Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Degeneracy Loci for Smooth Moduli of Sheaves

Yu Zhao

TL;DR

The paper advances Brill–Noether theory on moduli spaces of stable sheaves on a smooth projective surface by studying the degeneracy locus of the universal sheaf and proving that, under suitable conditions, these loci are either empty or irreducible Cohen–Macaulay of an expected dimension. It develops a general two‑term degeneracy theory for morphisms of locally free sheaves, relating degeneracy loci to Grassmannians and their derived structures, and establishing precise dimension bounds and equivalences of emptiness/irreducibility. The main application shows that for a fixed rank $r$, Chern data, and ample polarization, the degeneracy locus over the smooth moduli surface is nonempty exactly when a computable function $g(n,l)$ is nonnegative, with a closed formula for its dimension, thereby generalizing prior rank‑one results and linking to perverse coherent sheaf moduli. The results illuminate the geometry of degeneracy loci in moduli problems and connect Grassmannian geometry with derived structures to yield a coherent picture of when and how these loci populate the moduli spaces.

Abstract

Let S be a smooth projective surface over the complex field. Under certain technical assumptions, we prove that the degeneracy locus of the universal sheaf over the moduli space of stable sheaves is either empty or an irreducible Cohen-Macaulay variety of the expected dimension; we also give a criterion for when the degeneracy locus is nonempty. This result generalizes the work of Bayer, Chen, and Jiang for the Hilbert scheme of points on surfaces. The above statement is a special case of a more general phenomenon: for a two-term complex of locally free sheaves, the geometry of the degeneracy locus is closely related to the geometry of Grassmannians.

The Degeneracy Loci for Smooth Moduli of Sheaves

TL;DR

The paper advances Brill–Noether theory on moduli spaces of stable sheaves on a smooth projective surface by studying the degeneracy locus of the universal sheaf and proving that, under suitable conditions, these loci are either empty or irreducible Cohen–Macaulay of an expected dimension. It develops a general two‑term degeneracy theory for morphisms of locally free sheaves, relating degeneracy loci to Grassmannians and their derived structures, and establishing precise dimension bounds and equivalences of emptiness/irreducibility. The main application shows that for a fixed rank , Chern data, and ample polarization, the degeneracy locus over the smooth moduli surface is nonempty exactly when a computable function is nonnegative, with a closed formula for its dimension, thereby generalizing prior rank‑one results and linking to perverse coherent sheaf moduli. The results illuminate the geometry of degeneracy loci in moduli problems and connect Grassmannian geometry with derived structures to yield a coherent picture of when and how these loci populate the moduli spaces.

Abstract

Let S be a smooth projective surface over the complex field. Under certain technical assumptions, we prove that the degeneracy locus of the universal sheaf over the moduli space of stable sheaves is either empty or an irreducible Cohen-Macaulay variety of the expected dimension; we also give a criterion for when the degeneracy locus is nonempty. This result generalizes the work of Bayer, Chen, and Jiang for the Hilbert scheme of points on surfaces. The above statement is a special case of a more general phenomenon: for a two-term complex of locally free sheaves, the geometry of the degeneracy locus is closely related to the geometry of Grassmannians.
Paper Structure (8 sections, 11 theorems, 42 equations)

This paper contains 8 sections, 11 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. The Brill-Noether theory on the Moduli Space of Stable Sheaves
  3. The Degeneracy Locus and the Grassmannian
  4. Notations
  5. Acknowledgement
  6. Degeneracy Theory for Two-term Complexes
  7. Degeneracy Loci on the Smooth Moduli Surface of Coherent Sheaves over a Surface
  8. Moduli Space of Framed Coherent Sheaves on the Plane and ADHM Construction

Key Result

Theorem 1.1

We consider the following assumptions, which hold for a generic ample divisor when the surface $S$ is a del Pezzo, K3, or abelian surface: where $K_S$ is the canonical line bundle of $S$. If assumA holds, then we have

Theorems & Definitions (20)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Proposition 1.5: Proposition 4.6 of zhao2024hilbertschemesblowingups
  • Definition 2.1: Section 14.4 of MR0732620
  • Proposition 2.2: Theorem 14.4 (b)(c) of MR0732620.
  • Lemma 2.3
  • proof
  • proof : Proof of \ref{['thm:A']}
  • ...and 10 more