Singular loci of cominuscule Schubert varieties

TL;DR

This paper provides a uniform, representation-theoretic description of the singular loci of Schubert varieties in cominuscule rational homogeneous varieties, revealing conditions for smoothness and codimension properties.

Contribution

It introduces a type-independent characterization of singular loci using a recent combinatorial framework involving an integer A and marked Dynkin diagrams.

Findings

Variety is smooth iff A=0

Singular locus in Type ADE occurs in codimension at least three

Provides explicit description of irreducible components of singular loci

Abstract

Let X = G/P be a cominuscule rational homogeneous variety. (Equivalently, X admits the structure of a compact Hermitian symmetric space.) I give a uniform description (that is, independent of type) of the irreducible components of the singular locus of a Schubert variety Y in X in terms of representation theoretic data. The result is based on a recent characterization of the Schubert varieties by an non-negative integer A and a marked Dynkin diagram. Corollaries include: (1) the variety is smooth if and only if A=0; (2) if G of Type ADE, then the singular locus occurs in codimension at least three.