Affine vector space partitions and spreads of quadrics

Abstract

An affine spread is a set of subspaces of $\textrm{AG}(n, q)$ of the same dimension that partitions the points of $\textrm{AG}(n, q)$. Equivalently, an affine spread is a set of projective subspaces of $\textrm{PG}(n, q)$ of the same dimension which partitions the points of $\textrm{PG}(n, q) \setminus H_{\infty }$; here $H_{\infty }$ denotes the hyperplane at infinity of the projective closure of $\textrm{AG}(n, q)$. 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 $\textrm{PG}(n, q)$ 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 $\textrm{PG}(n, q)$ is equivalent to a spread of $\mathcal {Q}^+(n+1, q)$, $\mathcal {Q}(n+1, q)$ or $\mathcal {Q}^-(n+1, q)$, respectively.

1 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 $\textrm{PG}(n, q)$ is a set of projective subspaces of $\textrm{PG}(n, q)$ which partitions the points of $\textrm{PG}(n, q)$. A vector space partition whose subspaces are of the same dimension is called spread. It was shown by  André [1] and  Segre [13] that $\textrm{PG}(n, q)$ 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 $\textrm{PG}(n, q)$ is a member of one of the following classes: a symplectic space $\mathcal {W}(n, q)$, n odd, a parabolic quadric $\mathcal {Q}(n, q)$, n even, a hyperbolic quadric $\mathcal {Q}^+(n, q)$, n odd, an elliptic quadric $\mathcal {Q}^-(n, q)$, n odd, or a Hermitian variety $\mathcal {H}(n, q)$, q a square. A spread of a polar space $\mathcal {A}$ is a set of generators (i.e., maximal totally isotropic subspaces or maximal totally singular subspaces) of $\mathcal {A}$, which partitions the points of $\mathcal {A}$. A spread of $\mathcal {W}(n, q)$ is also a spread of $\textrm{PG}(n, q)$ into $\left( \frac{n-1}{2}\right)$-dimensional projective subspaces. It is easily seen that $\mathcal {Q}^+(n, q)$ has no spread if $n \equiv 1 \pmod {4}$. Many authors investigated spreads of polar spaces, see [3, 5, 9,10,11,12, 14, 15]. For q even, $\mathcal {Q}^+(n, q)$, $n \equiv -1\pmod {4}$, $\mathcal {Q}(n, q)$, $\mathcal {Q}^-(n, q)$ always have a spread. For q odd, with q a prime or $q \equiv 0$ or $2 \pmod {3}$, $\mathcal {Q}^+(7, q)$ and $\mathcal {Q}(6, q)$ have a spread. The parabolic quadric $\mathcal {Q}(n, q)$, with $n \equiv 0 \pmod {4}$ and q odd, has no spread. For q odd, $\mathcal {Q}^+(3, q)$ and $\mathcal {Q}^-(5, q)$ have a spread. The Hermitian varieties $\mathcal {H}(n, q)$, n odd and $\mathcal {H}(4, 4)$ do not have a spread. For existence results and open problem related to spreads of polar spaces, the reader is referred to [8, Sections 7.4 and 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 $\textrm{AG}(n, q)$, see [2]. Denote by $H_{\infty }$ the hyperplane at infinity of the projective closure of $\textrm{AG}(n, q)$. Then an affine vector space partition of $\textrm{PG}(n, q)$ is a set of projective subspaces of $\textrm{PG}(n, q)$ which partitions the points of $\textrm{PG}(n, q) \setminus H_{\infty }$. An affine spread is an affine vector space partition whose subspaces are of the same dimension. 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. Here, we are concerned with affine spreads $\mathcal {P}$ consisting of $(t+1)$-dimensional projective subspaces of $\textrm{PG}(n, q)$ such that

• 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 $\mathcal {Q}$.

