Geometry of Hessenberg varieties with applications to Newton-Okounkov bodies

TL;DR

This paper investigates the geometry of Hessenberg varieties in type A, providing explicit generators for their local ideals, establishing their local complete intersection property, and applying these results to compute Newton-Okounkov bodies, especially for the Peterson variety.

Contribution

It introduces explicit generators for local ideals of Hessenberg varieties, proves they are local complete intersections, and develops methods for computing Newton-Okounkov bodies.

Findings

Hessenberg varieties are local complete intersections.

Explicit generators for local ideals are provided.

Newton-Okounkov bodies are computed for the Peterson variety.

Abstract

In this paper, we study the geometry of various Hessenberg varieties in type A, as well as families thereof, with the additional goal of laying the groundwork for future computations of Newton-Okounkov bodies of Hessenberg varieties. Our main results are as follows. We find explicit and computationally convenient generators for the local defining ideals of indecomposable regular nilpotent Hessenberg varieties, and then show that all regular nilpotent Hessenberg varieties are local complete intersections. We also show that certain families of Hessenberg varieties, whose generic fibers are regular semisimple Hessenberg varieties and the special fiber is a regular nilpotent Hessenberg variety, are flat and have reduced fibres. This result further allows us to give a computationally effective formula for the degree of a regular nilpotent Hessenberg variety with respect to a Pl\"ucker embedding. Furthermore, we construct certain flags of subvarieties of a regular nilpotent Hessenberg variety, obtained by intersecting with Schubert varieties, which are suitable for computing Newton-Okounkov bodies. As an application of our results, we explicitly compute many Newton-Okounkov bodies of the two-dimensional Peterson variety with respect to Pl\"ucker embeddings.