Affine vector space partitions and spreads of quadrics

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