An affine spread $\mathcal {P}$ of $\textrm{PG}(n, q)$ satisfying the above properties is called hyperbolic, parabolic or elliptic, according as $\mathcal {Q}$ is hyperbolic, parabolic or elliptic, respectively. In [2, Section 5.2] the authors exhibited a particular hyperbolic affine spread of $\textrm{PG}(6, q)$, q even or $q \in \{3, 5\}$, and they conjecture that a hyperbolic affine spread of $\textrm{PG}(6, q)$ exists for all prime powers [2, Conjecture 2]. In this note it is shown that a hyperbolic, parabolic or elliptic affine spread of $\textrm{PG}(n, q)$ is equivalent to a spread of $\mathcal {Q}^+(n+1, q)$, $\mathcal {Q}(n+1, q)$ or $\mathcal {Q}^-(n+1, q)$, respectively.

2 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 $\mathcal {Q}_{n, e}$ denote a non-degenerate quadric of $\textrm{PG}(n, q)$ as indicated below:

$$\begin{aligned} \begin{array}{c||c|c|c} \mathcal {Q}_{n, e} &{} \mathcal {Q}^+(n, q) &{} \mathcal {Q}(n, q) &{} \mathcal {Q}^-(n, q) \\ \hline e &{} 0 &{} 1 &{} 2 \\ \end{array}, \end{aligned}$$

where n is odd if the quadric is hyperbolic or elliptic, whereas n is even if the quadric is parabolic. Associated with $\mathcal {Q}_{n, e}$, there is a polarity $\perp$ of $\textrm{PG}(n, q)$, which is non-degenerate except when $e = 1$ and q is even. In particular, the polarity $\perp$ is symplectic if $\mathcal {Q}_{n, e} \in \{\mathcal {Q}^+(n, q), \mathcal {Q}^-(n, q)\}$, q even, and orthogonal if $\mathcal {Q}_{n, e} \in \{\mathcal {Q}(n, q), \mathcal {Q}^+(n, q), \mathcal {Q}^-(n, q)\}$, q odd. For $\mathcal {Q}_{n, 1} = \mathcal {Q}(n, q)$, q even, the polarity $\perp$ is degenerate, indeed $N^{\perp } = \textrm{PG}(n, q)$, if N is the nucleus of $\mathcal {Q}(n, q)$, whereas $P^\perp$ is a hyperplane of $\textrm{PG}(n, q)$ for any other point P of $\textrm{PG}(n, q)$. A generator of $\mathcal {Q}_{n, e}$ is a projective space of maximal dimension contained in $\mathcal {Q}_{n, e}$ and generators of $\mathcal {Q}_{n, e}$ are $\left( \frac{n-e-1}{2} \right)$-spaces. A spread of $\mathcal {Q}_{n, e}$ is a set of $q^{\frac{n+e-1}{2}}+1$ pairwise disjoint generators of $\mathcal {Q}_{n, e}$. 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 $\textrm{PG}(n+1, q)$. An affine vector space partition (or abbreviated as avsp) of $\textrm{PG}(n+1,q)$ is a set $\mathcal {P}$ of subspaces of $\textrm{PG}(n+1,q)$ whose members are not contained in H and such that every point of $\textrm{PG}(n+1, q) \setminus H$ is contained in exactly one element of $\mathcal {P}$.

The type of an avsp $\mathcal {P}$ of $\textrm{PG}(n+1, q)$ is given by $(n+1)^{m_{n+1}} \dots 2^{m_2} 1^{m_1}$, where $m_i>0$ denotes the number of $(i-1)$-spaces of $\mathcal {P}$ for $1\le i \le n+1$. An avsp of type $i^{m_i}$ is also called an affine spread.

Definition 2.2

Let $\mathcal {P}$ be an avsp of $\textrm{PG}(n+1,q)$. Then $\mathcal {P}$ is said to be reducible if there exists a proper subspace U of $\textrm{PG}(n+1, q)$ not in $\mathcal {P}$ such that the members of $\mathcal {P}$ contained in U form an avsp of U. If $\mathcal {P}$ is not reducible, then it is said to be irreducible.

Definition 2.3

Let $\mathcal {P}$ be an avsp of $\textrm{PG}(n+1,q)$. Then $\mathcal {P}$ is said to be tight if no point of $\textrm{PG}(n+1, q)$ belongs to each of the members of $\mathcal {P}$.

