Affine vector space partitions and spreads of quadrics

An affine spread is a set of subspaces of $\mathrm{AG}(n, q)$ of the same dimension that partitions the points of $\mathrm{AG}(n, q)$. Equivalently, an {\em affine spread} is a set of projective subspaces of $\mathrm{PG}(n, q)$ of the same dimension which partitions the points of $\mathrm{PG}(n, q) \setminus H_{\infty}$; here $H_{\infty}$ denotes the hyperplane at infinity of the projective closure of $\mathrm{AG}(n, q)$. Let $\mathcal{Q}$ be a non degenerate quadric of $H_\infty$ and let $\Pi$ be a generator of $\mathcal{Q}$, where $\Pi$ is a $t$-dimensional projective subspace. An affine spread $\mathcal{P}$ consisting of $(t+1)$-dimensional projective subspaces of $\mathrm{PG}(n, q)$ is called hyperbolic, parabolic or elliptic (according as $\mathcal{Q}$ is hyperbolic, parabolic or elliptic) if the following hold: each member of $\mathcal{P}$ meets $H_\infty$ in a distinct generator of $\mathcal{Q}$ disjoint from $\Pi$; elements of $\mathcal{P}$ have at most one point in common; if $S, T \in \mathcal{P}$, $|S \cap T| = 1$, then $\langle S, T \rangle \cap \mathcal{Q}$ is a hyperbolic quadric of $\mathcal{Q}$. In this note it is shown that a hyperbolic, parabolic or elliptic affine spread of $\mathrm{PG}(n, q)$ is equivalent to a spread of $\mathcal{Q}^+(n+1, q)$, $\mathcal{Q}(n+1, q)$ or $\mathcal{Q}^-(n+1, q)$, respectively.


Introduction
Establishing the existence of a partition of a geometric structure into subspaces of the same dimension is a classical theme in finite geometry.A vector space partition of the projective space PG(n, q) is a set of projective subspaces of PG(n, q) which partitions the points of PG(n, q).A vector space partition whose subspaces are of the same dimension is called spread.It was shown by J. André [1] and B. Segre [13] that PG(n, q) admits a spread consisting of t-dimensional projective subspaces if and only if t + 1 divides n + 1.A finite classical polar space arises from a vector space of finite dimension over a finite field equipped with a non-degenerate reflexive sesquilinear or quadratic form.A non-degenerate polar space of PG(n, q) is a member of one of the following classes: a symplectic space W(n, q), n odd, a parabolic quadric Q(n, q), n even, a hyperbolic quadric Q + (n, q), n odd, an elliptic quadric Q − (n, q), n odd, or a Hermitian variety H(n, q), q a square.A spread of a polar space A is a set of generators (i.e., maximal totally S. Gupta: Department of Mathematics and Applications "R.Caccioppoli", University of Naples "Federico II", Via Cintia, Monte S. Angelo, I-80126 Naples, Italy; e-mail: somi.gupta@unina.it.
isotropic subspaces or maximal totally singular subspaces) of A, which partitions the points of A. A spread of W(n, q) is also a spread of PG(n, q) into n−1

2
-dimensional projective subspaces.It is easily seen that Q + (n, q) has no spread if n ≡ 1 (mod 4).Many authors investigated spreads of polar spaces, see [3,5,9,10,11,12,14,15].For q even, Q + (n, q), n ≡ −1 (mod 4), Q(n, q), Q − (n, q) always have a spread.For q odd, with q a prime or q ≡ 0 or 2 (mod 3), Q + (7, q) and Q(6, q) have a spread.The parabolic quadric Q(n, q), with n ≡ 0 (mod 4) and q odd, has no spread.For q odd, Q + (3, q) and Q − (5, q) have a spread.The Hermitian varieties H(n, q), n odd and H(4, 4) do not have a spread.For open problems related to spreads of polar spaces, the reader is referred to [8,Section 7.5] In this context it is natural to consider an "affine version" of a vector space partition as a set of proper affine subspaces that partitions the points of AG(n, q), see [2].Denote by H ∞ the hyperplane at infinity of the projective closure of AG(n, q).Then an affine vector space partition of PG(n, q) is a set of projective subspaces of PG(n, q) which partitions the points of PG(n, q) \ H ∞ .An affine spread is an affine vector space partition whose subspaces are of the same dimension.Let Q be a non degenerate quadric of H ∞ and let Π be a generator of Q, where Π is a t-dimensional projective subspace.Here, we are concerned with affine spreads P consisting of (t + 1)-dimensional projective subspaces of PG(n, q) such that • each member of P meets H ∞ in a distinct generator of Q disjoint from Π; • elements of P have at most one point in common; An affine spread P of PG(n, q) satisfying the above properties is called hyperbolic, parabolic or elliptic, according as Q is hyperbolic, parabolic or elliptic, respectively.In [2, Section 5.2] the authors exibhited a particular hyperbolic affine spread of PG(6, q), q even or q ∈ {3, 5}, and they conjecture that a hyperbolic affine spread of PG(6, q) exists for all prime powers [2, Conjecture 2].In this note it is shown that a hyperbolic, parabolic or elliptic affine spread of PG(n, q) is equivalent to a spread of Q + (n + 1, q), Q(n + 1, q) or Q − (n + 1, q), respectively.

