When are splitting loci Gorenstein?

TL;DR

This paper characterizes when splitting loci, important in vector bundle theory on projective lines, are Gorenstein or $ ext{Q}$-Gorenstein, using class group computations and canonical module formulas.

Contribution

It provides a complete characterization of the Gorenstein and $ ext{Q}$-Gorenstein properties of splitting loci as algebraic stacks, a novel result in this area.

Findings

Determined conditions for splitting loci to be Gorenstein.

Computed class groups of splitting loci in affine extension spaces.

Derived formulas for the canonical module of splitting loci.

Abstract

Splitting loci are certain natural closed substacks of the stack of vector bundles on $\mathbb{P}^1$, which have found interesting applications in the Brill-Noether theory of $k$-gonal curves. In this paper, we completely characterize when splitting loci, as algebraic stacks, are Gorenstein or $\mathbb{Q}$-Gorenstein. The main ingredients of the proof are a computation of the class groups of splitting loci in certain affine extension spaces, and a formula for the class of their canonical module.