In [2, Section 5.2] the authors studied a particular avsp $\{S_1, \dots , S_{q^3}\}$ of $\textrm{PG}(6, q)$, q even, of type $4^{q^3}$, such that the set $\{S_i \cap H \mid i = 1, \dots , q^3\}$ consists of the $q^3$ planes of a Klein quadric $\mathcal {Q}^+(5, q)$ that are disjoint from a fixed plane of $\mathcal {Q}^+(5, q)$. 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 $\mathcal {P}= \left\{ S_1, S_2, \dots , S_{q^{\frac{n+e+1}{2}}}\right\}$ be an avsp of $\textrm{PG}(n+1, q)$ of type $\left( \frac{n-e+3}{2}\right) ^{q^{\frac{n+e+1}{2}}}$. Then $\mathcal {P}$ is said hyperbolic, parabolic or elliptic if the set

$$\begin{aligned} \mathcal {G}= \left\{ \Pi _i = S_i \cap H \mid i = 1, \dots , q^{\frac{n+e+1}{2}}\right\} \end{aligned}$$

consists of $q^{\frac{n+e+1}{2}}$ generators of a quadric $\mathcal {Q}_{n, e} \subset H$, where $e = 0, 1, 2$, respectively, such that

• (1) elements of $\mathcal {G}$ are disjoint from a fixed generator $\Pi$ of $\mathcal {Q}_{n, e}$;

• (2) distinct members of $\mathcal {G}$ (or of $\mathcal {P}$) meet at most in one point;

• (3) if $|\Pi _i \cap \Pi _j| = 1$, then $\langle S_i, S_j \rangle \cap \mathcal {Q}_{n, e} = \mathcal {Q}^+(n - e, q)$.

Remark 2.5

Let n be odd, let $\mathcal {P}= \left\{ S_1, S_2, \dots , S_{q^{\frac{n+1}{2}}}\right\}$ be a hyperbolic avsp of $\textrm{PG}(n+1, q)$ and let $\Pi _i = S_i \cap H$, $i = 1, \dots , q^{\frac{n+1}{2}}$. Since members of $\mathcal {P}$ are $\left( \frac{n+1}{2} \right)$-spaces, then $\langle S_i, S_j \rangle = \textrm{PG}(n+1, q)$ and $|S_i \cap S_j| = 1$, if $i \ne j$. Hence $|\Pi _i \cap \Pi _j| = 1$, if $i \ne j$. Therefore, from the definition of a hyperbolic avsp the point (2) is equivalent to the requirement that distinct members of $\mathcal {G}$ pairwise intersect at one point, whereas the point (3) is redundant.

If $n \equiv -1 \pmod {4}$, then members of $\mathcal {G}$ and $\Pi$ belong to the same system of generators and hence $|\Pi _i \cap \Pi _j| = 0$, if $i \ne j$. Therefore hyperbolic avsp of $\textrm{PG}(n+1, q)$, $n \equiv -1\pmod {4}$, do not exist.

Remark 2.6

If $n = e+1$, then there are not $q^{e+1}$ many generators of $\mathcal {Q}_{n, e}$ disjoint from $\Pi$. Hence there is no hyperbolic, parabolic or elliptic avsp for $(n, e) \in \{(1, 0), (2, 1), (3, 2)\}$. If $n = e+3$, then requirement (2) is redundant.

Remark 2.7

A hyperbolic affine spread $\mathcal {P}$ of $\textrm{PG}(6, q)$ is a set of $q^3$ solids such that $S_i \cap H$, $i = 1, \dots q^3$, are planes of a Klein quadric $\mathcal {Q}^+(5, q)$ that are disjoint from a fixed plane of $\mathcal {Q}^+(5, q)$. Therefore the affine spreads studied in [2] are equivalent to hyperbolic affine spread of $\textrm{PG}(6, q)$ consisting of solids and the existence of a hyperbolic avsp of $\textrm{PG}(6, q)$ is equivalent to the “extension problem” formulated in [2, Conjecture 2].

In the following we will need the next result, that in the hyperbolic case has been proved in [4, Example 7.6].