Affine vector space partitions and quadrics
Throughout the paper we will use the term s-space to denote an s-dimensional projective subspace of the ambient projective space.Let Q r,e denote a non-degenerate quadric of PG(r, q) as indicated below: where r is odd if the quadric is hyperbolic or elliptic, whereas r is even if the quadric is parabolic.Associated with Q r,e , there is a polarity ⊥ of PG(r, q), which is non-degenerate except when e = 1 and q is even.In particular, the polarity ⊥ is symplectic if Q r,e ∈ {Q + (r, q), Q − (r, q)}, q even, and orthogonal if Q r,e ∈ {Q(r, q), Q + (r, q), Q − (r, q)}, q odd.For Q r,1 = Q(r, q), q even, the polarity ⊥ is degenerate, indeed N ⊥ = PG(r, q), if N is the nucleus of Q(r, q), whereas P ⊥ is a hyperplane of PG(r, q) for any other point P of PG(r, q).A generator of Q r,e is a projective space of maximal dimension contained in Q r,e and generators of Q r,e are r−e−1

2
-spaces.A spread of Q r,e is a set of q r+e−1 2 + 1 pairwise disjoint generators of Q r,e .More background information on the properties of the finite classical polar spaces can be found in [6,7,8].
Definition 2.1.Let H be a hyperplane of PG(r + 1, q).An affine vector space partition (or abbreviated as avsp) of PG(r + 1, q) is a set P of subspaces of PG(r + 1, q) whose members are not contained in H and such that every point of PG(r + 1, q) \ H is contained in exactly one element of P.
The type of an avsp P of PG(r + 1, q) is given by (r + 1) m r+1 . . . 2 m 2 1 m 1 , where m i denotes the number of (i − 1)-spaces of P for 1 ≤ i ≤ r + 1. Definition 2.2.Let P be an avsp of PG(r + 1, q).Then P is said to be reducible if there exists a proper subspace U of PG(r + 1, q) such that the members of P contained in U form an avsp of U .If P is not reducible, then it is said to be irreducible.Definition 2.3.Let P be an avsp of PG(r + 1, q).Then P is said to be tight if no point of PG(r + 1, q) belongs to each of the members of P.
In [2, Section 5.2] the authors studied a particular avsp {S 1 , . . ., S q 3 } of PG(6, q), q even, of type 4 q 3 , such that the set {S i ∩ H | i = 1, . . ., q 3 } consists of the q 3 planes of a Klein quadric Q + (5, q) that are disjoint from a fixed plane of Q + (5, q).Moreover, they conjecture that a similar construction can be realized for all prime powers [2, Conjecture 2].Motivated by their example, we introduce the definition of hyperbolic, parabolic, or elliptic avsp.Definition 2.4.Let P = S 1 , S 2 , . . ., S q r+e+1 2 be an avsp of PG(r+1, q) of type r−e+3 2 q r+e+1 2 .Then P is said hyperbolic, parabolic or elliptic if the set generators of a quadric Q r,e ⊂ H, where e = 0, 1, 2, respectively, such that 1. elements of G are disjoint from a fixed generator Π of Q r,e ; 2. distinct members of G (or of P) meet at most in one point; Remark 2.5.Let r be odd, let P = S 1 , S 2 , . . ., S q r+1 2 be a hyperbolic avsp of PG(r+1, q) and let Π i = S i ∩H, i = 1, . . ., q r+1 2 .Since members of P are r+1 2 -spaces, then S i , S j = PG(r+1, q) and Therefore for the definition of a hyperbolic avsp 2) is equivalent to require that distinct members of G pairwise intersect in one point, whereas requirement 3) is redundant.
In the following we will need the next result, that in the hyperbolic case has been proved in [4,

