A Characteristic Property of Geometric t-Spreads in Finite Projective Spaces

We prove the following combinatorial characterization of geometric spreads. Let ℒ be a family of t-dimensional subspaces of the projective space P = PG((t +1)r − 1 + k, q) with t⩾1, r⩾2 and k⩾0 such that every (tr + k)-dimensional subspace of Pcontains at least one element of Then |ℒ⩾(qt+1)r−1 +·+ qt+1 1, with equality iff ℒ is a geometric t-spread in a ((t + 1)r − 1)-dimensional subspace of P