Lemma 2.8

Let $\left\{ \Sigma _1, \dots , \Sigma _{q^{\frac{n+e+1}{2}}+1}\right\}$ be a spread of a quadric $\mathcal {Q}_{n+2, e}$ of $\textrm{PG}(n+2, q)$. Fix a point $P \in \Sigma _{q^{\frac{n+e+1}{2}}+1}$ and an n-space $H \subset P^\perp$ such that $P \notin H$. Set $\mathcal {Q}_{n, e} = H \cap \mathcal {Q}_{n+2, e}$ and

$$\begin{aligned} \Pi _i = \langle P, \Sigma _i \rangle \cap H, \quad i = 1, \dots , q^{\frac{n+e+1}{2}}. \end{aligned}$$

The following hold.

• (i) If $e \in \{1, 2\}$, i.e., $\mathcal {Q}_{n+2, e}$ is parabolic or elliptic, then $\Pi _i$ and $\Pi _j$, $i \ne j$, intersect in at most one point.

• (ii) If $e = 0$, i.e., $\mathcal {Q}_{n+2, 0}$ is hyperbolic and $n \equiv 1 \pmod {4}$, then $\Pi _i$ and $\Pi _j$, $i \ne j$, have exactly one point in common.

• (iii) Each point of $\mathcal {Q}_{n, e} \setminus \Sigma _{q^{\frac{n+e+1}{2}}+1}$ lies in precisely q members of $\left\{ \Pi _i \mid i = 1, \dots , q^{\frac{n+e+1}{2}}\right\}$.

Proof

Since $|(\Sigma _i\cap P^\perp )\cap (\Sigma _j\cap P^\perp )| = 0$, it follows that $\Pi _i\cap \Pi _j$ is at most one point. Assume that $e = 0$, i.e., $\mathcal {Q}_{n+2, 0}$ is hyperbolic and $n \equiv 1 \pmod {4}$. Consider the $\left( \frac{n+1}{2}\right)$-space $T_i=\langle P, \Sigma _i\cap P^\perp \rangle$. Then $T_i$ is a generator of $\mathcal {Q}_{n+2, 0}$ intersecting $\Sigma _i$ in an $\left( \frac{n-1}{2}\right)$-space. Therefore $T_i$ and $\Sigma _j$, $i \ne j$, are in different systems of generators and hence they have at least one point in common. It follows that $\Pi _i$ and $\Pi _j$, $i \ne j$, have at least one point in common.

Finally if R is a point of $\mathcal {Q}_{n, e} \setminus \Sigma _{q^{\frac{n+e+1}{2}}+1}$, since $P \in \Sigma _{q^{\frac{n+e+1}{2}}+1}$ and $R \notin \Sigma _{q^{\frac{n+e+1}{2}}+1}$, there are exactly q distinct members of $\left\{ \Sigma _1, \dots , \Sigma _{q^{\frac{n+e+1}{2}}+1}\right\}$ intersecting the line $\langle P, R \rangle$ in one point distinct from P. Hence, through a point of $\mathcal {Q}_{n, e} \setminus \Sigma _{q^{\frac{n+e+1}{2}}+1}$, there pass precisely q members of $\left\{ \Pi _i \mid i = 1, \dots , q^{\frac{n+e+1}{2}}\right\}$. $\square$

In what follows we observe that a hyperbolic, elliptic, or parabolic avsp of $\textrm{PG}(n+1, q)$ can be obtained starting from a spread of a hyperbolic, elliptic, or parabolic quadric of $\textrm{PG}(n+2, q)$.

Construction 2.9

Let $\left\{ \Sigma _1, \dots , \Sigma _{q^{\frac{n+e+1}{2}}+1}\right\}$ be a spread of a quadric $\mathcal {Q}_{n+2, e}$ of $\textrm{PG}(n+2, q)$. Fix a point $P \in \Sigma _{q^{\frac{n+e+1}{2}}+1}$ and a hyperplane $\mathcal {U} \cong \textrm{PG}(n+1,q)$ not containing the point P. Then $H = \mathcal {U} \cap P^{\perp }$ is an n-space of $\textrm{PG}(n+2, q)$ such that $\mathcal {Q}_{n, e} = H \cap \mathcal {Q}_{n+2,e}$. Let

$$\begin{aligned} S_i = \langle P, \Sigma _i\rangle \cap \mathcal {U}, \quad i = 1, \dots , q^{\frac{n+e+1}{2}}, \end{aligned}$$

and let $\mathcal {P}= \left\{ S_1, \dots , S_{q^{\frac{n+e+1}{2}}}\right\}$. Then $\mathcal {P}$ consists of $q^{\frac{n+e+1}{2}}$ $\left( \frac{n-e+1}{2}\right)$-spaces of $\mathcal {U}$, none of them is contained in H.

