} ∘ σ_{T} and δb_{S∘T} = (δb_{T})_{σ} + (δb_{S})_{T}. Note also that the 1-coboundary associated with the identity collineation of π is the zero 1-cochain, and that if T (O) ∈ ⟨P, Q⟩, then c(P, Q) = δ_{T} (T^{−1}(P), T^{−1}(Q)). We incorporate these ideas into the following theorem.

Theorem 5.5.1 Let π be an affine plane of order s, O a fixed point of π and A an additive abelian group of order s. Let G be a group of automorphisms of π that acts transitively on the points of π. For each T ∈ G let δb_{T} be a 1-coboundary on the simplicial complex of the point graph of π and σ_{T} an automorphism of A. Suppose that δb_{T} and σ_{T} satisfy the following conditions

1. If I denotes the identity collineation of π, then δb_{I} = 0

2. σ_{S∘T} = σ_{S} ∘ σ_{T} and δb_{S∘T} = (σ_{S})δb_{T} + (δb_{S})T, for all S, T ∈ G

3. If S(O), T (O) ∈ ⟨P, Q⟩ then δb_{T} (T^{−1}(P), T^{−1}(Q)) = δb_{S}(S^{−1}(P), S^{−1}(Q)).

If we define c(P, Q) = δb_{T} (T^{−1}(P), T^{−1}(Q)) for T ∈ G such that T (O) ∈ ⟨P, Q⟩, then c defines an algebraic s-fold cover of π over A. Further, c is normalised with respect to the point O and admits all elements of the group G with c_{T} = c_{(σ_{T})} + δb_{T}. Finally, if δb_{T} satisfies the following condition

4. If T (O) ∈ ⟨P, Q⟩, then

δb_{T} (T^{−1}(P), T^{−1}(Q)) = 0 if and only if O ∈ ⟨P, Q⟩,

then c satisfies the GQ condition.

Proof: Let P and Q be any pair of distinct points of π. Since there is an element T of G such that T (O) = P, it follows that c(P, Q) is defined. By condition 3 c is well defined.

Now since O ∈ ⟨O, P ⟩, it follows that c(O, P) = δb_{I}(O, P) = 0. Hence c is normalised with respect to the point O.

We now wish to show that c admits each element of G, so we consider the 1-cochain c_{T} for T ∈ G. Let P and Q be two distinct points of π and suppose that S(O) ∈ ⟨P, Q⟩ for some S ∈ G. Then by repeatedly using condition 2 we have the following:

c(T (P), T (Q)) = δb_{S}(S^{−1} ∘ T (P), S^{−1} ∘ T (Q))
= δb_{(S∘S^{−1}∘T)}(P, Q) − σ_{S} (δb_{(S^{−1}∘T)}(P, Q))
= δb_{T} (P, Q) − σ_{S} (σ_{(T^{−1}∘S)}^{−1} (δb_{(T^{−1}∘S)∘(S^{−1}∘T)}(P, Q)
−δb_{(T^{−1}∘S)}(S^{−1} ∘ T (P), S^{−1} ∘ T (Q))))
= δb_{T} (P, Q) − σ_{T} (−δb_{(T^{−1}∘S)}(S^{−1} ∘ T (P), S^{−1} ∘ T (Q)))
= δb_{T} (P, Q) + σ_{T} (c(P, Q)).

Thus, c_{T} = c_{(σ_{T})} + δb_{T}, for all T ∈ G and so c admits each element of G.

We now show that c defines an algebraic s-fold cover of π over A. Let P, Q, R be three distinct collinear points of π such that P, Q, R are incident with the line ℓ of π. Since G is transitive on the points of π there is an element T of G such that T (O) ∈ ℓ. In this case let T ∈ G be such that T (O) = P. Then we have

δc(P, Q, R) = c(P, Q) − c(P, R) + c(Q, R)
= δb_{T} (T^{−1}(P), T^{−1}(Q)) − δb_{T} (T^{−1}(P), T^{−1}(R)) + δb_{T} (T^{−1}(Q), T^{−1}(R))
= 0,

and so c defines an algebraic s-fold cover of π over A.

