Trialities and 1-systems of Q^{+}(7, q)

D. Luyckx  J. A. Thas

October 2002

Mathematics Subject Classification: 51A50, 51E12, 51E30

Proposed running head: Trialities and 1-systems of Q^{+}(7, q)

Abstract

The image of a 1-system of Q^{+}(7, q) under a triality of the D_{4}-geometry, attached to Q^{+}(7, q), will be investigated. Attention will mainly be paid to the case of a locally hermitian, semiclassical 1-system of a Q(6, q), embedded in Q^{+}(7, q). It is found that its image under a triality is always locally hermitian and semiclassical as well. Moreover, it is a proper 1-system of Q^{+}(7, q) whenever the original 1-system of Q(6, q) is not a spread of some generalized hexagon H(q) on Q(6, q). Finally, some results concerning isomorphisms will be obtained.

1 Definitions

1.1 Trialities, T-correspondences and generalized hexagons

Consider the projectively unique hyperbolic quadric Q^{+}(7, q) in PG(7, q). It is well known, see for instance [4, 22.4], that the generators of Q^{+}(7, q), which are 3-dimensional subspaces, can be subdivided in two subsets, often called families, in the following way. Two distinct generators are said to belong to the same family if and only if their intersection is empty or a line. Hence generators of different families intersect in a point or a plane and furthermore, every plane of Q^{+}(7, q) is contained in exactly two generators, namely one from each family. If we denote the two families of generators of Q^{+}(7, q) by F_{1} and F_{2}, the set of lines on Q^{+}(7, q) by L and the set of points of Q^{+}(7, q) by P, then a D_{4}- geometry Ω can be attached to Q^{+}(7, q), in the following way. The 0-points of Ω are the points of Q^{+}(7, q); the lines are just the lines of Q^{+}(7, q); the 1-points are the elements of one family of generators, say F_{1}; and the 2-points are the elements of the other family of generators. Incidence is symmetrized containment for i-points and lines, i = 0, 1, 2, and also for 0-points and j-points, j = 1, 2. A 1-point G_{1} ∈ F_{1} is said to be incident with a 2-point G_{2} ∈ F_{2} if and only if the intersection G_{1} ∩ G_{2} is a plane of Q^{+}(7, q). An important property of the geometry Ω is the fact that every permutation of the set {P, F_{1}, F_{2}} defines a geometry which is isomorphic to Ω.

Definitions A triality of the geometry Ω, attached to Q^{+}(7, q) as above, is a map τ:

τ : L → L, P → F_{1}, F_{1} → F_{2}, F_{2} → P

preserving the incidence in Ω and such that τ^{3} is the identity. A point p ∈ P is called an absolute point of τ if p is incident with p^{τ} ∈ F_{1}. A 1- or 2-point, that is, a generator G_{i} ∈ F_{i}, i = 1, 2, is said to be absolute for τ if it is incident with its image G_{i}^{τ}. A line L ∈ L is called absolute for τ if L = L^{τ}.

Although the condition “τ^{3} is the identity” is an explicit part of the definition of a triality, most of the results that will be obtained in this paper, hold under weaker conditions too. Often it is not necessary to require that τ^{3} is the identity. Therefore we introduce another less common definition, which is due to Tits [13].

Definition A T-correspondence of the geometry Ω is a map θ:

θ : L → L, P → F_{1}, F_{1} → F_{2}, F_{2} → P

which preserves incidence in Ω.

The definitions of absolute points and lines for a T-correspondence θ remain the same as for a triality τ.

Remark that the assignment of the names F_{1} and F_{2} to the families of generators of Q^{+}(7, q) is arbitrary, and hence the roles of F_{1} and F_{2} may be interchanged in the above definitions. As such, τ^{−1} = τ^{2} is also a triality and both θ^{−1} and θ^{2} are T-correspondences. For convenience, we agree on the convention that a triality, respectively a T-correspondence, is always a map as in the definitions above, unless otherwise mentioned, for some choice of F_{1} and F_{2} which we consider to be fixed throughout this paper.

Trialities are very interesting for the study of generalized hexagons, as some trialities produce generalized hexagons. This was shown by Tits in his celebrated paper “Sur la trialité et certains groupes qui s’en déduisent” [13]. Generalized hexagons are defined as follows.

Definitions A generalized hexagon is an incidence geometry Γ = (P, L, I) of points and lines such that the following three axioms are satisfied:

GH1 Γ contains no ordinary k-gon for 2 ≤ k ≤ 5.

GH2 Any two elements x, y ∈ P ∪ L are contained in some ordinary hexagon in Γ.

GH3 There exists an ordinary sevengon in Γ.

If Γ = (P, L, I) only has to satisfy conditions (i) and (ii), then it is called a weak generalized hexagon. A thick line of a weak generalized hexagon Γ is a line of Γ which contains at least three points of Γ.

The following theorem states the relation between trialities and generalized hexagons.

Theorem 1.1 (Tits [13]) Let τ be a triality of the geometry Ω. Suppose that one of the following equivalent hypotheses is satisfied:

(i) there exists at least one absolute i-point, for some i ∈ {0, 1, 2}, and every absolute i-point is incident with at least two absolute lines;

(ii) there exists a sequence of absolute lines (L_{1}, L_{2}, …, L_{d}), d > 2, such that L_{i} is concurrent with L_{i+1}, indices modulo d.

Then for every i ∈ {0, 1, 2}, the geometry Γ^{(i)} with point set P_{abs}^{(i)} the set of absolute i-points, with line set L_{abs} the set of absolute lines and with the natural incidence, is a weak generalized hexagon with thick lines.

Over the finite field GF(q), there exist essentially two examples of generalized hexagons arising from a triality. The first one is the so-called split Cayley hexagon, denoted by H(q), which exists for every q. This hexagon has order (q, q) and an interesting fact is that all points and lines of H(q) are contained in a hyperplane PG(6, q) of PG(7, q). Also, two points of H(q) are opposite (that is, at distance 6) in H(q) if and only if they are not joined by a line of Q^{+}(7, q) and moreover, the q + 1 lines through a point of H(q) form a flat pencil of lines in a totally singular plane of Q^{+}(7, q). The other example is the twisted triality hexagon T(q^{′3}, q′) and it exists only if q = q^{′3} for some prime power q′, but this example will not be important for the sequel. For more information about trialities and their relation to generalized hexagons, the reader is referred to the original paper of Tits [13], or to Chapter 2 of the monograph “Generalized Polygons” [14] by Van Maldeghem.

To finish this subsection, we give two easy properties of T-correspondences.

Property 1.2 Let θ be a T-correspondence of Ω. If L and M are two distinct concurrent lines, contained in a totally singular plane π on Q^{+}(7, q), then the same holds for L^{θ} and M^{θ}.

Proof. If L ∩ M is the point x and G_{1} ∈ F_{1} and G_{2} ∈ F_{2} are the two generators of Q^{+}(7, q) through π, then L^{θ} and M^{θ} are lines in the intersection of the two generators x^{θ} ∈ F_{1} and G_{1}^{θ} ∈ F_{2}, and they contain the point G_{2}^{θ}. □

Property 1.3 Let θ be a T-correspondence of Ω and let π be a totally singular plane on Q^{+}(7, q). Then the set of lines in π is mapped by θ onto the set of lines through a point in a generator of Q^{+}(7, q).