Proposition 2.10

The set $\mathcal {P}$ is a hyperbolic, parabolic or elliptic avsp of $\textrm{PG}(n+1, q)$ according as $\mathcal {Q}_{n+2, e}$ is hyperbolic, parabolic or elliptic, respectively.

Proof

In order to prove that $\mathcal {P}$ is an avsp of $\textrm{PG}(n+1, q)$ it is enough to show that for every point of $\mathcal {U}\setminus H$, there exists a member of $\mathcal {P}$ containing it. Let R be a point of $\mathcal {U}\setminus H$ and let $\ell$ be the line joining P and R. Since $\ell$ passes through P and is not contained in $P^{\perp }$, it is secant to $\mathcal {Q}_{n+2, e}$. Therefore there is a point, say $R'$, distinct from P, such that $R' \in \mathcal {Q}_{n+2, e}$. Moreover $R' \in \Sigma _j$, for some $j \in \left\{ 1, \dots , q^{\frac{n+e+1}{2}}\right\}$ and hence $R \in S_j$, by construction.

Set $\mathcal {G}= \left\{ \Pi _i = S_i \cap H \mid i = 1, \dots , q^{\frac{n+e+1}{2}}\right\}$. Then $\mathcal {G}$ consists of generators of $\mathcal {Q}_{n, e}$ disjoint from $\Sigma _{q^{\frac{n+e+1}{2}}+1} \cap H \subset \mathcal {Q}_{n, e}$. By Lemma 2.8, $|\Pi _i \cap \Pi _j|$, $i \ne j$, equals one, if $\mathcal {Q}_{n+2, e}$ is hyperbolic, or is at most one, otherwise.

Let $S_i, S_j \in \mathcal {P}$, with $|S_i \cap S_j| = 1$. Then $\langle S_i, S_j, P \rangle = \langle \Sigma _i, \Sigma _j \rangle$ is an $(n+2-e)$-space of $\textrm{PG}(n+2, q)$ containing two disjoint generators of $\mathcal {Q}_{n+2, e}$. Hence $\langle S_i, S_j, P \rangle \cap \mathcal {Q}_{n+2, e} = \mathcal {Q}_{n+2-e, 0} \simeq \mathcal {Q}^+(n+2-e, q)$. Such a quadric $\mathcal {Q}_{n+2-e, 0}$ meets $P^\perp$ in a cone having as vertex the point P and as base a $\mathcal {Q}_{n-e, 0} \simeq \mathcal {Q}^+(n-e, q)$. It follows that $\langle S_i, S_j \rangle \cap \mathcal {Q}_{n, e} = \mathcal {Q}^+(n-e, q)$, as required. $\square$

We have seen that a spread of a hyperbolic, elliptic, or parabolic quadric of $\textrm{PG}(n+2, q)$ gives rise to a hyperbolic, elliptic, or parabolic avsp of $\textrm{PG}(n+1, q)$, respectively. The converse also holds true, as shown below.

Theorem 2.11

If $\mathcal {P}$ is a hyperbolic, parabolic or elliptic avsp of $\textrm{PG}(n+1, q)$, then there is a spread of $\mathcal {Q}_{n+2, e}$, where $\mathcal {Q}_{n+2, e}$ is a hyperbolic, parabolic or elliptic quadric, respectively.

