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When are splitting loci Gorenstein?

Feiyang Lin

TL;DR

The paper classifies when splitting loci $\overline{\Sigma}_{\vec{e}}$ inside the stack $\mathcal{B}_{r,d}$ of vector bundles on $\mathbb{P}^1$ are Gorenstein or $\mathbb{Q}$-Gorenstein. It develops a systematic framework based on the $A^1$-class group, a canonical-module adjunction formula, and a torus action on extension spaces to control the grading of splitting-locus ideals. A core result is that $\overline{\Sigma}_{\vec{e}}$ is Gorenstein if and only if $\vec{e}$ is a block arithmetic progression with difference $0,1$ or $2$, or is contiguous; it is $\mathbb{Q}$-Gorenstein (but not Gorenstein) precisely when $\vec{e}$ is an arithmetic progression with difference $t\ge 3$, with the smallest $N$ equal to $t$ if $t$ is odd and $t/2$ if $t$ is even. The method unifies codimension-one Chow calculations, an explicit excision sequence for $A^1$, and a detailed canonical-class computation, and also aligns with known results about Hankel determinantal loci in low rank. These results supply a sharp structural understanding of singularities for splitting loci and have implications for Hurwitz-Brill-Noether-type problems and potential rational-singularity questions in the tame-to-wild transition regime. Overall, the work provides a complete, computable criterion for Gorenstein and $\mathbb{Q}$-Gorenstein singularities in this geometric context, with broader relevance to moduli problems for vector bundles on curves.

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.

When are splitting loci Gorenstein?

TL;DR

The paper classifies when splitting loci inside the stack of vector bundles on are Gorenstein or -Gorenstein. It develops a systematic framework based on the -class group, a canonical-module adjunction formula, and a torus action on extension spaces to control the grading of splitting-locus ideals. A core result is that is Gorenstein if and only if is a block arithmetic progression with difference or , or is contiguous; it is -Gorenstein (but not Gorenstein) precisely when is an arithmetic progression with difference , with the smallest equal to if is odd and if is even. The method unifies codimension-one Chow calculations, an explicit excision sequence for , and a detailed canonical-class computation, and also aligns with known results about Hankel determinantal loci in low rank. These results supply a sharp structural understanding of singularities for splitting loci and have implications for Hurwitz-Brill-Noether-type problems and potential rational-singularity questions in the tame-to-wild transition regime. Overall, the work provides a complete, computable criterion for Gorenstein and -Gorenstein singularities in this geometric context, with broader relevance to moduli problems for vector bundles on curves.

Abstract

Splitting loci are certain natural closed substacks of the stack of vector bundles on , which have found interesting applications in the Brill-Noether theory of -gonal curves. In this paper, we completely characterize when splitting loci, as algebraic stacks, are Gorenstein or -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.

Paper Structure

This paper contains 10 sections, 15 theorems, 60 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A Torus Action On Successive Extension Spaces
  4. Gorenstein and N-Gorenstein Algebraic Stacks
  5. Reduction to a class calculation
  6. Some class calculations
  7. Codimension One Chow Group of Universal Splitting Loci
  8. Proof of Theorem \ref{['thm:excision']}
  9. The Class of the Canonical Module
  10. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.2

A splitting locus $\mathbf{\overline{\Sigma}}_{\vec{e}}$ is Gorenstein if and only if $\vec{e}$ is a block arithmetic progression of difference $0, 1$ or $2$, or if $\vec{e}$ is contiguous. $\mathbf{\overline{\Sigma}}_{\vec{e}}$ is $\mathbb{Q}$-Gorenstein (but not Gorenstein) if and only if it is an

Figures (1)

  • Figure 1: Building $\operatorname{Ext}^1(O(D), \mathcal{O}(-M)^{\oplus r-1})$ from $\mathcal{B}_{r,d}$

Theorems & Definitions (31)

  • Example 1.1
  • Theorem 1.2
  • Theorem 1.3: Theorem \ref{['prop:classCanonical']}
  • Theorem 1.4
  • Remark 1.5
  • Proposition 2.1
  • proof
  • Example 2.2
  • Example 2.3
  • Lemma 2.4
  • ...and 21 more