Small resolutions of minuscule Schubert varieties

Abstract

Let X be a minuscule Schubert variety. In this paper, we associate a quiver with X and use the combinatorics of this quiver to describe all relative minimal models � : � X → X. We prove that all the morphismsare small and give a combinatorial criterion for � X to be smooth and thus a small resolution of X. We describe in this way all small resolutions of X. As another application of this description of relative minimal models, we obtain the following more intrinsic statement of the main result of Perrin, J. Algebra 294 (2005), 431-462. Let α ∈ A1(X) be an effective 1-cycle class. Then the irreducible components of the scheme Homα(p 1 ,X ) of morphisms from P 1 to X and of class α are indexed by the set: ne(α )= {β ∈ A1( � X) | β is effective and � π∗β = α} which is independent of the choice of a relative minimal model � X of X.