+1
be a spread of a quadric Q r+2,e of PG(r + 2, q).Fix a

+1
and an r-space H ⊂ P ⊥ such that P / ∈ H. Set Q r,e = H ∩ Q r+2,e and .
The following hold.
Proof.Since |(Σ i ∩P ⊥ )∩(Σ j ∩P ⊥ )| = 0, it follows that Π i ∩Π j is at most one point.Assume that e = 0, i.e., Q r+2,0 is hyperbolic and r ≡ 1 (mod 4).Consider the r+1 2 -space T i = P, Σ i ∩ P ⊥ .Then T i is a generator of Q r+2,0 intersecting Σ i in an r−1 2 -space.Therefore T i and Σ j , i = j, are in different systems of generators and hence they have at least one point in common.It follows that Π i and Π j , i = j, have at least one point in common.
Finally, it is easily observed that any line passing through P meets precisely q elements of the spread distinct from Σ q r+e+1 2

+1
there pass precisely q members of G.

+1
be a spread of a quadric Q r+2,e of PG(r + 2, q).

+1
and a hyperplane U ∼ = PG(r + 1, q) not containing the point P .Then , and let P = S 1 , . . ., S . Then P consists of q r+e+1 2 r−e+1 2 -spaces of U , none of them is contained in H.
Proposition 2.10.The set P is a hyperbolic, parabolic or elliptic avsp of PG(r + 1, q) according as Q r+2,e is hyperbolic, parabolic or elliptic, respectively.
Proof.In order to prove that P is an avsp of PG(r + 1, q) it is enough to show that for every point of U \ H, there exists a member of P containing it.Let R be a point of U \ H and let ℓ be the line joining P and R. Since ℓ passes through P and is not contained in P ⊥ , it is secant to Q r+2,e .Therefore there is a point, say R ′ , distinct from P , such that R ′ ∈ Q r+2,e .Moreover R ′ ∈ Σ j , for some j ∈ 1, . . ., q r+e+1 2 and hence R ∈ S j , by construction.
. Then G consists of generators of Q r,e disjoint from Σ is hyperbolic, or is at most one, otherwise.
Let S i , S j ∈ P, with |S i ∩ S j | = 1.Then S i , S j , P = Σ i , Σ j is an (r + 2 − e)-space of PG(r+2, q) containing two disjoint generators of Q r+2,e .Hence S i , S j , P ∩Q r+2,e = Q r+2−e,0 ≃ Q + (r + 2 − e, q).Such a quadric Q r+2−e,0 meets P ⊥ in a cone having as vertex the point P and as base a Q r−e,0 ≃ Q + (r − e, q).It follows that S i , S j ∩ Q r,e = Q + (r − e, q), as required.
We have seen that a spread of a hyperbolic, elliptic, or parabolic quadric of PG(r + 2, q) gives rise to a hyperbolic, elliptic, or parabolic avsp of PG(r + 1, q), respectively.The converse also holds true, as shown below.
Theorem 2.11.If P is a hyperbolic, parabolic or elliptic avsp of PG(r + 1, q), then there is a spread of Q r+2,e , where Q r+2,e is a hyperbolic, parabolic or elliptic quadric, respectively.
Then there exists a hyperplane H of PG(r + 1, q), a quadric Q r,e of H, that is hyperbolic, parabolic or elliptic, respectively, and a fixed generator Π of Q r,e , such that Π i = S i ∩ H is a generator of Q r,e disjoint from Π. Embed PG(r + 1, q) as a hyperplane section, say U , of PG(r + 2, q) and fix a quadric Q r+2,e of PG(r + 2, q) in such a way that H ∩ Q r+2,e = Q r,e .Let P be one of the two points of Q r+2,e on H ⊥ , where ⊥ is the polarity of PG(r + 2, q) associated with Q r+2,e .For i = 1, . . ., q r+e+1 2 , consider the following r−e+3 2 -space Hence F i meets P ⊥ in the r−e+1

