Decompositions of Schubert Varieties and Small Resolutions
Scott Larson
TL;DR
This paper introduces a method for constructing small resolutions of Schubert varieties using BP decompositions and pattern avoidance, providing explicit isomorphisms and connecting to Gelfand-MacPherson resolutions.
Contribution
It offers a new approach to small resolutions of Schubert varieties via explicit isomorphisms and decompositions, expanding the toolkit for resolving singularities.
Findings
Many new small resolutions of Schubert varieties are constructed.
Resolutions are expressed explicitly in terms of pattern avoidance in type A.
Resolutions are shown to be Gelfand-MacPherson resolutions.
Abstract
We provide a method for gluing (small) resolutions of singularities of Schubert varieties \(X_w\). An explicit isomorphism of \(X_w\) with an (iterated) bundle is constructed when \(w\) has an (iterated) BP decomposition. Combined with the first result this gives many new small resolutions of Schubert varieties. In type A, this can be expressed in terms of pattern avoidance. Also we show resolutions of Schubert varieties constructed quite generally are in fact Gelfand-MacPherson resolutions.
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Taxonomy
TopicsAdvanced Algebra and Geometry · Nonlinear Waves and Solitons · Algebraic Geometry and Number Theory