Computing the intersection of regular Hessenberg varieties with Schubert cells

Abstract

In his PhD thesis Erik Insko gave the conditions in which the intersection of any Schubert cell with a regular nilpotent Hessenberg variety is smooth. In this thesis we relax the nilpotent condition and aim to extend his method to describe regular Hessenberg varieties without the nilpotent restriction. We conclude that one specific intersection is smooth at the origin.