Now suppose that the δb_{T} satisfy condition 4, which is equivalent to c(P, Q) ≠ 0 if O ∉ ⟨P, Q⟩. We show that c satisfies the GQ condition. Let P, Q, R be three distinct non-collinear points of π. Let ℓ = ⟨P, Q⟩ and T be an element of G such that T (O) = P. Then

δc(P, Q, R) = σ_{T} (δc(T^{−1}(P), T^{−1}(Q), T^{−1}(R))) since T is admitted by c
= σ_{T} (δc(O, T^{−1}(Q), T^{−1}(R)))
= σ_{T} (c(T^{−1}(Q), T^{−1}(R))),

and so by 4 we have c(T^{−1}(P), T^{−1}(Q)) ≠ 0. Thus δc(P, Q, R) ≠ 0 and c satisfies the GQ condition.

□

Note that in Theorem 5.5.1 we do not actually need G to be transitive on the pointset of π for c to be defined for all pairs of points of π. In fact we could relax the condition to require that the orbit of G on the pointset of π containing O have a point incident with each line of π and still have c well defined and defining an algebraic s-fold cover of π. However, this would entail strengthening condition 4 to force c to satisfy the GQ condition.

5.6 Covers associated with T_{2}(O)^{∧}

Let O be an oval of PG(2, q), q even, with nucleus N. In Section 5.2.2 we calculated the algebraic q-fold cover of AG(2, q) arising from T_{2}(O) and its regular point (∞). From [49, 3.3.2(i)], all lines of type (b) of T_{2}(O) are regular (lines of type (b) are the points of O), thus in the dual GQ T_{2}(O)^{∧}, the points of type (b) of T_{2}(O) are regular points. In this section we show that for each of these regular points of T_{2}(O)^{∧} there is an associated affine plane isomorphic to AG(2, q) and an algebraic q-fold cover of AG(2, q). For each of these q + 1 algebraic q-fold covers of AG(2, q) we calculate an explicit 1-cochain which defines the cover.

We represent T_{2}(O)^{∧} by swapping the labels point and line in the construction of T_{2}(O) (see Section 1.4.3). Thus we will have points of type (a) and (b) and lines of type (i), (ii) and (iii). The regular points we are considering are q + 1 points of type (b).

Let T_{2}(O) be constructed from Σ = PG(3, q), H = PG(2, q) contained in Σ and the oval O of H, which has nucleus N. Let X be a point of type (b) in T_{2}(O)^{∧} = (P, B, I) and π_{X} the affine plane of order q constructed from T_{2}(O)^{∧} and X as in Theorem 5.0.1. The lines of T_{2}(O)^{∧} incident with X are the planes of Σ meeting H in the line ⟨N, X⟩ and the line of type (i), which we will denote by [∞]. The points of X^{⊥} are X, the lines of Σ incident with X but not contained in H, and the points of O \ {X}.

Let ℓ ∈ P \ X^{⊥}. Then ℓ is a point of type (a) of T_{2}(O)^{∧} not incident with X. That is, a line of Σ such that ℓ ⊄ H and ℓ is incident with a point of O \ {X}^{⊥}, Y say. In T_{2}(O)^{∧}, {X, ℓ} consists of the point Y of type (b) and the q points of type (a) that are the q lines of Σ contained in the plane ⟨ℓ, X⟩, incident with X but not contained in H. This set, {X, ℓ}^{⊥}, is a point of π and the q points of T_{2}(O)^{∧} that cover {X, ℓ} are the q points of type (a) that are lines of the plane ⟨X, ℓ⟩, incident with Y but not X.

