The uniqueness of the 1-system of Q^{−}(7, q), q even

D. Luyckx^{∗} J. A. Thas

Ghent University, Dept. of Pure Mathematics and Computer Algebra,

Galglaan 2, B-9000 Ghent, Belgium

          dluyckx@cage.rug.ac.be, jat@cage.rug.ac.be

March 2003

Abstract

For q odd, it was shown in [3] that the elliptic quadric Q^{−}(7, q) possesses a unique 1-system, the so-called classical 1-system. Here, the same result will be obtained for even q.

1 Basic properties of 1-systems of Q^{−}(7, q)

A 1-system M of the elliptic quadric Q^{−}(7, q) is a set {L_{0}, L_{1}, …, L_{q^{4}}} of q^{4} + 1 lines of Q^{−}(7, q) with the property that every plane of Q^{−}(7, q) containing a line L_{i} of M has an empty intersection with (L_{0} ∪ L_{1} ∪ … ∪ L_{q^{4}}) \ L_{i}. We denote the union of all elements of M by M̃. Concerning the generators of Q^{−}(7, q), which are planes, the following result is shown by Shult and Thas in [5] in a more general context; here it is stated for 1-systems of Q^{−}(7, q) in particular.

Theorem 1.1 (Shult and Thas [5]) If M is a 1-system of the elliptic quadric Q^{−}(7, q), then every generator of Q^{−}(7, q) contains exactly q + 1 points of M̃.

In [2] a similar result for lines of Q^{−}(7, q) is obtained, as it is shown that every line of Q^{−}(7, q) has 0, 1, 2 or q + 1 points in common with M̃, where the latter occurs if and only if the line belongs to M. In combination with Theorem 1.1, this implies that every totally singular plane of Q^{−}(7, q) either contains a line of M, or a (q + 1)-arc of points of M̃.

Let M be an arbitrary 1-system of Q^{−}(7, q). If L_{1}, L_{2} and L_{3} are arbitrary lines of M, then ⟨L_{1}, L_{2}⟩ is 3-dimensional and it intersects Q^{−}(7, q) in a hyperbolic quadric Q^{+}(3, q). This hyperbolic quadric Q^{+}(3, q) contains no points of M̃, apart from the points on L_{1} and L_{2}, for otherwise there would exist a line of Q^{+}(3, q) meeting L_{1} and L_{2} and containing at least three points of M̃, a contradiction since such a line cannot be a line of M. Hence L_{3} has no point in common with ⟨L_{1}, L_{2}⟩ and it follows that ⟨L_{1}, L_{2}, L_{3}⟩ is 5-dimensional for every three distinct lines L_{1}, L_{2} and L_{3} of M.

∗The first author is Research Assistant of the Fund for Scientific Research – Flanders (Belgium) (F.W.O.).

Both for q even and for q odd, there is exactly one 1-system of Q^{−}(7, q) known, which is the unique 1-system of Q^{−}(7, q) if q is odd, see [3]. It is called the classical 1-system of Q^{−}(7, q) and can be constructed as follows. In the extension PG(7, q^{2}) of PG(7, q) = ⟨Q^{−}(7, q)⟩, there exist two disjoint 3-spaces γ and \overline{γ}, which are conjugate with respect to the extension GF(q^{2}) of GF(q), polar with respect to the polarity defined by the extension Q^{+}(7, q^{2}) of Q^{−}(7, q), and such that γ ∩ Q^{+}(7, q^{2}) is an elliptic quadric Q^{−}(3, q^{2}). The classical 1-system M of Q^{−}(7, q) then consists of all lines x\overline{x}∩PG(7, q), where x varies on Q^{−}(3, q^{2}) and \overline{x} is its conjugate, so is a point of \overline{γ} ∩ Q^{+}(7, q^{2}). In Shult and Thas [5], one can find a proof that the set of lines M = \{x\overline{x} ∩ PG(7, q) | x ∈ Q^{−}(3, q^{2})\} is indeed a 1-system of Q^{−}(7, q).

From now on it is assumed that q is even and it will be shown that also in this case, the classical 1-system is the unique 1-system of Q^{−}(7, q).