Proof

Let $\mathcal {P}= \left\{ S_1, \dots , S_{q^{\frac{n+e+1}{2}}}\right\}$ be a hyperbolic, parabolic or elliptic avsp of $\textrm{PG}(n+1, q)$. Then there exists a hyperplane H of $\textrm{PG}(n+1, q)$, a quadric $\mathcal {Q}_{n, e}$ of H, that is hyperbolic, parabolic or elliptic, respectively, and a fixed generator $\Pi$ of $\mathcal {Q}_{n, e}$, such that $\Pi _i = S_i \cap H$ is a generator of $\mathcal {Q}_{n, e}$ disjoint from $\Pi$. Embed $\textrm{PG}(n+1, q)$ as a hyperplane section, say $\mathcal {U}$, of $\textrm{PG}(n+2, q)$ and fix a quadric $\mathcal {Q}_{n+2, e}$ of $\textrm{PG}(n+2, q)$ in such a way that $H \cap \mathcal {Q}_{n+2, e} = \mathcal {Q}_{n, e}$ and $H^\perp$ is a line secant to $\mathcal {Q}_{n+2, e}$. Here $\perp$ is the polarity of $\textrm{PG}(n+2, q)$ associated with $\mathcal {Q}_{n+2, e}$. Let P be one of the two points of $\mathcal {Q}_{n+2, e}$ on $H^\perp$. For $i = 1, \dots , q^{\frac{n+e+1}{2}}$, consider the following $\left( \frac{n-e+3}{2}\right)$-space

$$\begin{aligned} \mathcal {F}_i = \langle P, S_i \rangle . \end{aligned}$$

Hence $\mathcal {F}_i$ meets $P^\perp$ in the $\left( \frac{n-e+1}{2}\right)$-space spanned by P and $\Pi _i$, that is a generator of $\mathcal {Q}_{n+2, e}$. We claim that $\mathcal {F}_i$ contains a further generator of $\mathcal {Q}_{n+2, e}$ besides $\langle P, \Pi _i \rangle$. Indeed, if $\langle P, \Pi _i \rangle$ were the unique generator of $\mathcal {Q}_{n+2, e}$ contained in $\mathcal {F}_i$, then $\mathcal {F}_i \subset \langle P, \Pi _i \rangle ^\perp = P^\perp \cap \Pi _i^\perp \subset P^\perp$, contradicting the fact that $\mathcal {F}_i \cap P^\perp$ is an $\left( \frac{n-e+1}{2}\right)$-space. Therefore $\mathcal {F}_i$ has to contain a further generator of $\mathcal {Q}_{n+2, e}$, say $\Sigma _i$. Denote by $\Sigma _{q^{\frac{n+e+1}{2}}+1}$ the generator of $\mathcal {Q}_{n+2, e}$ spanned by P and $\Pi$. We claim that

$$\begin{aligned} \left\{ \Sigma _1, \dots , \Sigma _{q^{\frac{n+e+1}{2}}+1}\right\} \end{aligned}$$

is a spread of $\mathcal {Q}_{n+2, e}$. Assume by contradiction that there is a point $Q' \in \Sigma _i \cap \Sigma _{q^{\frac{n+e+1}{2}}+1}$, for some $i \in \left\{ 1, \dots , q^{\frac{n+e+1}{2}}\right\}$. Let $Q = \langle P, Q' \rangle \cap H$, then $Q \in \Pi _i \cap \Pi$, a contradiction. Assume by contradiction that $|\Sigma _i \cap \Sigma _j| > 0$, for some $1 \le i < j \le q^{\frac{n+e+1}{2}}$. Then necessarily $\Sigma _i \cap \Sigma _j$ is a point, otherwise $|S_i \cap S_j| > 1$. Moreover, we infer that such a point, say $R' = \Sigma _i \cap \Sigma _j$, must be in $P^\perp$. Indeed, if $R' \notin P^\perp$, then the point $R = \langle P, R' \rangle \cap \mathcal {U}$ belongs to both $S_i \setminus H$ and $S_j \setminus H$, contradicting the fact that $S_1, \dots , S_{q^{\frac{n+e+1}{2}}}$ is an avsp. In particular, $|S_i \cap S_j| = 1$.

If $e = 0$, by Remark 2.5, we may assume that $n \equiv 1 \pmod {4}$. Hence $n+2 \equiv -1 \pmod {4}$ and $\Sigma _k$ and $\Sigma _{q^{\frac{n+e+1}{2}}+1}$ belong to the same system of generators of $\mathcal {Q}_{n+2, 0}$, for $k = 1, \dots , q^{\frac{n+e+1}{2}}$. It follows that if $|\Sigma _i \cap \Sigma _j| > 0$, then they have at least a line in common, a contradiction.