Let H′ be a line of type (ii) of T_{2}(O)^{∧} such that X is not incident with H′. Then H′ is a plane of Σ containing a unique point of O \ {X}, Z say. Now Z ∈ X^{⊥} (since Z I [∞] I X) and so Z is the line of π_{X} covered by H′. The q lines of T_{2}(O)^{∧} that cover Z (as a point of π_{X}) are the q lines of type (ii) that are planes of Σ meeting H in the line ⟨N, Z⟩.

Let P be a line of type (i) of T_{2}(O)^{∧} such that P ∉ X^{⊥}. That is, P is a point of Σ \ H. The line of π_{X} that is covered by P is the line ⟨P, X⟩, and the q lines of T_{2}(O)^{∧} covering the line ⟨P, X⟩ are the q points ⟨P, X⟩ \ {X} of type (i).

Now a point of π_{X} is a set of q coplanar lines, so we will identify the point of π_{X} with this plane. We will identify a line of π_{X} which is a point of Y ∈ O \ {X} with the line ⟨Y, X⟩, and a line of π_{X} that is a line of Σ, not contained in H and incident with X, will be identified with itself. With these identifications π_{X} is the incidence structure Points: Planes of Σ, not H, containing X, but not N.

Lines: Lines of Σ containing X, but not N.

Incidence: Inherited from Σ.

Hence,

π_{X} ≅ (\frac{Σ}{X} \ \frac{⟨X, N⟩}{X})^{D} ≅ AG(2, q).

Now we introduce coordinates by letting Σ = PG(3, q), H the plane with equation x_{0} = 0 and letting O have the canonical form

O = {(1, t, F (t)) : t ∈ GF(q)} ∪ {(0, 1, 0)},

where F is a polynomial, as in [24, 8.4.2 Theorem]. In this case, the nucleus of O is the point (0, 0, 1).

We consider the various possibilities for the point X and construct 1-cochains corresponding to the q-fold covers of AG(2, q).

5.6.1 Case (a) X = (0, 0, 1, 0)

The pointset of π_{X} is the set of planes of Σ containing (0, 0, 1, 0), not including H. That is, the planes with coordinates [x_{1}, x_{2}, 0, 1] with x_{1}, x_{2} ∈ GF(q). Note that if we let the point [x_{1}, x_{2}, 0, 1] of π_{X} be represented by the pair (x_{1}, x_{2}), then π_{X} assumes the canonical form of AG(2, q).

The plane [x_{1}, x_{2}, 0, 1] contains X and a second point of Y of O. Now the point (0, 1, t, F (t)) of O \ {X} is contained on the plane [x_{1}, x_{2}, 0, 1] of Σ if and only if t = F^{−1}(x_{2}) (F^{−1} is defined since F is a permutation of GF(q)), and so Y = (0, 1, F^{−1}(x_{2}), x_{2}). The covers of the point (x_{1}, x_{2}) of AG(2, q) are the lines of Σ contained in the plane [x_{1}, x_{2}, 0, 1] that are incident with Y, but not X. In order to show that the s-fold cover of π_{X} defined by T_{2}(O)^{∧} is algebraic we must first represent the covers of [x_{1}, x_{2}, 0, 1] as the set {((x_{1}, x_{2}), α) : α ∈ GF(q)} and then define an appropriate 1-cochain. Now the set of covers of (x_{1}, x_{2}) may be described as the intersection of the plane [x_{1}, x_{2}, 0, 1] with the set of planes {[α, F^{−1}(x_{2}), 1, 0] : α ∈ GF(q)}. So let ((x_{1}, x_{2}), α) represent the cover of (x_{1}, x_{2}), [x_{1}, x_{2}, 0, 1] ∩ [α, F^{−1}(x_{2}), 1, 0].

Consider the points of T_{2}(O)^{∧} represented by ((x_{1}, x_{2}), α) and ((y_{1}, y_{2}), β). If x_{2} ≠ y_{2}, then ((x_{1}, x_{2}), α) and ((y_{1}, y_{2}), β) are collinear in T_{2}(O)^{∧} if and only if [x_{1}, x_{2}, 0, 1]∩[α, F^{−1}(x_{2}), 1, 0] and [y_{2}, y_{1}, 0, 1] ∩ [β, F^{−1}(y_{2}), 1, 0] are concurrent lines of Σ. This is the case if and only if

α + β = \frac{x_{1} + y_{1}}{x_{2} + y_{2}} (F^{−1}(x_{2}) + F^{−1}(y_{2})).

