Small resolutions of minuscule Schubert varieties

Let X be a minuscule Schubert variety. In this article, we use the combinatorics of quivers to define new quasi-resolutions of X . We describe in particular all relative minimal models π̂ : X̂ → X of X and prove that all the morphisms π̂ are small (in the sense of intersection cohomology). In particular, all small resolutions of X are given by the smooth relative minimal models X̂ and we describe all of them. As another application of this decription of relative minimal models, we give a more intrinsic statement of the main result of [Pe1].