2 A 1-system of Q^{−}(7, q) is an egg of PG(7, q)

In this section, it will be shown that every 1-system of Q^{−}(7, q), q even, is an egg of the ambient space PG(7, q) of Q^{−}(7, q), so we start with the general definition of an egg O(n, 2n, q). An egg O(n, 2n, q) of PG(4n−1, q) is a set of q^{2n}+1 (n−1)-dimensional subspaces π_{0}, π_{1}, …, π_{q^{2n}} of PG(4n−1, q), every three of which generate a PG(3n−1, q) and such that each element π_{i} of O(n, 2n, q) is contained in a PG^{(i)}(3n − 1, q), having no point in common with (π_{0} ∪ π_{1} ∪ ⋯ ∪ π_{q^{2n}}) \ π_{i}. The space PG^{(i)}(3n − 1, q) is called the tangent space of O(n, 2n, q) at π_{i}. An egg O(n, 2n, q) is called good at an element π_{i} if for all distinct π_{j}, π_{k}, j ≠ i ≠ k, the space generated by π_{i}, π_{j} and π_{k} contains exactly q^{n} + 1 elements of O(n, 2n, q). An egg O(n, 2n, q) is called regular if it is constructed in the following way.

Consider the algebraic extension GF(q^{n}) of GF(q) and the corresponding extension PG(4n − 1, q^{n}) of PG(4n − 1, q). Consider n 3-dimensional subspaces PG^{(1)}(3, q^{n}), PG^{(2)}(3, q^{n}), …, PG^{(n)}(3, q^{n}) of PG(4n − 1, q^{n}), which generate the space PG(4n − 1, q^{n}) and constitute a conjugate n-tuple with respect to the extension GF(q^{n}) of GF(q). Let O be an ovoid of PG^{(1)}(3, q^{n}). With every point p^{(1)} of O, there correspond n − 1 points p^{(2)}, p^{(3)}, …, p^{(n)} such that the points p^{(1)}, p^{(2)}, …, p^{(n)} constitute a conjugate n-tuple with respect to the extension GF(q^{n}) of GF(q). The points p^{(1)}, p^{(2)}, …, p^{(n)} define an (n−1)-dimensional subspace of PG(4n − 1, q). If we let p^{(1)} vary in O, we obtain q^{2n} + 1 such (n − 1)- dimensional subspaces of PG(4n − 1, q), which form a regular egg O(n, 2n, q). Regular eggs are good at each of their elements. In [6], Thas shows that the converse also holds.

Theorem 2.1 (Thas [6]) Every egg O(n, 2n, q) of PG(4n − 1, q) which is good at each of its elements, is regular.

In the special case where the ovoid O of PG^{(1)}(3, q^{n}) is an elliptic quadric Q^{−}(3, q^{n}), the egg is said to be classical.

Concerning 1-systems of Q^{−}(7, q), with q even, a first result is obtained in the following lemma.

Lemma 2.2 Let π be a totally singular plane of Q^{−}(7, q), containing a (q + 1)- arc O of points of M̃. If x ∈ O is arbitrary and M ∈ M is the line of M containing x, then for any point z ∈ π \ O, the plane ⟨z, M⟩ is totally singular if and only if z lies on the tangent line of O at x.

Proof. If the plane ⟨z, M⟩ is totally singular and xz is a secant line of O, then the generator ⟨z, M⟩ of Q^{−}(7, q) contains the line M of M and a point of M̃ not on M, namely the point (O ∩ xz) \ {x}. This contradicts the definition of a 1-system of Q^{−}(7, q) and it follows that xz must be the tangent line of O at x.

To prove the converse, consider an arbitrary point y of M \ {x}. Then y^{⊥} intersects π in some line K through x. If K were a secant of O, then the totally singular plane ⟨y, K⟩ would again contain the line M ∈ M, and some other point of M̃, not on M, a contradiction. Hence K must be the unique tangent of O at x and one concludes that the plane ⟨z, M⟩ is totally singular for every point z of K \ {x}. □

In Shult and Thas [5] it is proved that a 1-system of Q^{−}(7, q) has two intersection numbers with respect to hyperplanes. In particular, a hyperplane H of PG(7, q) contains exactly one line of M if and only if it is the tangent hyperplane of Q^{−}(7, q) at some point of M̃. Otherwise, H contains exactly q^{2} + 1 lines of M. In the next lemma, it is stated that whenever H is the tangent hyperplane of Q^{−}(7, q) at a point p ∉ M̃, then the q^{2} + 1 lines of M in H lie in fact in a 5-dimensional subspace of PG(7, q).

Lemma 2.3 For every point p of Q^{−}(7, q) \ M̃, the q^{2} + 1 lines of M in p^{⊥} are contained in a 5-dimensional subspace of PG(7, q).

