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.