If x_{1} = y_{1}, then ((x_{1}, x_{2}), α) and ((y_{1}, y_{2}), β) are incident with the same point of type (b) of T_{2}(O)^{∧}, and are collinear as points of T_{2}(O)^{∧} if and only if the plane they span contains the point N. This is the case if and only if α + β = 0.

Thus the 1-cochain

c_{∞}((x_{1}, x_{2}), (y_{1}, y_{2})) = \{ \begin{matrix}\frac{x_{1} + y_{1}}{x_{2} + y_{2}} (F^{−1}(x_{2}) + F^{−1}(y_{2})) & for x_{2} ≠ −y_{2} \\ 0 & for x_{2} = y_{2} \end{matrix}

defines the q-fold cover of AG(2, q) constructed from T_{2}(O)^{∧} and the regular point (0, 0, 1, 0).

5.6.2 Case (b) X = (0, 1, t, F (t)), t ∈ GF(q)

At this point we note that the only property of F that we used in deriving c_{∞} was that O = {(0, 1, s, F (s)) : s ∈ GF(q)} ∪ {(0, 0, 1, 0)} is an oval with nucleus (0, 0, 0, 1). Suppose that we can find a collineation T of H = PG(2, q) that maps O onto an oval O′, such that (0, 1, t, F (t)) is mapped to (0, 0, 1, 0) and (0, 0, 0, 1) is fixed. Then T (O) = O′ = {(0, 1, u, G(u)) : u ∈ GF(q)} ∪ {(0, 0, 1, 0)}, for some polynomial G. Since T induces an isomorphism from T_{2}(O)^{∧} to T_{2}(O′)^{∧}, it follows from Corollary 5.3.4 that the derivation in Section 5.6.1 gives a 1-cochain equivalent to that defining the q-fold cover given by T_{2}(O)^{∧} and the regular point (0, 1, t, F (t)). In what follows we define such collineations and calculate the corresponding 1-cochains. Note that we still assume that F satisfies the conditions of [24, 8.4.2 Theorem].

Now let T be the collineation of H that acts by T (x_{0}, x_{1}, x_{2}, x_{3}) = (x_{0}, x_{2} + tx_{1}, x_{1}, x_{3} + F (t)x_{1}). Then T (0, 0, 0, 1) = (0, 0, 0, 1), T (0, 1, t, F (t)) = (0, 1, t, F (t)) and

T (O) = {(0, s + t, 1, F (s) + F (t)) : s ∈ GF(q)} ∪ {(0, 1, 0, 0)}

= \{(0, 1, \frac{1}{s + t}, \frac{F (s) + F (t)}{s + t}) : s ∈ GF(q) \ \{t\}\}

= \{(0, 1, u, \frac{\frac{1}{u} + F (t)}{\frac{1}{u}}) : u ∈ GF(q) \ \{0\}\} ∪ \{(0, 1, 0, 0), (0, 0, 1, 0)\}
= {(0, 1, u, f_{t}(u)) : u ∈ GF(q)} ∪ {(0, 0, 1, 0)},

where

f_{t}(u) = \{ \begin{matrix}(F (\frac{1}{u} + t) + F (t))/\frac{1}{u} & if u ≠ 0 \\ 0 & if u = 0 \end{matrix} .

Now if we define the permutation f on GF(q) by

f(u) = \{ \begin{matrix}\frac{1}{u} & if u ≠ 0 \\ 0 & if u = 0 \end{matrix} ,

which may be written as the polynomial f(x) = x^{q−2}, then f_{t} = F_{t} ∘ f where F_{t} is the permutation s ↦ (F (t + s) + F (t))/s defined in [24, 8.4.2 Theorem].

