Intersections of Schubert varieties and smooth $T$-stable subvarieties of flag varieties

TL;DR

This paper investigates the smoothness and irreducibility of Hessenberg Schubert varieties, which are closures of Bialynicki-Birula cells in Hessenberg varieties, providing new criteria for their smoothness.

Contribution

It introduces criteria for the smoothness of Hessenberg Schubert varieties by analyzing their intersections with Schubert varieties in flag varieties.

Findings

Identifies conditions for the smoothness of Hessenberg Schubert varieties.

Establishes irreducibility criteria for intersections with Schubert varieties.

Provides a framework to understand singularities of cell closures in Hessenberg varieties.

Abstract

A smooth projective variety with an action of a torus admits a cell decomposition, called the Bialynicki-Birula decomposition. Singularities of the closures of these cells are not well-known. One of the examples of such closures is a Schubert variety in a flag variety G/B, and there are several criteria for the smoothness of Schubert varieties. In this paper, we focus on the closures of Bialynicki-Birula cells in regular semisimple Hessenberg varieties Hess(s,h), called Hessenberg Schubert varieties. We first consider the intersection of the Schubert varieties with Hess(s,h) and investigate the irreducibility and the smoothness of this intersection, from which we get a sufficient condition for a Hessenberg Schubert variety to be smooth.