Affine vector space partitions and spreads of quadrics

An affine spread is a set of subspaces of AG(n, q) of the same dimension that partitions the points of AG(n, q). Equivalently, an { affine spread} is a set of projective subspaces of PG(n, q) of the same dimension which partitions the points of PG(n, q) H∞; here H∞ denotes the hyperplane at infinity of the projective closure of AG(n, q). Let Q be a non degenerate quadric of H_∞ and let Π be a generator of Q, where Π is a t-dimensional projective subspace. An affine spread P consisting of $(t+1)$-dimensional projective subspaces of PG(n, q) is called hyperbolic, parabolic or elliptic (according as Q is hyperbolic, parabolic or elliptic) if the following hold: each member of P meets H_∞ in a distinct generator of Q disjoint from Π; elements of P have at most one point in common; if S, T ∈ P, |S ∩ T| = 1, then S, T ∩ Q is a hyperbolic quadric of Q. In this note it is shown that a hyperbolic, parabolic or elliptic affine spread of PG(n, q) is equivalent to a spread of Q⁺(n+1, q), Q(n+1, q) or Q⁻(n+1, q), respectively.