On the Gorensteinization of Schubert Varieties via Boundary Divisors

TL;DR

This paper introduces a one-step Gorensteinization process for Schubert varieties through boundary divisor blow-ups, utilizing degenerations to affine toric schemes to establish Gorenstein properties.

Contribution

It presents a novel Gorensteinization method for Schubert varieties by leveraging degenerations to toric schemes and boundary blow-ups, extending Gorenstein properties to general cases.

Findings

The blow-up along the boundary divisor in the toric degeneration is Gorenstein.

A degeneration of the blow-up of Kazhdan-Lusztig varieties to a Gorenstein scheme is constructed.

The method potentially characterizes the non-Gorenstein locus of Schubert varieties.

Abstract

We will describe a one-step "Gorensteinization" process for a Schubert variety by blowing-up along its boundary divisor. The local question involves Kazhdan-Lusztig varieties which can be degenerated to affine toric schemes defined using the Stanley-Reisner ideal of a subword complex. The blow-up along the boundary in this toric case is in fact Gorenstein. We show that there exists a degeneration of the blow-up of the Kazhdan-Lusztig variety to this Gorenstein scheme, allowing us to extend this result to Schubert varieties in general. The potential use of this one-step Gorensteinization to describe the non-Gorenstein locus of Schubert varieties is discussed, as well as the relationship between Gorensteinizations and the convergence of the Nash blow-up process in the toric case.