Proof. Let again G_{1} ∈ F_{1} and G_{2} ∈ F_{2} denote the two generators through the plane π. Then all lines of π are mapped by θ onto all lines in the generator G_{1}^{θ} ∈ F_{2} through the point G_{2}^{θ}. □

1.2 1-Systems

Consider a quadric Q ∈ {Q^{−}(5, q), Q(6, q), Q^{+}(7, q)}. A 1-system of Q is a set M of q^{3} + 1 lines L_{0}, L_{1}, …, L_{q^{3}} with the property that every generator of Q which contains a line L_{i} of M, is disjoint from all lines L_{j} ∈ M, j ≠ i. Thus, if Q = Q^{−}(5, q), a 1-system of Q is nothing but a spread of Q^{−}(5, q). The set of all points on the lines of M will be denoted by M̃, so M̃ is the union of all elements of M.

As an example, we give the definition of a hermitian spread of Q^{−}(5, q). Let Q^{+}(5, q^{2}) be the extension to GF(q^{2}) of Q^{−}(5, q). Then there exist two disjoint and conjugate planes π and \overline{π} on Q^{+}(5, q^{2}), which contain no point of Q^{−}(5, q). The set of lines of Q^{−}(5, q), the extensions to GF(q^{2}) of which have a point in common with both π and \overline{π}, forms a spread S of Q^{−}(5, q); see for instance [11]. Since the common points of π and the extensions to GF(q^{2}) of the lines of S form a hermitian curve H(2, q^{2}) in π, this spread is called a hermitian spread of Q^{−}(5, q).

Clearly, every 1-system of a Q^{−}(5, q) or a Q(6, q) which is embedded in Q^{+}(7, q), is also a 1-system of Q^{+}(7, q). If a 1-system of Q^{+}(7, q) is not contained in a hyperplane of the ambient space PG(7, q) of Q^{+}(7, q), then it is said to be a proper 1-system of Q^{+}(7, q). Similarly, a 1-system of Q(6, q) ⊆ PG(6, q) is proper if and only if it is not contained in a hyperplane of PG(6, q).

Let M be a 1-system of Q, with again Q ∈ {Q^{−}(5, q), Q(6, q), Q^{+}(7, q)}. Then M is locally hermitian at a line L ∈ M if and only if for every line M ∈ M \ {L}, the regulus of lines containing L of the hyperbolic quadric ⟨L, M⟩∩Q = Q^{+}(3, q), is completely contained in M. Hence a locally hermitian 1-system of Q consists of q^{2} reguli through a special line L, and these reguli will often be denoted by R_{1}, R_{2}, …, R_{q^{2}}. If Q is either a Q(6, q) or a Q^{+}(7, q), then it holds for every locally hermitian 1-system of Q that ⟨R_{i}, R_{j}⟩ is 5-dimensional if i ≠ j, and moreover, ⟨R_{i}, R_{j}⟩∩Q is an elliptic quadric Q_{ij}^{−}(5, q), for the following reason. If Q_{ij} := ⟨R_{i}, R_{j}⟩ ∩ Q is not elliptic, then it contains totally singular planes. If M ≠ L is a line of R_{i}, then there exists in particular a totally singular plane α of Q_{ij} through the line M. But α∩⟨R_{j}⟩ is at least a point, which implies that α meets at least one line of R_{j} in a point. This contradicts the definition of a 1-system of Q and we conclude that Q_{ij} is indeed an elliptic quadric Q_{ij}^{−}(5, q).

Suppose next that M is a 1-system of Q, with Q ∈ {Q(6, q), Q^{+}(7, q)}, which is locally hermitian at the line L ∈ M, and let x be an arbitrary point on L. Consider a 5-dimensional subspace γ of the tangent hyperplane T_{x}(Q) of Q at x, with the property that x ∉ γ, so that γ ∩ Q := Q′ is a parabolic quadric Q(4, q), respectively a hyperbolic quadric Q^{+}(5, q). Every regulus R_{i} of M through L has a unique transversal through the point x, which meets Q′ in a point n_{i}. Together with the point l := L ∩ γ, we thus obtain q^{2} + 1 points on Q′. If two points l and n_{i}, respectively n_{i} and n_{j} with i ≠ j, were collinear on Q′, then the totally singular plane ⟨x, l, n_{i}⟩, respectively ⟨x, n_{i}, n_{j}⟩, would contain at least 2q + 1 points of M̃. This is a contradiction to the elementary properties of 1systems, see Shult and Thas [10, Theorem 8]. It follows that the q^{2} + 1 points l, n_{1}, n_{2}, …, n_{q^{2}} form an ovoid of Q′. This ovoid is called the projection along reguli of M from x onto γ and denoted by O_{x}. If the ovoid O_{x} is the classical ovoid Q^{−}(3, q) of Q′ for all points x on L, then M is said to be semiclassical.

Finally, let M be a 1-system of Q(6, q) and suppose that M is locally hermitian at the line L ∈ M. Denote the q^{2} reguli of M through L by R_{1}, R_{2}, …, R_{q^{2}} and consider two distinct reguli R_{i}, R_{j} of M through L. Then ⟨R_{i}, R_{j}⟩∩Q(6, q) is an elliptic quadric Q_{ij}^{−}(5, q), as has been explained above. This quadric Q_{ij}^{−}(5, q) defines a generalized quadrangle which is the point-line dual of the generalized quadrangle H(3, q^{2}) arising from a non-singular hermitian variety in PG(3, q^{2}), see Payne and Thas [9]. With R_{i} and R_{j} correspond point sets W_{i} and W_{j} on H(3, q^{2}), which are Baer sublines of lines in PG(3, q^{2}) and have exactly one point of H(3, q^{2}) in common. So W_{i} and W_{j} are contained in exactly one common plane which intersects H(3, q^{2}) in a hermitian curve H(2, q^{2}) of H(3, q^{2}). This implies that the reguli R_{i} and R_{j} uniquely define a hermitian spread S_{ij} of Q_{ij}^{−}(5, q). In PG(5, q) = ⟨Q_{ij}^{−}(5, q)⟩, consider a 3-dimensional subspace PG(3, q) skew to L. If S_{ij} = {L, M_{1}, M_{2}, …, M_{q^{3}}} and τ := T_{L}(Q_{ij}^{−}(5, q)) is the tangent space of Q_{ij}^{−}(5, q) at L, then the lines ⟨L, M_{i}⟩∩PG(3, q), i = 1, 2, …, q^{3}, together with τ ∩ PG(3, q) := L′ form a regular spread S of PG(3, q); see Bloemen, Thas and Van Maldeghem [1]. Let x be an arbitrary point on L and denote the q^{2} reguli of S_{ij} through L by R′_{1}, R′_{2}, …, R′_{q^{2}}. Then each of these q^{2} reguli has a unique transversal through x, which must be a generating line of the cone T_{x}(Q_{ij}^{−}(5, q)) ∩ Q_{ij}^{−}(5, q) := xQ_{x}^{−}(3, q). Together with the line L, this yields all q^{2} + 1 lines containing x of the cone xQ_{x}^{−}(3, q). For every point z ∈ Q_{x}^{−}(3, q), z ∉ L, the plane ⟨L, z⟩ intersects PG(3, q) in a point. In this way, we obtain q^{2} points of PG(3, q) which form a plane π_{x} together with the line L′. This plane π_{x} is in fact the projection from L onto PG(3, q) of ⟨xQ_{x}^{−}(3, q)⟩ \ L. Each conic of Q_{x}^{−}(3, q) through l′ := L∩Q_{x}^{−}(3, q) corresponds to a line of π_{x}, different from L′; this line belongs to the opposite regulus of a regulus of S through L′. Conversely, any such line of π_{x} corresponds to a conic of Q_{x}^{−}(3, q) on l′. Hence each regulus of S containing L′ defines a conic of Q_{x}^{−}(3, q) through l′, and conversely. The set of q reguli of S_{ij} defined by a regulus of S containing L′ will be called an R-conic of Q(6, q). If x′ ∈ L \ {x}, then the plane π_{x′} defines the same R-conics as the plane π_{x}. In particular, for every point y on L, the q transversals through y of the elements of an R-conic, together with the line L, form a cone with vertex y and a non-singular conic C_{y} ⊆ Q_{y}^{−}(3, q) through the point L ∩ Q_{y}^{−}(3, q) as base.