2
-space spanned by P and Π i , that is a generator of Q r+2,e .We claim that F i contains a further generator of Q r+2,e besides P, Π i .Indeed, if P, Π i were the unique generator of Q r+2,e contained in F i , then

+1
the generator of Q r+2,e spanned by P and Π.We claim that

+1
is a spread of Q r+2,q .Assume by contradiction that there is a point

+1
, for some i ∈ 1, . . ., q . Then necessarily Σ i ∩ Σ j is a point, otherwise |S i ∩ S j | > 1.Moreover, we infer that such a point, say R ′ = Σ i ∩ Σ j , must be in P ⊥ .Indeed, if R ′ / ∈ P ⊥ , then the point R = P, R ′ ∩ U belongs to both S i \ H and S j \ H, contradicting the fact that S 1 , . . ., S q r+e+1 2 is an avsp.In particular, |S i ∩ S j | = 1.
It follows that if |Σ i ∩ Σ j | > 0, then they have at least a line in common, a contradiction.If e ∈ {1, 2}, observe that F i , F j is a PG(r + 2 − e, q) containing the cone having as vertex the point P and as base the hyperbolic quadric Q + (r − e, q) = S i , S j ∩ Q r,e .Furthermore, two more generators of Q r+2,e , namely Σ i , Σ j , are contained in F i , F j and do not pass through P .Therefore necessarily we have that F i , F j ∩Q r+2,e = Q r+2−e,0 .Let us denote by Q + (r+2−e, q) the hyperbolic quadric F i , F j ∩Q r+2,e , so that generators of Q + (r+2−e, q) are r−e+1 2 -spaces.Since Σ i ∩ P, Π i and Σ j ∩ P, Π j are r−e−1

2
-spaces, we have that Σ i and P, Π i lie in distinct systems of generators of Q + (r+2−e).Similarly for Σ j and P, Π j .Two possibilities arise: either r + 2 − e ≡ −1 (mod 4) or r + 2 − e ≡ 1 (mod 4).Since P, Π i ∩ P, Π j is a line, if the former case occurs, then P, Π i , P, Π j belong to the same system of generators of Q + (r + 2 − e, q).Hence Σ i , Σ j belong to the same system of generators of Q + (r + 2 − e, q) and if |Σ i ∩ Σ j | > 0, then they have at least a line in common, a contradiction.In the latter case, P, Π i , P, Π j are in different systems of generators of Q + (r + 2 − e, q).Therefore Σ i , P, Π j belong to the same system of generators of Q + (r + 2 − e, q).Similarly for Σ j , P, Π i .It follows that Σ i , Σ j belong to different systems of generators of Q + (r + 2 − e, q) and again, if |Σ i ∩ Σ j | > 0, then they have at least a line in common, a contradiction.
Proof.Let P = S 1 , S 2 , . . ., S q r+e+1 2 be a hyperbolic, parabolic or elliptic avsp of PG(r + 1, q).If e = 0, in order to prove that P is irreducible, it is enough to observe that, if i = j, then the span of S i and S j is the whole PG(r + 1, q).Similarly, if e = 1 and |S i ∩ S j | = 0, then S i , S j = PG(r + 1, q).If either e = 1 and |S i ∩ S j | = 1 or e = 2, we claim that the elements of P contained in S i , S j , where i = j, do not cover all the points of S i , S j \ H.By Theorem 2.11, there is a quadric Q r+2,e with a spread S = Σ 1 , . . ., Σ q r+e+1 2

+1
such that P can be obtained via Construction 2.9.Then the number of elements of P contained in S i , S j equals the number of elements of S contained in P, Σ i , Σ j .Since P, Σ i , Σ j ∩ Q r+2,e is either a hyperbolic quadric Q + (r + 2 − e, q), if |S i ∩ S j | = 1, or a parabolic quadric Q(r + 1, q), if |S i ∩ S j | = 0 and e = 2, such a number cannot exceed q + 1 by Lemma 2.12.It follows that P is irreducible.By Lemma 2.8 iii), through a point of H there pass precisely q members P. Hence tightness follows.