Small resolutions of minuscule Schubert varieties

By N. Perrin. Compositio Mathematica, 2006

49 citations recorded by the source corpus.

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 $X$.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/4e1846b8ee78827737d816b5f76f0e1291c89d27

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Citations in the corpus

A jeu de taquin theory for increasing tableaux, with applications to K-theoretic Schubert calculus (2007, result): “Associated to each minuscule G/P is a planar poset (3G/P ,≺), obtained as a subposet of the poset of positive roots G∨ for the dual root system of G; this fact has been known for some time, and recently has been exploited by various authors; see, for example, [Perrin 2007; Purbhoo and Sottile 2008]”
K-theory of minuscule varieties (2013, background): “Several papers illustrate that many aspects of the geometry and combinatorics of (co)minuscule varieties are natural generalizations of Grassmannians of type A [24, 26, 22, 27, 8].”
A combinatorial rule for (co)minuscule Schubert calculus (2006, background): “We focus on the latter, explaining the adjustments for the former as necessary.) These G/P’s and their Schubert varieties are of significant and fundamental interest in geometry and representation theory, see, e.g., [BiLa00, Chapter 9], [Kos61] and the references therein, as well as, e.g., [ Pe06 , ”
Quantum Cohomology of Minuscule Homogeneous Spaces (2006, methodology): “The set WX can be identified with the set of partitions contained in this rectangle, and the quiver Qw, for any w ∈ WX, is the complement in the rectangle of the partition defining X(w) (see [ Pe2 ] and examples 3.26 and 3.30 below).”