In [6] and [7], it is shown that the locally hermitian, semiclassical 1-systems of Q(6, q) have a special property, which is formulated in the following lemma.

Lemma 1.4 ([6], [7]) Let M be a proper 1-system of Q(6, q), which is locally hermitian at some line L ∈ M and in addition semiclassical. Denote the q^{2} reguli of M through L by R_{1}, R_{2}, …, R_{q^{2}}. Then for every i ≠ j, the elliptic quadric Q_{ij}^{−}(5, q) = ⟨R_{i}, R_{j}⟩ ∩ Q(6, q) contains exactly q reguli of M through L and these reguli form an R-conic.

To conclude this section, a construction is given for locally hermitian spreads of the elliptic quadric Q^{−}(5, q).

Let S be a spread of Q^{−}(5, q), which is locally hermitian at some line L ∈ S. Consider the tangent space γ := T_{L}(Q^{−}(5, q)) of Q^{−}(5, q) at the line L and denote the q^{2} reguli of S through L by R_{1}, R_{2}, …, R_{q^{2}}. Then for every regulus R_{i}, it holds that ⟨R_{i}⟩^{⊥}, with ⊥ the polarity of Q^{−}(5, q), is a line L_{i} in γ, and the q^{2} + 1 lines L, L_{1}, L_{2}, …, L_{q^{2}} form a line spread S in γ. Conversely, one can reconstruct a locally hermitian spread of Q^{−}(5, q) from every line spread S′ in γ which contains the line L, just by reversing the above construction. If a locally hermitian spread of Q^{−}(5, q) is obtained in this way from a line spread S of γ through L, it will be denoted by S(S). It can be shown, see [2], that S(S) is hermitian if and only if the spread S is the regular line spread in γ, which is determined by two conjugate lines T and \overline{T} with respect to GF(q^{2}), such that ⟨L, T ⟩ and ⟨L, \overline{T} ⟩ are the generators containing L of the extension Q^{+}(5, q^{2}) of Q^{−}(5, q) to GF(q^{2}).

2 T-correspondences and locally hermitian 1-systems of Q(6, q)

Before atention is paid to the image under a T-correspondence (and hence also a triality) of a locally hermitian 1-system of an induced Q(6, q) ⊆ Q^{+}(7, q), it is first shown that T-correspondences map 1-systems of Q^{+}(7, q) onto 1-systems, in the following easy, but basic, theorem.

Theorem 2.1 Let θ be a T-correspondence of Ω and consider an arbitrary 1system M of Q^{+}(7, q). Then M^{θ} is also a 1-system of Q^{+}(7, q).

Proof. Suppose that the claim is false and M^{θ} is not a 1-system of Q^{+}(7, q). This implies that there exist a line M ∈ M^{θ} and a generator G of Q^{+}(7, q) on M such that G meets some line N of M^{θ} \ {M} at least in a point x. In that case ⟨M, x⟩ is a totally singular plane on Q^{+}(7, q) and as such contained in exactly two generators G_{1} and G_{2} of Q^{+}(7, q); assume without loss of generality that G = G_{1} ∈ F_{1} and G_{2} ∈ F_{2}. It then holds that N^{θ^{−1}} and M^{θ^{−1}} are lines of M, the generator x^{θ^{−1}} ∈ F_{2} contains N^{θ^{−1}}, G_{1}^{θ^{−1}} is a point of M^{θ^{−1}} and as x ∈ G_{1}, it follows that the point G_{1}^{θ^{−1}} is contained in x^{θ^{−1}}. In other words, x^{θ^{−1}} is a generator of Q^{+}(7, q) through the line N^{θ^{−1}} ∈ M, which has the point G_{1}^{θ^{−1}} in common with the line M^{θ^{−1}} ∈ M \ {N^{θ^{−1}}}. This is a contradiction to the fact that M is a 1-system of Q^{+}(7, q), which proves the theorem. □

Since locally hermitian 1-systems are characterized by the fact that they consist of a number of reguli through some common line, it is important to understand the action of a T-correspondence on a regulus.

Lemma 2.2 Let θ be a T-correspondence of Ω and consider a 3-dimensional subspace γ of PG(7, q) such that γ ∩ Q^{+}(7, q) = Q^{+}(3, q). Then θ maps each regulus of lines of Q^{+}(3, q) onto a regulus of lines of some Q^{+}(3, q)′ = γ′ ∩ Q^{+}(7, q), with γ′ also a 3-dimensional subspace of PG(7, q).

