Hessenberg varieties of codimension one in the flag variety

TL;DR

This paper investigates the geometric and topological properties of codimension one Hessenberg varieties in type A flag varieties, providing formulas, characterizations, and new insights into their structure and singularities.

Contribution

It introduces a formula for the Poincaré polynomial, characterizes irreducibility, and shows all such varieties are reduced schemes, with a novel point-counting method for their Poincaré polynomials.

Findings

Derived a formula for the Poincaré polynomial of codimension one Hessenberg varieties.

Characterized when these varieties are irreducible.

Demonstrated that all codimension one Hessenberg varieties are reduced schemes.

Abstract

We study geometric and topological properties of Hessenberg varieties of codimension one in the type A flag variety. Our main results: (1) give a formula for the Poincar\'e polynomial, (2) characterize when these varieties are irreducible, and (3) show that all are reduced schemes. We prove that the singular locus of any nilpotent codimension one Hessenberg variety is also a Hessenberg variety. A key tool in our analysis is a new result applying to all (type A) Hessenberg varieties without any restriction on codimension, which states that their Poincar\'e polynomials can be computed by counting the points in the corresponding variety defined over a finite field. The results below were originally motivated by work of the authors in [arXiv:2107.07929] studying the precise relationship between Hessenberg and Schubert varieties, and we obtain a corollary extending the results from that paper to all codimension one (type A) Schubert varieties.