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.