Proof. Denote the two reguli of the hyperbolic quadric Q^{+}(3, q) by {K_{0}, K_{1}, …, K_{q}} and {T_{0}, T_{1}, …, T_{q}}. Since for i ≠ j, any two lines K_{i} and K_{j}, respectively T_{i} and T_{j}, are disjoint and not contained in a generator of Q^{+}(7, q), the same holds for their images K_{i}^{θ} and K_{j}^{θ}, respectively T_{i}^{θ} and T_{j}^{θ}. On the other hand, every K_{i} meets every T_{j} in a point, which implies that K_{i}^{θ} and T_{j}^{θ} are contained in a generator of Q^{+}(7, q), for i, j ∈ {0, 1, …, q}. If K_{i}^{θ} ∩ T_{j}^{θ} were a point y, then K_{i} and T_{j} would be contained in the generator y^{θ^{−1}} of Q^{+}(7, q), clearly a contradiction. So K_{i}^{θ} is disjoint from T_{j}^{θ}, for all i, j ∈ {0, 1, …, q} and the situation is such that every T_{j}^{θ} is contained in (K_{i}^{θ})^{⊥}, for all i = 0, 1, …, q. Consequently, the q + 1 lines T_{j}^{θ}, j = 0, 1, …, q, must form a regulus of Q^{+}(3, q)′′ := ⟨K_{0}^{θ}, K_{1}^{θ}⟩^{⊥} ∩ Q^{+}(7, q) and this in turn implies that the lines K_{0}^{θ}, K_{1}^{θ}, …, K_{q}^{θ} form a regulus of ⟨Q^{+}(3, q)′′⟩^{⊥} ∩ Q^{+}(7, q) := Q^{+}(3, q)′, where Q^{+}(3, q)′ = ⟨K_{0}^{θ}, K_{1}^{θ}⟩ ∩ Q^{+}(7, q). This proves the lemma. □

This lemma ensures that every T-correspondence θ maps a 1-system M of Q^{+}(7, q) that is locally hermitian at some line L, onto a 1-system M^{θ} that is locally hermitian at the line L^{θ}. The next theorem we will prove, tells something about the dimension of ⟨M^{θ}⟩, but first we need another lemma.

Lemma 2.3 Suppose that θ is a T-correspondence of Ω and let A be the line set of a generalized hexagon H(q) on Q^{+}(7, q). Then the set A^{θ} is the set of lines of a generalized hexagon H(q)′ on Q^{+}(7, q).

Proof. The hexagon H(q) consists of the absolute points and lines of some triality τ of the D_{4}-geometry Ω, attached to Q^{+}(7, q), which satisfies the hypotheses of Theorem 1.1. Let L be a line of H(q), that is, L is absolute for τ. Then the line L^{θ} satisfies

(L^{θ})^{θ^{−1}τθ} = L^{τθ} = L^{θ},

so L^{θ} coincides with its image under the map θ^{−1}τθ. It is obvious that θ^{−1}τθ is also a triality of Ω, but to prove the claim completely, we must ensure that θ^{−1}τθ satisfies one of the hypotheses of Theorem 1.1. We will check the second of the two equivalent conditions from that theorem. On H(q) there certainly exists an ordinary hexagon, consisting of the sequence of lines (L_{1}, L_{2}, L_{3}, L_{4}, L_{5}, L_{6}), where L_{i} is concurrent with L_{i+1}, indices modulo 6. Then (L_{1}^{θ}, L_{2}^{θ}, L_{3}^{θ}, L_{4}^{θ}, L_{5}^{θ}, L_{6}^{θ}) is a sequence of absolute lines for θ^{−1}τθ. Since L_{i} is concurrent with L_{i+1} on H(q) for i = 1, 2, …, 6 and indices taken modulo 6, ⟨L_{i}, L_{i+1}⟩ is a totally singular plane of Q^{+}(7, q). Now by Property 1.2 this implies that L_{i}^{θ} and L_{i+1}^{θ} have an absolute point for θ^{−1}τθ in common, for i = 1, 2, …, 6 and indices taken modulo 6. Hence the triality θ^{−1}τθ satisfies (ii) of Theorem 1.1 and so its absolute points and lines are the points and lines of a generalized hexagon. Finally, the generalized hexagon of the absolute points and lines of θ^{−1}τθ must be isomorphic to the one defined by τ, because we have that M^{θ^{−1}τθ} = M if and only if (M^{θ^{−1}})^{τ} = M^{θ^{−1}}. So a line M of Q^{+}(7, q) is absolute for θ^{−1}τθ if and only its inverse image M^{θ^{−1}} is absolute for τ. Hence both hexagons have the same order (q, q). This proves the lemma completely. □

Using this lemma, we can prove the following interesting theorem.

Theorem 2.4 Let A be a set of lines on a Q(6, q) ⊆ Q^{+}(7, q) and consider a Tcorrespondence θ of Ω. Then the lines of A^{θ} lie in an induced Q(6, q)′ ⊆ Q^{+}(7, q) if and only if the lines of A are lines of a generalized hexagon H(q) on Q(6, q).

Proof. Denote by B the set of all lines of Q(6, q) and suppose that A^{θ} ⊆ Q(6, q)′ for some parabolic quadric Q(6, q)′ ⊆ Q^{+}(7, q). We investigate which lines of B^{θ} are lines of this Q(6, q)′. If p is an arbitrary point of Q(6, q)′, then p^{θ^{−1}} is a generator of Q^{+}(7, q). So p^{θ^{−1}} intersects Q(6, q) in a totally singular plane, containing q^{2} + q + 1 lines of B. By Property 1.3, this implies that the q^{2} + q + 1 lines of B^{θ} through the point p are exactly the lines on p contained in some generator G of Q^{+}(7, q) through p. But G meets Q(6, q)′ in a totally singular plane α of Q(6, q)′, and hence the common lines of B^{θ} and Q(6, q)′ through the point p are exactly the q + 1 lines on p in the totally singular plane α. In other words, the common lines of B^{θ} and Q(6, q)′ through an arbitrary point of Q(6, q)′ form a pencil of lines in some totally singular plane of Q(6, q)′ containing this point. Now, by Theorem 3.7 of [12], this implies that the common lines of B^{θ} and Q(6, q)′ either form the set of all lines in the q^{3} + 1 planes of a spread of Q(6, q)′, or they form the line set of a generalized hexagon H(q)′ on Q(6, q)′. The first case cannot occur in the current situation, because in that case, (B^{θ})^{θ^{−1}} = B would contain q^{2} + q + 1 lines through a point of Q^{+}(7, q) in a generator of Q^{+}(7, q). This is of course a contradiction, as the rank of Q(6, q) is 3. So the lines of B^{θ} on Q(6, q)′ form the set of lines of a generalized hexagon H(q)′ on Q(6, q)′. Since A^{θ} ⊆ B^{θ} and the lines of A^{θ} lie on Q(6, q)′ by assumption, all lines of A^{θ} must be lines of H(q)′. If we now apply Lemma 2.3 to the T-correspondence θ^{−1}, we find that all lines of A are lines of a generalized hexagon H(q) on Q(6, q).

As the converse has already been shown in Lemma 2.3, the proof of the theorem is complete. □

Suppose as a special case that M is a locally hermitian 1-system of Q(6, q) ⊆ Q^{+}(7, q), not isomorphic to a spread of a generalized hexagon H(q) on Q(6, q). Then by the above results, the image M^{θ} of M under a T-correspondence θ is a locally hermitian 1-system of Q^{+}(7, q) that is not contained in a non-tangent hyperplane of PG(7, q). If it were contained in a tangent hyperplane of Q^{+}(7, q), this would imply the existence of a set of q^{3}+1 lines on Q^{+}(5, q) with the property that every plane of Q^{+}(5, q) which contains one of these lines, is disjoint from the other q^{3} lines. From [10, Theorem 4], it follows that this is a contradiction. We conclude that M^{θ} is a proper locally hermitian 1-system of Q^{+}(7, q). In the next section we will have a closer look at M^{θ} if the 1-system M of Q(6, q) is not only locally hermitian, but also semiclassical.

