The singular locus of semisimple Hessenberg varieties

TL;DR

This paper investigates the structure of semisimple Hessenberg varieties, identifying their irreducible components, their smoothness properties, and the nature of their singular loci, with specific examples illustrating exceptions to general patterns.

Contribution

It explicitly describes the irreducible components of semisimple Hessenberg varieties for the standard Hessenberg space and characterizes their intersections and singular loci.

Findings

Irreducible components of standard semisimple Hessenberg varieties are smooth.

The intersections of these components form the singular locus.

Some semisimple Hessenberg varieties are singular and irreducible, contrary to common cases.

Abstract

Although regular semisimple Hessenberg varieties are smooth and irreducible, semisimple Hessenberg varieties are not necessarily smooth in general. In this paper we determine the irreducible components of semisimple Hessenberg varieties corresponding to the standard Hessenberg space. We prove that these irreducible components are smooth and give an explicit description of their intersections, which constitute the singular locus. We conclude with an example of a semisimple Hessenberg variety corresponding to another Hessenberg space which is singular and irreducible, showing that results of this nature do not hold for all semisimple Hessenberg varieties.