The Hilton-Milner theorem for the distance-regular graphs of bilinear forms

Let V be an (n+l)-dimensional vector space over the finite field Fq with l≥n>0, and W be a fixed l-dimensional subspace of V. Suppose F is a non-trivial intersecting family of n-dimensional subspaces U of V with U∩W=0. In this paper, we give the tight upper bound for the size of F, and describe the structure of F which reaches the upper bound.