3 T-correspondences and locally hermitian, semiclassical 1-systems of Q(6, q)

Suppose again that θ is a T-correspondence of Ω. We now examine the special case where M is a locally hermitian 1-system of a Q(6, q) ⊆ Q^{+}(7, q), which is in addition semiclassical. Let L ∈ M be a line at which M is locally hermitian and denote the q^{2} reguli of M through L by R_{1}, R_{2}, …, R_{q^{2}}, as usually. We already know that M^{θ} is locally hermitian at the line L^{θ}, but one would also like to know whether M^{θ} is semiclassical or not.

Let x be a point on L^{θ}. Then the q^{2} transversals of the reguli of M^{θ} through L^{θ} are the images under θ of q^{2} lines in the generator x^{θ^{−1}} ∈ F_{2} through L. From the proof of Lemma 2.2, it follows that if T is a transversal on x of a regulus R_{i}^{θ}, with R_{i}^{θ} the image under θ of some regulus R_{i} of M containing L, then T^{θ^{−1}} is a line of ⟨R_{i}⟩^{⊥}, with ⊥ the polarity of Q^{+}(7, q). As x is a point of T, the line T^{θ^{−1}} is thus the unique line in the intersection of x^{θ^{−1}} and ⟨Q_{i}^{+}(3, q)⟩ = ⟨R_{i}⟩^{⊥}. So the transversals through x of the reguli of M^{θ} containing L^{θ}, are the images under θ of the q^{2} lines ⟨R_{i}⟩^{⊥} ∩ x^{θ^{−1}}, i = 1, 2, …, q^{2}. These q^{2} lines in x^{θ^{−1}} are pairwise disjoint, since no two transversals through x of the reguli of M^{θ} containing L^{θ}, are contained in a totally singular plane of Q^{+}(7, q). For the same reason, all of them are also disjoint from L, so that {L} ∪ {⟨R_{i}⟩^{⊥} ∩ x^{θ^{−1}} | i = 1, 2, …, q^{2}} is a spread of the 3-dimensional projective space x^{θ^{−1}}. If x^{θ^{−1}} := G, then we denote this spread by S_{G}. Apparently, the structure of S_{G} determines the configuration of the transversals on x of the reguli of M^{θ} through L^{θ}.

Lemma 3.1 The 1-system M^{θ} is semiclassical if and only if for every generator G ∈ F_{2} containing L, the spread S_{G} is regular.

Proof. Let x on L^{θ} be fixed and consider in x^{θ^{−1}} := G the spread S_{G}. Suppose that {K_{0}, K_{1}, …, K_{q}} is a regulus of lines of S_{G} and denote its opposite regulus by {T_{0}, T_{1}, …, T_{q}}. Then K_{0}^{θ}, K_{1}^{θ}, …, K_{q}^{θ} and T_{0}^{θ}, T_{1}^{θ}, …, T_{q}^{θ} are lines through the point x. As K_{i} and K_{j} are disjoint for i ≠ j, the lines K_{i}^{θ} and K_{j}^{θ} are not contained in a totally singular plane of Q^{+}(7, q). On the other hand, K_{i} ∩ T_{j} is a point for all i, j ∈ {0, 1, …, q}, so that ⟨K_{i}^{θ}, T_{j}^{θ}⟩ must be a totally singular plane of Q^{+}(7, q), by Property 1.2. Hence the only possible configuration is that ⟨K_{0}^{θ}, K_{1}^{θ}, …, K_{q}^{θ}⟩ ∩ Q^{+}(7, q) is a quadratic cone xC with generating lines K_{0}^{θ}, K_{1}^{θ}, …, K_{q}^{θ} and similarly ⟨T_{0}^{θ}, T_{1}^{θ}, …, T_{q}^{θ}⟩∩Q^{+}(7, q) is a quadratic cone xC′, but such that ⟨xC⟩^{⊥} ∩ Q^{+}(7, q) = xC′. One immediately sees that the converse also holds: if K_{0}^{θ}, K_{1}^{θ}, …, K_{q}^{θ}, respectively T_{0}^{θ}, T_{1}^{θ}, …, T_{q}^{θ}, are the generators of a quadratic cone xC, respectively xC′, with ⟨xC⟩^{⊥} ∩ Q^{+}(7, q) = xC′ as an additional property, then {K_{0}, K_{1}, …, K_{q}} is a regulus in G with opposite regulus {T_{0}, T_{1}, …, T_{q}}. It now easily follows that M^{θ} is semiclassical if and only if the spread S_{G} is a regular spread of G, for all generators G ∈ F_{2} through the line L. □

Thanks to this lemma, we are able to show that M^{θ} is semiclassical whenever M itself is semiclassical, by relying on the properties of locally hermitian, semiclassical 1-systems of Q(6, q). We start with the case of a proper 1-system of Q(6, q).

Theorem 3.2 Let M be a proper 1-system of a Q(6, q) ⊆ Q^{+}(7, q), which is locally hermitian at the line L ∈ M, and consider a T-correspondence θ of Ω. If M is semiclassical, then M^{θ} is semiclassical too.

Proof. Suppose that M is locally hermitian at the line L and furthermore semiclassical, and let the q^{2} reguli of M through L be denoted by R_{1}, R_{2}, …, R_{q^{2}}, as always. Consider two arbitrary reguli R_{i} and R_{j} of M through L. Then we know from Lemma 1.4 that the elliptic quadric ⟨R_{i}, R_{j}⟩ ∩ Q^{+}(7, q) = Q_{ij}^{−}(5, q) contains exactly q reguli of M through L. To make the notation easier, let i = 1 and j = 2 and denote the q reguli of M through L in Q_{12}^{−}(5, q) by R_{1}, R_{2}, …, R_{q}. Let G ∈ F_{2} be an arbitrary generator of Q^{+}(7, q) through the line L. We shall show that the lines ⟨R_{i}⟩^{⊥} ∩ G, i = 1, 2, …, q, together with L, form a regulus of lines of S_{G}. Of course, the proof holds for all R_{i}, R_{j}, i, j ∈ {1, 2, …, q^{2}} with i ≠ j, and not only for R_{1} and R_{2}.

