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. / Thesis / Master of Science (MSc)

A Hessenberg variety is a subvariety of the flag variety parametrized by two maps: a Hessenberg function on $[n]$ and a linear map on $\C^n$. We study regular nilpotent Hessenberg varieties in Lie type A by focusing on the Hessenberg function $h=(n-1,n,\ldots,n)$. We first state a formula for the $f^w_{n,1}$ which generates the local defining ideal $J_{w,h}$ for any $w\in\Ss_n$. Second, we prove that there exists a convenient monomial order so that $\lead(J_{w,h})$ is squarefree. As a consequence, we conclude that each codimension-1 regular nilpotent Hessenberg variety is locally Frobenius split (in positive characteristic). / Thesis / Master of Science (MSc)