On singularity and normality of regular nilpotent Hessenberg varieties

TL;DR

This paper provides a combinatorial criterion to identify singular points in regular nilpotent Hessenberg varieties and characterizes when these varieties are normal, enhancing understanding of their geometric properties.

Contribution

It introduces a new combinatorial characterization for singular points and normality in regular nilpotent Hessenberg varieties, linking combinatorics with geometric properties.

Findings

Identified combinatorial conditions for singular points.

Characterized normality of Hessenberg varieties.

Connected permutation flags with geometric singularities.

Abstract

Regular nilpotent Hessenberg varieties form an important family of subvarieties of the flag variety, which are often singular and sometimes not normal varieties. Like Schubert varieties, they contain distinguished points called permutation flags. In this paper, we give a combinatorial characterization for a permutation flag of a regular nilpotent Hessenberg variety to be a singular point. We also apply this result to characterize regular nilpotent Hessenberg varieties which are normal algebraic varieties.