Connectedness of intersections of special Schubert varieties

Abstract

Let Gr l,n be the Grassmann variety of l -dimensional subspaces of an n -dimensional vector space V over an algebraically closed field k . Let σ(W)={Λ∈Gr l,n : Λ∩W≠0} denote the special Schubert variety associated to a subspace W of V . The main theorem of the paper is the following: The intersection ⋂ m j=1 σ(V j ) of the special Schubert varieties associated to subspaces V j , j=1,2,⋯,m , of dimension n−l−a j +1 such that l(n−l)−∑ m j=1 a j >0 is connected. Moreover, the intersection is irreducible of dimension l(n−l)−∑ m j=1 a j for a general choice of V j . The authors conjecture that the irreducibility holds for intersections of arbitrary Schubert varieties, when they are in general position with nonempty intersection. For a related connectivity result the authors refer to a paper of J. P. Hansen [Amer. J. Math. 105 (1983), no. 3, 633–639].