Resolution of singularities of Schubert cycles

AbstractLet G(r,n) be the Grassmannian of r-dimensional subspaces of Kn. With each sequence t = (t1,…,tr) of positive integers such that 1<-t1<t2<⋯<tr<-n a set St is associated and called a Schubert cycle. In general, the Schubert cycles are not smooth, i.e. they have singular points, and our purpose is to eliminate these points of singularity by blowing up certain sub-Schubert cycles; to be more precise, we order the sub-Schubert cycles, say σ1≤σ2≤⋯ σk of the Schubert cycle St1,…,tr by dimension, then the main result of this article guarantees that the singularities of St1,…,tr can be resolved by blowing up first σ1, then by blowing up the strict transform of σ2, then of σ3, etc