As the reguli R_{1}, R_{2}, …, R_{q} form an R-conic of Q_{12}^{−}(5, q) by Lemma 1.4, the transversals of R_{1}, R_{2}, …, R_{q} through x, together with the line L, form a quadratic cone xC on Q_{12}^{−}(5, q) for every point x on L. Hence ⟨xC⟩^{⊥} ∩Q^{+}(7, q) is a quadratic cone xC′ through x and contained in L^{⊥}, but not containing L. The 3-space ⟨xC′⟩ meets every generator through L in a line, which is a generating line of xC′. Denote the line xC′ ∩ G by A_{x}. We also know that ⟨xC⟩ meets each ⟨R_{i}⟩, i = 1, 2, …, q, in a plane through L. As a consequence, ⟨xC′⟩ ∩ ⟨Q_{i}^{+}(3, q)⟩ is a plane as well, for all i = 1, 2, …, q, where Q_{i}^{+}(3, q) := ⟨R_{i}⟩^{⊥} ∩ Q^{+}(7, q). Since every Q_{i}^{+}(3, q) is disjoint from L and every line of xC′ meets L in the point x, the quadratic cone xC′ and Q_{i}^{+}(3, q) have a non-singular conic C_{i} in common. But C_{i} shares exactly one point with the line A_{x}, which shows that A_{x} has a unique point in common with Q_{i}^{+}(3, q), for i = 1, 2, …, q. In other words: A_{x} meets L and the q lines ⟨R_{i}⟩^{⊥} ∩ G, i = 1, 2, …, q, in a point. Suppose that two such lines A_{x} and A_{y}, for distinct points x and y on L, share a point u, say u ∈ Q_{1}^{+}(3, q). Since x ≠ y, ⟨A_{x}, A_{y}⟩ must then be a plane that contains L and the q − 1 lines ⟨R_{i}⟩^{⊥} ∩ G, i = 2, 3, …, q. But this means that L and ⟨R_{i}⟩^{⊥} are not disjoint for i = 2, 3, …, q, a contradiction. So every two lines A_{x} and A_{y}, for distinct x, y ∈ L, are disjoint. By the above, it follows that {L} ∪ {⟨R_{i}⟩^{⊥} ∩ G | i = 1, 2, …, q} is a regulus of lines of S_{G}, with opposite regulus {A_{x} | x ∈ L}. This property does not only hold for the q reguli R_{1}, R_{2}, …, R_{q}, but for any q reguli of M through L which are contained in an elliptic quadric Q_{ij}^{−}(5, q). As such, we have found that for any two lines ⟨R_{i}⟩^{⊥} ∩ G and ⟨R_{j}⟩^{⊥} ∩ G, i ≠ j, the regulus defined by these two lines and L completely consists of lines of S_{G}. Consequently, there are q(q + 1) reguli through L in S_{G} and by Theorem 3.1 of Gevaert, Johnson and Thas [3], it follows that S_{G} is regular. Finally, since the generator G ∈ F_{2} through L was chosen arbitrarily, the desired result follows from Lemma 3.1. □

Remark that the previous theorem is not valid for locally hermitian spreads of an elliptic quadric Q^{−}(5, q), which is embedded in Q^{+}(7, q). The following result solves the corresponding question in this case: it is possible to show that M^{θ} is semiclassical if and only if the spread M of Q^{−}(5, q) arises from a regular line spread of PG(3, q) by the construction, explained in Section 1.2.

Theorem 3.3 Let M be a spread of an elliptic quadric Q^{−}(5, q), embedded in Q^{+}(7, q), which is locally hermitian at some line L ∈ M, and consider a Tcorrespondence θ of Ω. Then the image M^{θ} of M under θ is semiclassical if and only if M = S(S) for a regular line spread S of T_{L}(Q^{−}(5, q)).

Proof. Suppose that M is locally hermitian at the line L ∈ M and denote the q^{2} reguli of M through L by R_{1}, R_{2}, …, R_{q^{2}}. By Lemma 3.1, M^{θ} is semiclassical if and only if for every generator G ∈ F_{2} of Q^{+}(7, q) through L, the spread S_{G} is a regular line spread of G. As M is locally hermitian, we can write M = S(S) for some line spread of the tangent space β := T_{L}(Q^{−}(5, q)) of Q^{−}(5, q) at L. Let T be the line ⟨Q^{−}(5, q)⟩^{⊥}, with ⊥ the polarity of Q^{+}(7, q). Then ⟨β, T ⟩ is the 5-dimensional subspace L^{⊥}, with ⊥ again the polarity of Q^{+}(7, q). Also, both β and ⟨L, T ⟩ = β^{⊥} only intersect Q^{+}(7, q) in L. Now the spread S in β consists of L and the q^{2} lines ⟨R_{i}⟩^{⊥} ∩ β, and every subspace ⟨R_{i}⟩^{⊥} contains the line T. So the spread S is in fact the projection from T onto β of the spaces ⟨L, T ⟩ \ T and ⟨R_{i}⟩^{⊥} \ T, i = 1, 2, …, q^{2}. If G ∈ F_{2} is a generator of Q^{+}(7, q) through L, then β ∩ G is the line L. It also holds that the spread S_{G} consists of L and the q^{2} lines ⟨R_{i}⟩^{⊥} ∩ G, so it is the projection from T onto G of ⟨L, T ⟩ \ T and ⟨R_{i}⟩^{⊥} \ T, i = 1, 2, …, q^{2}. Hence it is clear that S is a regular line spread of β if and only if S_{G} is a regular line spread in G. This holds for all generators G ∈ F_{2} of Q^{+}(7, q) containing the line L, so that the theorem follows. □

By now, we have quite extensive knowledge on what M^{θ} looks like if M ⊆ Q(6, q) is locally hermitian and semiclassical and θ is a T-correspondence of Ω. If M is a spread of an induced elliptic quadric Q^{−}(5, q) ⊆ Q(6, q), then Theorem 3.3 provides a criterion to decide whether M^{θ} is semiclassical or not. Also, M^{θ} spans the whole PG(7, q) whenever M is not hermitian. If M is a proper locally hermitian, semiclassical 1-system of Q(6, q), then M^{θ} is also locally hermitian and semiclassical. Moreover, it is not contained in a hyperplane of PG(7, q), provided that M is not a spread of some H(q) on Q(6, q). In particular, it then follows that M and M^{θ} are not isomorphic.

In the next section, we will have a look at M^{θ} in view of the results of [8], since a characterization of the proper locally hermitian, semiclassical 1-systems of Q^{+}(7, q) was obtained there.

4 M^{θ} compared to the known locally hermitian 1systems of Q^{+}(7, q)

It is the aim of this section to compare the 1-systems M^{θ}, with M and θ as in the foregoing section, to the other previously known proper locally hermitian 1-systems of Q^{+}(7, q), namely the ones that arise from an ovoid O of a hermitian polar space H(3, q^{2}). This means the following.

Consider the quadratic extension Q^{+}(7, q^{2}) of Q^{+}(7, q) and let γ and \overline{γ} be two disjoint generators of Q^{+}(7, q^{2}), which are conjugate with respect to the extension GF(q^{2}) of GF(q) and contain no point of PG(7, q). Denote by T the set of lines of Q^{+}(7, q), the extensions of which to GF(q^{2}) have a point in common with both γ and \overline{γ}. It is known that the set T constitutes a partition of the point set of Q^{+}(7, q) and that the intersection points of the extensions to GF(q^{2}) of the lines of T with γ, form a hermitian variety H(3, q^{2}) in γ, and similarly for \overline{γ}. Let O be an ovoid of H(3, q^{2}) ⊆ γ and let \overline{O} ⊆ \overline{γ} be its conjugate. Then by Shult and Thas [10], the set of lines M := \{x\overline{x} ∩ PG(7, q) | x ∈ O\} is a 1-system of Q^{+}(7, q).

The next theorem states under which conditions M^{θ}, with M and θ as in the previous section, was previously known.

