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 nondegenerate quadric of \(H_\infty \) and let \(\Pi \) be a generator of \(\mathcal {Q}\), where \(\Pi \) is a tdimensional 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.
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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 tdimensional 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 nondegenerate reflexive sesquilinear or quadratic form. A nondegenerate 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 \(\left( \frac{n1}{2}\right) \)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 nondegenerate quadric of \(H_\infty \) and let \(\Pi \) be a generator of \(\mathcal {Q}\), where \(\Pi \) is a tdimensional 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 sspace to denote an sdimensional projective subspace of the ambient projective space. Let \(\mathcal {Q}_{n, e}\) denote a nondegenerate quadric of \(\textrm{PG}(n, q)\) as indicated below:
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 nondegenerate 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 \(\left( \frac{ne1}{2} \right) \)spaces. A spread of \(\mathcal {Q}_{n, e}\) is a set of \(q^{\frac{n+e1}{2}}+1\) 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 \((i1)\)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 \(\left( \frac{ne+3}{2}\right) ^{q^{\frac{n+e+1}{2}}}\). Then \(\mathcal {P}\) is said hyperbolic, parabolic or elliptic if the set
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 nspace \(H \subset P^\perp \) such that \(P \notin H\). Set \(\mathcal {Q}_{n, e} = H \cap \mathcal {Q}_{n+2, e}\) and
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 \(\left( \frac{n1}{2}\right) \)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 nspace of \(\textrm{PG}(n+2, q)\) such that \(\mathcal {Q}_{n, e} = H \cap \mathcal {Q}_{n+2,e}\). Let
and let \(\mathcal {P}= \left\{ S_1, \dots , S_{q^{\frac{n+e+1}{2}}}\right\} \). Then \(\mathcal {P}\) consists of \(q^{\frac{n+e+1}{2}}\) \(\left( \frac{ne+1}{2}\right) \)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 \((n+2e)\)space of \(\textrm{PG}(n+2, q)\) containing two disjoint generators of \(\mathcal {Q}_{n+2, e}\). Hence \(\langle S_i, S_j, P \rangle \cap \mathcal {Q}_{n+2, e} = \mathcal {Q}_{n+2e, 0} \simeq \mathcal {Q}^+(n+2e, q)\). Such a quadric \(\mathcal {Q}_{n+2e, 0}\) meets \(P^\perp \) in a cone having as vertex the point P and as base a \(\mathcal {Q}_{ne, 0} \simeq \mathcal {Q}^+(ne, q)\). It follows that \(\langle S_i, S_j \rangle \cap \mathcal {Q}_{n, e} = \mathcal {Q}^+(ne, q)\), 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 \(\left( \frac{ne+3}{2}\right) \)space
Hence \(\mathcal {F}_i\) meets \(P^\perp \) in the \(\left( \frac{ne+1}{2}\right) \)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 \(\left( \frac{ne+1}{2}\right) \)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
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 \(\textrm{PG}(n+2e, q)\) containing the cone having as vertex the point P and as base the hyperbolic quadric \(\mathcal {Q}^+(ne, q) = \langle S_i, S_j \rangle \cap \mathcal {Q}_{n, e}\). 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 \(\langle \mathcal {F}_i, \mathcal {F}_j \rangle \cap \mathcal {Q}_{n+2, e} = \mathcal {Q}_{n+2e, 0}\). Let us denote by \(\mathcal {Q}^+(n+2e, q)\) the hyperbolic quadric \(\langle \mathcal {F}_i, \mathcal {F}_j \rangle \cap \mathcal {Q}_{n+2, e}\), so that generators of \(\mathcal {Q}^+(n+2e, q)\) are \(\left( \frac{ne+1}{2}\right) \)spaces. Since \(\Sigma _i \cap \langle P, \Pi _i \rangle \) and \(\Sigma _j \cap \langle P, \Pi _j \rangle \) are \(\left( \frac{ne1}{2}\right) \)spaces, we have that \(\Sigma _i\) and \(\langle P, \Pi _i \rangle \) lie in distinct systems of generators of \(\mathcal {Q}^+(n+2e, q)\). Similarly for \(\Sigma _j\) and \(\langle P, \Pi _j \rangle \). Two possibilities arise: either \(n+2e \equiv 1 \pmod {4}\) or \(n+2e \equiv 1 \pmod {4}\). 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 \(\mathcal {Q}^+(n+2e, q)\). Hence \(\Sigma _i\), \(\Sigma _j\) belong to the same system of generators of \(\mathcal {Q}^+(n+2e, q)\) 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 \(\mathcal {Q}^+(n+2e, q)\). Therefore \(\Sigma _i\), \(\langle P, \Pi _j \rangle \) belong to the same system of generators of \(\mathcal {Q}^+(n+2e, q)\). Similarly for \(\Sigma _j\), \(\langle P, \Pi _i \rangle \). It follows that \(\Sigma _i\), \(\Sigma _j\) belong to different systems of generators of \(\mathcal {Q}^+(n+2e, q)\) 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 \(\mathcal {Q}_{n+1,e1} \subset \mathcal {Q}_{n+2, e}\).
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 \(\mathcal {Q}^+(n+2e, q)\), 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 \(2 \pmod {3}\).
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Acknowledgements
This work was supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA– INdAM).
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Gupta, S., Pavese, F. Affine vector space partitions and spreads of quadrics. Des. Codes Cryptogr. (2024). https://doi.org/10.1007/s10623024014471
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DOI: https://doi.org/10.1007/s10623024014471