If $e \in \{1, 2\}$, observe that $\langle \mathcal {F}_i, \mathcal {F}_j \rangle$ is a $\textrm{PG}(n+2-e, q)$ containing the cone having as vertex the point P and as base the hyperbolic quadric $\mathcal {Q}^+(n-e, q) = \langle S_i, S_j \rangle \cap \mathcal {Q}_{n, e}$. Furthermore, two more generators of $\mathcal {Q}_{n+2, e}$, namely $\Sigma _i, \Sigma _j$, are contained in $\langle \mathcal {F}_i, \mathcal {F}_j \rangle$ and do not pass through P. Therefore necessarily we have that $\langle \mathcal {F}_i, \mathcal {F}_j \rangle \cap \mathcal {Q}_{n+2, e} = \mathcal {Q}_{n+2-e, 0}$. Let us denote by $\mathcal {Q}^+(n+2-e, q)$ the hyperbolic quadric $\langle \mathcal {F}_i, \mathcal {F}_j \rangle \cap \mathcal {Q}_{n+2, e}$, so that generators of $\mathcal {Q}^+(n+2-e, q)$ are $\left( \frac{n-e+1}{2}\right)$-spaces. Since $\Sigma _i \cap \langle P, \Pi _i \rangle$ and $\Sigma _j \cap \langle P, \Pi _j \rangle$ are $\left( \frac{n-e-1}{2}\right)$-spaces, we have that $\Sigma _i$ and $\langle P, \Pi _i \rangle$ lie in distinct systems of generators of $\mathcal {Q}^+(n+2-e, q)$. Similarly for $\Sigma _j$ and $\langle P, \Pi _j \rangle$. Two possibilities arise: either $n+2-e \equiv -1 \pmod {4}$ or $n+2-e \equiv 1 \pmod {4}$. Since $\langle P, \Pi _i \rangle \cap \langle P, \Pi _j \rangle$ is a line, if the former case occurs, then $\langle P, \Pi _i \rangle$, $\langle P, \Pi _j \rangle$ belong to the same system of generators of $\mathcal {Q}^+(n+2-e, q)$. Hence $\Sigma _i$, $\Sigma _j$ belong to the same system of generators of $\mathcal {Q}^+(n+2-e, q)$ and if $|\Sigma _i \cap \Sigma _j| > 0$, then they have at least a line in common, a contradiction. In the latter case, $\langle P, \Pi _i \rangle$, $\langle P, \Pi _j \rangle$ are in different systems of generators of $\mathcal {Q}^+(n+2-e, q)$. Therefore $\Sigma _i$, $\langle P, \Pi _j \rangle$ belong to the same system of generators of $\mathcal {Q}^+(n+2-e, q)$. Similarly for $\Sigma _j$, $\langle P, \Pi _i \rangle$. It follows that $\Sigma _i$, $\Sigma _j$ belong to different systems of generators of $\mathcal {Q}^+(n+2-e, q)$ and again, if $|\Sigma _i \cap \Sigma _j| > 0$, then they have at least a line in common, a contradiction. $\square$

Following the proof of Theorem 2.11, it can be observed that for any hyperbolic, parabolic, or elliptic affine spread $\mathcal {P}$, it is possible to construct a spread $\mathcal {S}$ of a hyperbolic, parabolic, or elliptic quadric, respectively, such that if we apply Construction 2.9 starting from $\mathcal {S}$, we obtain $\mathcal {P}$. As pointed out in [2, Section 5.2, Theorem 2], tightness and irreducibility follow immediately for a hyperbolic avsp of $\textrm{PG}(5, q)$. We will show that the same occurs in higher dimensions and for parabolic, elliptic or hyperbolic avsp.

Lemma 2.12

Let $e \in \{1, 2\}$, let $\mathcal {S}= \left\{ \Sigma _1, \dots , \Sigma _{q^{\frac{n+e+1}{2}}+1} \right\}$ be a spread of $\mathcal {Q}_{n+2, e}$. Then at most $q+1$ members of $\mathcal {S}$ are contained in a quadric $\mathcal {Q}_{n+1,e-1} \subset \mathcal {Q}_{n+2, e}$.

