Affine vector space partitions and spreads of quadrics

Abstract

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.