Large nontrivial intersecting families in Gr(k,mathbb{F}_q^{2k})

We consider intersecting families of $k$-dimensional subspaces of $\mathbb{F}_q^{n}$. It is known that for n>2k the largest families with these properties are point-pencils, i.e. families of all subspaces that contain a fixed line. For n = 2k any extremal example is either a point-pencil, or its dual.

We are interested in a Hilton—Milner type of problem: what are the size and structure of largest families that are not contained in a point-pencil or its dual? This problem was previously solved for n > 2k+1. During the talk we will solve it for n=2k and sufficiently large q, using the spread approximation technique and association schemes.