Proof

If $e = 1$, then exactly 2 members of $\mathcal {S}$ are contained in a hyperbolic hyperplane section $\mathcal {Q}_{n+1, 0} \simeq \mathcal {Q}^+(n+1, q)$ of $\mathcal {Q}_{n+2, 1} \simeq \mathcal {Q}(n+2, q)$, whereas, if $e = 2$, then there are $q+1$ elements of $\mathcal {S}$ contained in a parabolic hyperplane section $\mathcal {Q}_{n+1, 1} \simeq \mathcal {Q}(n+1, q)$ of $\mathcal {Q}_{n+2, 2} \simeq \mathcal {Q}^-(n+2, q)$. $\square$

Proposition 2.13

Let $\mathcal {P}$ be a hyperbolic, parabolic or elliptic avsp of $\textrm{PG}(n+1, q)$, then $\mathcal {P}$ is tight and irreducible.

Proof

Let $\mathcal {P}=\left\{ S_1, S_2, \dots , S_{q^{\frac{n+e+1}{2}}}\right\}$ be a hyperbolic, parabolic or elliptic avsp of $\textrm{PG}(n+1, q)$. If $e = 0$, in order to prove that $\mathcal {P}$ is irreducible, it is enough to observe that, if $i \ne j$, then the span of $S_i$ and $S_j$ is the whole $\textrm{PG}(n+1, q)$. Similarly, if $e = 1$ and $|S_i \cap S_j| = 0$, then $\langle S_i, S_j \rangle = \textrm{PG}(n+1, q)$. If either $e = 1$ and $|S_i \cap S_j| = 1$ or $e = 2$, we claim that the elements of $\mathcal {P}$ contained in $\langle S_i, S_j \rangle$, where $i \ne j$, do not cover all the points of $\langle S_i, S_j \rangle \setminus H$. By Theorem 2.11, there is a quadric $\mathcal {Q}_{n+2, e}$ with a spread $\mathcal {S}= \left\{ \Sigma _1, \dots , \Sigma _{q^{\frac{n+e+1}{2}}+1} \right\}$ such that $\mathcal {P}$ can be obtained via Construction 2.9. Then the number of elements of $\mathcal {P}$ contained in $\langle S_i, S_j \rangle$ equals the number of elements of $\mathcal {S}$ contained in $\langle P, \Sigma _i, \Sigma _j \rangle$. Since $\langle P, \Sigma _i, \Sigma _j \rangle \cap \mathcal {Q}_{n+2, e}$ is either a hyperbolic quadric $\mathcal {Q}^+(n+2-e, q)$, if $|S_i \cap S_j| = 1$, or a parabolic quadric $\mathcal {Q}(n+1, q)$, if $|S_i \cap S_j| = 0$ and $e = 2$, such a number cannot exceed $q+1$ by Lemma 2.12. It follows that $\mathcal {P}$ is irreducible.

By Lemma 2.8 (iii), through a point of H there pass precisely q members $\mathcal {P}$. Hence tightness follows. $\square$

3 Conclusion

We have seen that a hyperbolic, parabolic or elliptic affine spread of $\textrm{PG}(n, q)$ is equivalent to a spread of $\mathcal {Q}^+(n+1, q)$, $\mathcal {Q}(n+1, q)$ or $\mathcal {Q}^-(n+1, q)$, respectively. Furthermore, such an affine spread is tight and irreducible. Based on Proposition 2.10 and Theorem 2.11, the extension problem formulated in [2, Conjecture 2] is equivalent to that of the existence of a spread of the triality quadric $\mathcal {Q}^+(7, q)$, which in turn is equivalent to that of the existence of an ovoid of $\mathcal {Q}^+(7, q)$. We remark that the existence of a spread of $\mathcal {Q}^+(7, q)$ has been established in the cases when q is even or when q is odd, with q a prime or $q \equiv 0$ or $2 \pmod {3}$. Therefore, a positive answer to the extension problem follows in the cases when q is even or when q is odd, with q a prime or $q \equiv 0$ or $2 \pmod {3}$.