Affine vector space partitions and spreads of quadrics

Abstract

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 ∞ ; here $$H_{\infty }$$ H ∞ 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 ∞ 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 ∞ 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 ∈ P , $$|S \cap T| = 1$$ | S ∩ T | = 1 , then $$\langle S, T \rangle \cap \mathcal {Q}$$ ⟨ S , T ⟩ ∩ 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.