The Nash blow-up of a cominuscule Schubert variety

TL;DR

This paper computes the Nash blow-up of cominuscule Schubert varieties, showing it is isomorphic to another Schubert variety, and explores implications for smoothness, resolutions, and torus actions.

Contribution

It provides a new algebraic characterization of Nash blow-ups for cominuscule Schubert varieties and relates them to other Schubert varieties of the same Lie type.

Findings

Nash blow-up is isomorphic to another Schubert variety of the same Lie type.

Characterization of the smooth locus via Nash blow-up.

Criteria for Nash blow-up to be a resolution in Grassmannian cases.

Abstract

We compute the Nash blow-up of a cominuscule Schubert variety. In particular, we show that the Nash blow-up is algebraically isomorphic to another Schubert variety of the same Lie type. As a consequence, we give a new characterization of the smooth locus and, for Grassmannian Schubert varieties, determine when the Nash blow-up is a resolution of singularities. We also study the induced torus action on the Nash blow-up and give a bijection between its torus fixed points and Peterson translates on the Schubert variety.