Singularities of Affine Schubert Varieties

TL;DR

This paper investigates the singularities of affine Schubert varieties in the affine Grassmannian, identifying singular loci and tangent spaces for specific classes, and providing partial results for general cases.

Contribution

It explicitly determines singular loci and tangent spaces for certain classes of affine Schubert varieties, advancing understanding of their geometric structure.

Findings

Singular loci identified for two classes of affine Schubert varieties

Explicit tangent spaces computed at singular points for one class

Partial results on singular loci for general affine Schubert varieties

Abstract

This paper studies the singularities of affine Schubert varieties in the affine Grassmannian (of type $\mathrm{A}^{(1)}_\ell$). For two classes of affine Schubert varieties, we determine the singular loci; and for one class, we also determine explicitly the tangent spaces at singular points. For a general affine Schubert variety, we give partial results on the singular locus.