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. [ABSTRACT FROM AUTHOR]