Affine vector space partitions and spreads of quadrics

期刊论文
出版模式：

开放获取
发表日期：

2024年6月27日

查看全文

Somi Gupta,
Francesco Pavese

摘要

An affine spread is a set of subspaces of $\textrm{AG}(n, q)$ $AG (n, q)$ of the same dimension that partitions the points of $\textrm{AG}(n, q)$ $AG (n, q)$ . Equivalently, an affine spread is a set of projective subspaces of $\textrm{PG}(n, q)$ $PG (n, q)$ of the same dimension which partitions the points of $\textrm{PG}(n, q) \setminus H_{\infty }$ $PG (n, q) \ H_{\infty}$ ; here $H_{\infty }$ $H_{\infty}$ denotes the hyperplane at infinity of the projective closure of $\textrm{AG}(n, q)$ $AG (n, q)$ . Let $\mathcal {Q}$ $Q$ be a non-degenerate quadric of $H_\infty $ $H_{\infty}$ and let $\Pi $ $Π$ be a generator of $\mathcal {Q}$ $Q$ , where $\Pi $ $Π$ is a t-dimensional projective subspace. An affine spread $\mathcal {P}$ $P$ consisting of $(t+1)$ $(t + 1)$ -dimensional projective subspaces of $\textrm{PG}(n, q)$ $PG (n, q)$ is called hyperbolic, parabolic or elliptic (according as $\mathcal {Q}$ $Q$ is hyperbolic, parabolic or elliptic) if the following hold:

Each member of $\mathcal {P}$ $P$ meets $H_\infty $ $H_{\infty}$ in a distinct generator of $\mathcal {Q}$ $Q$ disjoint from $\Pi $ $Π$ ;

Elements of $\mathcal {P}$ $P$ have at most one point in common;

If $S, T \in \mathcal {P}$ $S, T \in P$ , $|S \cap T| = 1$ $| S \cap T | = 1$ , then $\langle S, T \rangle \cap \mathcal {Q}$ $⟨ S, T ⟩ \cap Q$ is a hyperbolic quadric of $\mathcal {Q}$ $Q$ .

In this note it is shown that a hyperbolic, parabolic or elliptic affine spread of $\textrm{PG}(n, q)$

PG (n, q)

is equivalent to a spread of $\mathcal {Q}^+(n+1, q)$

Q^{+} (n + 1, q)

, $\mathcal {Q}(n+1, q)$

Q (n + 1, q)

or $\mathcal {Q}^-(n+1, q)$

Q^{-} (n + 1, q)

, respectively.