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 $\widehat{\pi}:{\widehat{X}}\to X$. We prove that all the morphisms $\widehat{\pi}$ are small and give a combinatorial criterion for $\widehat{X}$ 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 $\alpha\in A_1(X)$ be an effective 1-cycle class. Then the irreducible components of the scheme Hom$_{\alpha}(p^1,X)$ of morphisms from $\mathbb{P}^1$ to $X$ and of class $\alpha$ are indexed by the set: ${\mathfrak{ne}}(\alpha)=\{\beta\in A_1(\widehat{X}) \mid \beta$ is effective and $\widehat{\pi}_*\beta=\alpha\}$ which is independent of the choice of a relative minimal model $\widehat{X}$ of $X$.