Proof. Consider a totally singular plane π of Q^{−}(7, q) through the point p, and such that π meets M̃ in a (q + 1)-arc O. As q is even, the (q + 1)-arc O has a nucleus n, which is the intersection of all tangent lines of O. If p = n, then p lies on all q +1 tangents of O and by the previous lemma, it follows that ⟨p, M⟩ is a totally singular plane for all lines M ∈ M containing a point of O. Hence the q +1 lines of M which intersect π in a point, are contained in p^{⊥}. If p is not the nucleus n of O, then p lies on exactly one tangent line of O, say at the point x ∈ O. Again by the previous lemma, the line M ∈ M through x is the unique line of M intersecting π with the property that ⟨p, M⟩ is a totally singular plane. It follows that this line M is the only line of M containing a point of O, which is completely contained in p^{⊥}.

Let γ ⊆ p^{⊥} be a 5-dimensional subspace not through p, so that γ ∩ Q^{−}(7, q) is an elliptic quadric Q^{−}(5, q). We project the q^{2} + 1 lines of M in p^{⊥} from p onto γ; this yields a set M_{p} of q^{2} +1 mutually disjoint lines of Q^{−}(5, q). Starting from this set M_{p}, one can define a substructure S = (P, B, I) of the generalized quadrangle Q^{−}(5, q) as follows. The point set P is the set of all points on the lines of M_{p}, the line set B consists of all lines of Q^{−}(5, q) consisting entirely of points of P, and incidence I is the incidence of Q^{−}(5, q). We will show that S is a subquadrangle of Q^{−}(5, q). To that end, consider an arbitrary line L of Q^{−}(5, q), which does not belong to M_{p}. Then ⟨p, L⟩ is a totally singular plane of Q^{−}(7, q) containing a (q + 1)-arc O of points of M̃. By the previous paragraph it follows that either exactly one, or all q + 1 points of L lie on a line of M_{p}, so either L is a line of B, or it meets P in a unique point. Since this holds for all lines L of Q^{−}(5, q) not in M_{p}, it follows that every line of Q^{−}(5, q) containing at least two distinct points of P belongs to B. This implies that the substructure S = (P, B, I) satisfies the conditions of [4, Theorem 2.3.1] and since the lines of B are pairwise disjoint, one concludes from [4, Theorem 2.3.1] that S is a subquadrangle of Q^{−}(5, q). Moreover, the lines of M_{p} partition the point set of this subquadrangle and hence M_{p} is a spread of S, so that |P| = (q + 1)(q^{2} + 1). Consequently, S has order (q, q) and thus it is a generalized quadrangle Q(4, q). It readily follows that the q^{2} + 1 lines of M in p^{⊥} all lie in a 5-dimensional subspace ε, where ε ∩ Q^{−}(7, q) is a cone pQ(4, q) with p as vertex and a parabolic quadric Q(4, q) as base. □

We are now ready to show the main result of this section.

Theorem 2.4 If q is even, then every 1-system of Q^{−}(7, q) is an egg O(2, 4, q) of PG(7, q), which is good at each of its elements.

Proof. By definition, all q^{4} + 1 lines of a 1-system M of Q^{−}(7, q) are pairwise disjoint and in Section 1 it has been explained that every three distinct elements of M span a PG(5, q). Moreover, the quadric Q^{−}(7, q) has a unique 5-dimensional tangent space at each line L of M, and this tangent space is disjoint from all lines of M \ {L} by definition. Hence the lines of M form an egg O(2, 4, q) of PG(7, q) and it only remains to show that M, considered as an egg O(2, 4, q), is good at each of its elements.

On Q^{−}(7, q), there are (q^{4} +1)q^{2} points p which do not belong to M̃. By the previous lemma, it holds for every such point p that the q^{2} + 1 lines of M in p^{⊥} span a 5-dimensional subspace of PG(7, q), which intersects Q^{−}(7, q) in a cone pQ(4, q); also, for any three such lines L_{i}, L_{j}, L_{k}, we have that ⟨L_{i}, L_{j}, L_{k}⟩^{⊥} ∩ Q^{−}(7, q) = {p}, and so p is uniquely defined by {L_{i}, L_{j}, L_{k}}. It follows that there exist