Given the above, we have that

c_{t}((x_{1}, x_{2}), (y_{1}, y_{2})) = \{ \begin{matrix}\frac{x_{1} + y_{1}}{x_{2} + y_{2}} (f_{t}^{−1}(x_{2}) + f_{t}^{−1}(y_{2})) & if x_{2} ≠ −y_{2} \\ 0 & if x_{2} = y_{2} \end{matrix} ,

is a 1-cochain that defines the q-fold cover of AG(2, q) given by T_{2}(O)^{∧} and the regular point (0, 1, t, F (t)).

By a similar approach to that taken in Section 5.4.3, it is possible to show that if

c((x_{1}, x_{2}), (y_{1}, y_{2})) = \{ \begin{matrix}\frac{x_{1} + y_{1}}{x_{2} + y_{2}} (G(x_{2}) + G(y_{2})) & if x_{2} ≠ y_{2} \\ 0 & if x_{2} = y_{2} \end{matrix}

is a 1-cochain of the simplicial complex of the point graph of AG(2, q), then c satisfies the GQ condition if and only if the set {(1, t, G(t)) : t ∈ GF(q)} ∪ {(0, 1, 0)} is an oval of PG(2, q).

5.7 Remarks

The work in this chapter raises many questions and possibilities for future research. An obvious line of enquiry is to investigate 1-cochains of the simplicial complex of the point graph of a non-desarguesian affine plane π of order s. Any such 1-cochain that satisfies the GQ condition gives rise to a GQ of order s that is new, since all of the known GQs of order s with a regular point have associated affine plane AG(2, s) (as in Theorem 5.0.1) for s a prime power. (This was shown for W (q) in Section 5.2.1. For T_{2}(O) it was shown in Section 5.2.2 that the affine plane associated with the regular point (∞) is AG(2, q). For T_{2}(O)^{∧} the affine plane associated with each of the regular points of type (b) was shown to be AG(2, q) in Section 5.6. Note that these results are already well known! (See [49, Chapter 8], for example). By [49, 3.3.3 (i)] the only regular points of T_{2}(O)^{∧} are the points of type (b) and by [49, 12.4.6] if T_{2}(O) has a regular point not (∞), then O is a translation oval and so by [49, 12.5.3 (i)] is isomorphic to T_{2}(O′)^{∧} for some oval O′. Thus the work in Sections 5.2.1, 5.2.2 and 5.6 covers all regular points of known GQ of order s).

By Sections 5.2.1, 5.2.2 and 5.6 and by Theorem 5.3.7 we know all 1-cochains of the simplicial complex of the point graph of AG(2, q) that give rise to the known GQ of order q. It would be interesting to investigate the 1-cochains to see if others satisfying the GQ condition could be found. A possible approach is to consider a particular group G of AΓL(3, q) and see it it is possible to find a 1-cochain admitting G and satisfying the GQ condition by the construction in Theorem 5.5.1.

It was observed in Section 1.6 that an s-fold cover of a geometry is equivalent to an sfold cover of the dual geometry. Thus the construction in Theorem 5.1.2 can be equivalently rewritten in terms of an s-fold cover of the dual affine plane. Presumably the algebraic s-fold covers of an affine plane π are equivalent to the algebraic s-fold covers of the dual geometry of π, and so the corresponding theories regarding GQs of order s with a regular point are presumably also equivalent. However, it may be that covering the dual affine plane offers some computational advantages. Note that even though the point graph of a dual affine plane is not complete it is relatively straight-forward to show that the first cohomology group of the corresponding simplicial complex is trivial, and so the cohomological aspects of the theory remain simple.

In Section 5.1 the theory of GQs with a regular point and covers of the associated affine plane was introduced for a general cover rather than an algebraic cover. It would be interesting to try and find other “natural” constructions of covers, apart from the algebraic covers, and investigate those with respect to the construction in Theorem 5.1.2.

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