Theorem 4.1 Suppose that M is a locally hermitian, semiclassical 1-system of a Q(6, q) ⊆ Q^{+}(7, q) and let θ be a T-correspondence of Ω. Then M^{θ} arises from an ovoid of H(3, q^{2}) if and only if M is a spread of an induced Q^{−}(5, q) ⊆ Q(6, q).

Proof. For this proof, let PG(7, q^{2}) be the extension of PG(7, q) to GF(q^{2}); the quadric Q^{+}(7, q) extends to a hyperbolic quadric Q^{+}(7, q^{2}).

First assume that M is a spread of an induced Q^{−}(5, q) ⊆ Q(6, q). Then the extension of Q^{−}(5, q) to GF(q^{2}) is a hyperbolic quadric Q^{+}(5, q^{2}) and it holds that ⟨Q^{+}(5, q^{2})⟩^{⊥} ∩ Q^{+}(7, q^{2}) is a secant line meeting Q^{+}(7, q^{2}) in two points x and \overline{x}, which are conjugate with respect to the extension GF(q^{2}) of GF(q). As every line of M, considered as a line over GF(q^{2}), lies in a totally singular plane of Q^{+}(7, q^{2}) through x, and in another one through \overline{x}, every line of M^{θ}, again considered over GF(q^{2}), has a point in common with the disjoint generators x^{θ} and \overline{x}^{θ} of Q^{+}(7, q^{2}). This implies that M^{θ} = \{y\overline{y} ∩ PG(7, q) | y ∈ O\} for some ovoid O of the hermitian variety H(3, q^{2}) which is defined in x^{θ} by the lines of Q^{+}(7, q), the extensions of which meet both x^{θ} and \overline{x}^{θ}.

Conversely, assume that there exist two generators G and \overline{G} of Q^{+}(7, q^{2}), which are disjoint and conjugate with respect to the extension GF(q^{2}) of GF(q). Further, suppose that M^{θ} = \{x\overline{x} ∩ PG(7, q) | x ∈ O\} for some ovoid O of the hermitian variety H(3, q^{2}), defined in G by the lines of Q^{+}(7, q), the extensions of which to GF(q^{2}) meet both G and \overline{G} in a point. There are two possibilities: either G and \overline{G} are elements of the first family of generators of Q^{+}(7, q^{2}), or they both belong to the second family of generators. If G and \overline{G} are elements of F_{1}, then G^{θ^{−1}} and \overline{G}^{θ^{−1}} are points of Q^{+}(7, q^{2}) \ Q^{+}(7, q). Moreover, every line of M^{θ} has a point in common with G and \overline{G}, and consequently every line of M, if considered over GF(q^{2}), lies in a totally singular plane of Q^{+}(7, q^{2}) through G^{θ^{−1}} and in another one containing \overline{G}^{θ^{−1}}. But this means that the line ⟨G^{θ^{−1}}, \overline{G}^{θ^{−1}}⟩ ∩ PG(7, q) is an external line of Q^{+}(7, q), such that all lines of M are contained in the elliptic quadric ⟨G^{θ^{−1}}, \overline{G}^{θ^{−1}}⟩^{⊥} ∩ Q^{+}(7, q) = Q^{−}(5, q). In the case where G, \overline{G} ∈ F_{2}, the spaces G^{θ^{−1}} and \overline{G}^{θ^{−1}} are generators of Q^{+}(7, q^{2}) of the first family. Consider an arbitrary line M ∈ M^{θ} and denote the common point of the extension of M to GF(q^{2}) and G by x. Then M^{θ^{−1}} is a line in the generator x^{θ^{−1}} ∈ F_{2}. But x is a point of G, which means that x^{θ^{−1}} ∩ G^{θ^{−1}} is a plane, so that the extension to GF(q^{2}) of M^{θ^{−1}} has a point y in common with G^{θ^{−1}}. Similarly, the extension to GF(q^{2}) of M^{θ^{−1}} also has the point \overline{y} in common with \overline{G}^{θ^{−1}}. As M ∈ M^{θ} was chosen arbitrarily, this implies that M too arises from an ovoid O′ of the hermitian polar space H(3, q^{2})′ in G^{θ^{−1}}, by the usual construction. On the other hand, M is by assumption contained in a hyperplane of PG(7, q), so ⟨O′⟩ cannot coincide with the 3-dimensional subspace G^{θ^{−1}}. Thus ⟨O′⟩ must be a plane in G^{θ^{−1}} and it follows that M is contained in the 5-dimensional subspace ⟨O′, \overline{O′}⟩, considered over GF(q). We conclude that M is a spread of the elliptic quadric ⟨O′, \overline{O′}⟩ ∩ Q^{+}(7, q) = Q^{−}(5, q). □

By the information of this theorem, it is clear that M^{θ} is not isomorphic to an example of a locally hermitian 1-system of Q^{+}(7, q) which was known before, provided that ⟨M⟩ = PG(6, q) and M is not a spread of a generalized hexagon H(q) on Q(6, q). Thus we have obtained an alternative description to the one in [8], of a subclass in the class of all proper locally hermitian, semiclassical 1systems of Q^{+}(7, q), namely the subclass of all such 1-systems of the form M^{θ}, with M and θ as above.

5 Isomorphism results for M^{θ}

Another question that appears naturally, is the following. In [6] and [7], it has been shown that there exist several orbits under the action of PΓO(7, q) in the set of the proper locally hermitian, semiclassical 1-systems of Q(6, q). It seems obvious in this context to wonder whether two non-isomorphic 1-systems M_{1} and M_{2} of Q(6, q) yield non-isomorphic 1-systems M_{1}^{θ} and M_{2}^{θ} of Q^{+}(7, q), with θ a T-correspondence as is usual in this chapter. It will turn out that this is indeed the case.

We start with two easy cases; the first one is a corollary of Theorem 4.1.

Corollary 5.1 Let θ be a T-correspondence of Ω. A spread M of an induced Q^{−}(5, q) ⊆ Q^{+}(7, q) is a hermitian spread of Q^{−}(5, q) if and only if M^{θ} is a hermitian spread of some Q^{−}(5, q)′ ⊆ Q^{+}(7, q).

Proof. M is a hermitian spread of a Q^{−}(5, q) on Q^{+}(7, q) if and only if there hold two things: firstly M must be a spread of a Q^{−}(5, q) and secondly M must arise from an ovoid O of H(3, q^{2}), which is in this case the classical ovoid O ≅ H(2, q^{2}). By applying Theorem 4.1 to θ, it follows from the first property that M^{θ} arises from some ovoid O′ of a hermitian variety H(3, q^{2})′. On the other hand, M = (M^{θ})^{θ^{−1}} arises from an ovoid of H(3, q^{2}), so that, again by Theorem 4.1 which we now apply to the T-correspondence θ^{−1}, M^{θ} is a 1-system of a Q^{−}(5, q)′ ⊆ Q^{+}(7, q). By combining both results, the statement of the corollary follows. □

Corollary 5.2 Let M_{1} and M_{2} be locally hermitian, semiclassical spreads of the hexagon H(q), embedded in a Q(6, q) ⊆ Q^{+}(7, q). Then M_{1} ≅ M_{2} under the action of PΓO^{+}(8, q) if and only if M_{1}^{θ} ≅ M_{2}^{θ} under the action of PΓO^{+}(8, q), with θ a T-correspondence of Ω.