\frac{q^{2}(q^{4} + 1) · (q^{2} + 1)q^{2}(q^{2} − 1)}{3 · 2 · 1} = \frac{1}{6}(q^{4} + 1)q^{4}(q^{4} − 1)

triples (L_{i}, L_{j}, L_{k}) of distinct lines of M with the property that ⟨L_{i}, L_{j}, L_{k}⟩ ∩ Q^{−}(7, q) is some cone pQ(4, q) containing exactly q^{2} + 1 lines of M. On the other hand, the fact that |M| = q^{4} + 1 implies that \frac{1}{6}(q^{4} + 1)q^{4}(q^{4} − 1) is also the total number of triples (L_{i}, L_{j}, L_{k}) consisting of three distinct lines of M. Hence it holds for every triple of distinct lines L_{i}, L_{j}, L_{k} of M that ⟨L_{i}, L_{j}, L_{k}⟩ is a 5-dimensional subspace of PG(7, q) which contains exactly q^{2} + 1 lines of M. So the 1-system M, considered as an egg O(2, 4, q) of PG(7, q), is good at each of its elements. □

As has been mentioned in Theorem 2.1, every egg which is good at each of its elements, is regular. This implies that there exist two disjoint and conjugate 3-dimensional subspaces ρ and \overline{ρ} in the extension PG(7, q^{2}) of PG(7, q) to GF(q^{2}), and ovoids O ⊆ ρ, respectively \overline{O} ⊆ \overline{ρ}, such that M = \{x\overline{x} ∩ PG(7, q) | x ∈ O\}. In the next section, it will be deduced from this property that M is classical.

3 The uniqueness result

Using the same notation as above, there remain two more properties to show. Firstly, it must hold that the ovoid O is the elliptic quadric Q^{−}(3, q^{2}) = ρ ∩ Q^{+}(7, q^{2}), and similarly for its conjugate \overline{O}. Secondly, one must prove that \overline{ρ} = ρ^{⊥}, so the conjugate \overline{ρ} of ρ must coincide with the image ρ^{⊥} of ρ under the polarity ⊥ of Q^{+}(7, q^{2}). These properties will be shown in the following theorem.

Theorem 3.1 If q is even, the elliptic quadric Q^{−}(7, q) has a unique 1-system up to a projectivity.

Proof. Let M be a 1-system of Q^{−}(7, q), q even. Then by Theorem 2.4, M is a regular egg of the ambient space PG(7, q) of Q^{−}(7, q). Denote the extensions to GF(q^{2}) of PG(7, q) and Q^{−}(7, q) by PG(7, q^{2}), respectively Q^{+}(7, q^{2}). In PG(7, q^{2}) there exist two disjoint and conjugate 3-spaces ρ and \overline{ρ} such that M = \{x\overline{x}∩PG(7, q) | x ∈ O}, for some ovoid O ⊆ ρ.

Since all extensions of lines of M to GF(q^{2}) are lines of Q^{+}(7, q^{2}), the ovoid O must be contained in ρ∩Q^{+}(7, q^{2}). Moreover O is an ovoid of ρ, which implies that a totally singular line, respectively plane, of ρ ∩ Q^{+}(7, q^{2}) may contain at most 2, respectively q+2 points of O. These observations exclude the possibilities that ρ ∩ Q^{+}(7, q^{2}) is a hyperbolic quadric Q^{+}(3, q^{2}), or a cone xQ(2, q^{2}), or a cone LQ^{+}(1, q^{2}) for some point x or some line L of Q^{+}(7, q^{2}); also ρ ∩ Q^{+}(7, q^{2}) cannot be of type LQ^{−}(3, q^{2}) nor can be a unique plane. Furthermore, one easily counts that the number of generators of Q^{+}(7, q^{2}) which intersect Q^{−}(7, q) in a plane equals 2(q^{2} + 1)(q^{3} + 1)(q^{4} + 1), that every generator of Q^{+}(7, q^{2}) which contains a line of Q^{−}(7, q) also contains a plane of Q^{−}(7, q), and that there exist 2(q^{2} + 1)(q^{4} + 1)q^{3}(q^{3} − 1) generators of Q^{+}(7, q^{2}) having exactly one point in common with Q^{−}(7, q). As 2(q^{2} + 1)(q^{3} + 1)(q^{4} + 1) + 2(q^{2} + 1)(q^{4} + 1)q^{3}(q^{3} − 1) equals the total number of generators of Q^{+}(7, q^{2}), it follows that ρ must contain at least one point of Q^{−}(7, q) if it is a generator of Q^{+}(7, q^{2}). But this contradicts the fact that ρ∩\overline{ρ} = ∅ and consequently ρ∩Q^{+}(7, q^{2}) must be an elliptic quadric Q^{−}(3, q^{2}), which thus coincides with the ovoid O.

We proceed to show that \overline{ρ} = ρ^{⊥}. First, let x be an arbitrary point of O. Then the image x^{⊥} of x with respect to the polarity of Q^{+}(7, q^{2}) is a 6dimensional subspace containing L^{⊥}, with L the line x\overline{x} ∩ PG(7, q) of M. But L^{⊥} ∩ PG(7, q) is the tangent space T_{L}(Q^{−}(7, q)) of Q^{−}(7, q) at L, so that L^{⊥} ∩ Q^{−}(7, q) = LQ^{−}(3, q). If x^{⊥} contained a point of Q^{−}(7, q) \ L^{⊥}, then it would contain a 6-dimensional subspace of PG(7, q), a contradiction. So x^{⊥} intersects Q^{−}(7, q) exactly in LQ^{−}(3, q). If y is an arbitrary point of O \ {x}, then x^{⊥} and the line y\overline{y} have a point in common, say p, which is a point of Q^{+}(7, q^{2})\Q^{−}(7, q) by the above observation. Let C be any conic on O containing x and y. Now the number of points on y\overline{y} which do not belong to PG(7, q) equals q^{2} − q, while C \ {y} has q^{2} points. Hence there must exist distinct points u, u′ on C \ {y} such that u^{⊥} ∩ y\overline{y} = u′^{⊥} ∩ y\overline{y} = \{w\}. But then the point w is a point of u^{⊥} ∩ y^{⊥} ∩ u′^{⊥} = ⟨C⟩^{⊥}, and so z^{⊥} ∩ y\overline{y} = \{w\}(= \{p\}) for any point z on C \ \{y\}. Since C is any conic on O containing x and y, it follows that p is a point of ρ^{⊥} and consequently ρ^{⊥} has a point in common with all lines y\overline{y}, y ∈ O.

Suppose next that \overline{ρ} ∩ ρ^{⊥} is at least a point r. If \overline{ρ} ≠ ρ^{⊥}, then these two 3-spaces have at most a plane in common and ⟨\overline{ρ}, ρ^{⊥}⟩ is at most 6-dimensional. But both \overline{ρ} and ρ^{⊥} contain a point of each line x\overline{x}, x ∈ O, and as \overline{ρ} ∩ ρ^{⊥} is at most a plane, this implies that ⟨\overline{ρ}, ρ^{⊥}⟩ ∩ PG(7, q), so also any hyperplane of PG(7, q) through ⟨\overline{ρ}, ρ^{⊥}⟩ ∩ PG(7, q), contains at least q^{4} − q^{2} lines of M. This is a contradiction to the fact that a hyperplane of PG(7, q) contains either 1 or q^{2} + 1 lines of M and one concludes that \overline{ρ} ∩ ρ^{⊥} is either empty, or \overline{ρ} and ρ^{⊥} coincide.

Finally, assume that \overline{ρ} ∩ ρ^{⊥} is empty. Then there exist 3 pairwise disjoint 3dimensional subspaces ρ, \overline{ρ} and ρ^{⊥}, having a point in common with the extensions to GF(q^{2}) of all lines of M. By Theorem 25.6.1 of [1], this implies that all lines x\overline{x}, x ∈ O, are elements of a system of maximal spaces of a Segre variety 𝒮_{1;3} in PG(7, q^{2}). On the other hand, the lines of M are lines of PG(7, q), and the restriction of 𝒮_{1;3} to PG(7, q) is a Segre variety 𝒮′_{1;3} of PG(7, q). This is nevertheless impossible because the system of maximal lines of the Segre variety 𝒮′_{1;3} in PG(7, q) contains q^{3} + q^{2} + q + 1 lines and hence cannot contain all q^{4} + 1 lines of M. It follows that \overline{ρ} and ρ^{⊥} are not disjoint either. Consequently, \overline{ρ} = ρ^{⊥} and M is the classical 1-system of Q^{−}(7, q). □

References

[1] J. W. P. Hirschfeld and J. A. Thas. General Galois Geometries. Oxford University Press, Oxford, 1991, 407pp.

[2] D. Luyckx. m-systems of polar spaces and SPG reguli. Bull. Belg. Math. Soc., 9(2):177–183, 2002.

[3] D. Luyckx and J. A. Thas. The Uniqueness of the 1-System of Q^{−}(7, q), q Odd. J. Combin. Theory Ser. A, 98(2):253–267, 2002.

[4] S. E. Payne and J. A. Thas. Finite Generalized Quadrangles. Pitman, London, 1984, 312pp.

[5] E. E. Shult and J. A. Thas. m-Systems of Polar Spaces. J. Combin. Theory Ser. A, 68(1):184–204, 1994.

[6] J. A. Thas. Geometric characterization of the [n−1]-ovaloids of the projective space PG(4n − 1, q). Simon Stevin, 47:97–106, 1973/74.