Proof. This is an immediate consequence of Lemma 2.3, Corollary 5.1 and the fact that both for q odd and for q even, there exist at most two non-isomorphic locally hermitian, semiclassical spreads of H(q): the hermitian spread of a Q^{−}(5, q) and S_{[9]} if q is odd and q ≡ 1 mod 3, see [1], respectively S_{[δ]} if q is even and q = 2^{2e}, see [5]. □

The next theorem handles the general case.

Theorem 5.3 Let M_{1} and M_{2} be two proper locally hermitian, semiclassical 1-systems of a Q(6, q) ⊆ Q^{+}(7, q) and let θ be a T-correspondence of Ω. Then M_{1}^{θ} and M_{2}^{θ} are isomorphic under the action of PΓO^{+}(8, q) if and only if M_{1} and M_{2} are isomorphic under the action of PΓO^{+}(8, q).

Proof. In the proof, we will need the element μ_{p} ∈ PGO^{+}(8, q), see Hirschfeld and Thas [4, Lemma 22.6.3], with p the point ⟨Q(6, q)⟩^{⊥} and ⊥ the polarity of Q^{+}(7, q), where μ_{p} stabilizes the points of Q(6, q) and maps an arbitrary point r of Q^{+}(7, q) \ Q(6, q) onto the unique second common point of the line pr and the quadric Q^{+}(7, q). It can be shown that this map can be extended to an element of PGO^{+}(8, q), which is also denoted by μ_{p}. Let G_{1} ∈ F_{1} be a generator of Q^{+}(7, q). Then G_{1} ∩ Q(6, q) is a totally singular plane, say π. As π is fixed by μ_{p}, the image of G_{1} under μ_{p} must be a generator, different from G_{1}, through π. This generator is uniquely defined as the generator G_{2} through π which belongs to F_{2}. So G_{1}^{μ_{p}} = G_{2} and similarly G_{2}^{μ_{p}} = G_{1} and thus μ_{p} interchanges the families of generators F_{1} and F_{2}.

First assume that M_{1} ≅ M_{2} under the action of PΓO^{+}(8, q), so there exists an element α ∈ PΓO^{+}(8, q) such that M_{1}^{α} = M_{2}. If α interchanges F_{1} and F_{2}, then αμ_{p} ∈ PΓO^{+}(8, q) fixes both families of generators and it also maps M_{1} onto M_{2}. So we may assume that α preserves the families F_{1} and F_{2}. In that case we have that

(M_{1}^{θ})^{θ^{−1}αθ} = M_{1}^{αθ} = M_{2}^{θ},

so that θ^{−1}αθ maps M_{1}^{θ} onto M_{2}^{θ}. But, as α preserves the families of generators, the map θ^{−1}αθ stabilizes the sets P, L, F_{1} and F_{2}, preserves the incidence between these sets and stabilizes Q^{+}(7, q). Hence θ^{−1}αθ is an element of PΓO^{+}(8, q), which shows that M_{1}^{θ} and M_{2}^{θ} are also isomorphic under PΓO^{+}(8, q).

To prove the converse, assume that M_{1}^{θ} and M_{2}^{θ} are isomorphic under the action of PΓO^{+}(8, q), with isomorphism α ∈ PΓO^{+}(8, q). Then it holds that M_{1}^{θα} = M_{2}^{θ}, from which one derives that M_{1}^{θαθ^{−1}} = M_{2}. Here one should distinguish between the case where α preseves F_{1} and F_{2} and the case where it interchanges both families of generators. If α preseves the families F_{1} and F_{2}, one easily sees that θαθ^{−1} is itself an element of PΓL(8, q), stabilizing Q^{+}(7, q). So it immediately follows that M_{1} is isomorphic to M_{2} under PΓO^{+}(8, q), with isomorphism θαθ^{−1}. Suppose next that α interchanges F_{1} and F_{2}. It still holds that the map θαθ^{−1} maps M_{1} onto M_{2}, but now θαθ^{−1} is not a collineation. It is easily seen that θαθ^{−1} still maps lines onto lines, but unfortunately it interchanges points and generators of the first family, while preserving the second family of generators. Now the element μ_{p} ∈ PGO^{+}(8, q) comes into play, as we consider the map θαθ^{−1}μ_{p}. Clearly this map acts as follows on M_{1}:

M_{1}^{θαθ^{−1}μ_{p}} = M_{2}^{μ_{p}} = M_{2},

so θαθ^{−1}μ_{p} also maps M_{1} onto M_{2}. But by the fact that μ_{p} interchanges F_{1} and F_{2}, θαθ^{−1}μ_{p} maps points onto elements of F_{2}, elements of F_{2} onto elements of F_{1} and elements of F_{1} onto points. As it also preserves the incidence of Ω, θαθ^{−1}μ_{p} is a T-correspondence of Ω. Since M_{1} and M_{2} are 1-systems of a Q(6, q) ⊆ Q^{+}(7, q), it now follows from Theorem 2.4 that M_{1} and M_{2} are spreads of a generalized hexagon H(q) on Q(6, q). Consequently Corollary 5.2 yields that M_{1} ≅ M_{2} under PΓO^{+}(8, q). This proves the theorem. □

Theorem 5.3 learns how many orbits under the action of PΓO^{+}(8, q) there are in the set of all 1-systems of Q^{+}(7, q) of the form M^{θ}, with M a proper locally hermitian, semiclassical 1-system of Q(6, q) ⊆ Q^{+}(7, q) and θ a fixed T-correspondence or triality of the D_{4}-geometry Ω, attached to Q^{+}(7, q). This number of orbits equals the number of orbits under the action of PΓO(7, q) in the set of all proper locally hermitian, semiclassical 1-systems of Q(6, q). This number is known: for q odd, it equals the number of orbits of Aut(GF(q)) in the set of squares of GF(q) \ {0, 1}; see the remark following Theorem 7.2 of [6]. If q is even, it equals the number of orbits of Aut(GF(q)) in the set of all elements of GF(q) \ {0} with trace zero; see the remark following Theorem 6.2 in [7].

Acknowledgements

The first author is Research Assistant of the Fund for Scientific Research – Flanders (Belgium) (F.W.O.). The authors would like to thank Prof. G. Lunardon, Università di Napoli “Frederico II”, for suggesting to study the image of a locally hermitian, semiclassical 1-system of Q(6, q) under a triality of Ω.

References

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Addresses of the authors:

D. Luyckx (author to communicate with) Department of Pure Mathematics and Computer Algebra – Ghent University Galglaan 2 BE-9000 Gent, Belgium dluyckx@cage.rug.ac.be

J. A. Thas Department of Pure Mathematics and Computer Algebra – Ghent University Galglaan 2 BE-9000 Gent, Belgium jat@cage.rug.ac.be