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There are no exotic compact moduli of sheaves on a curve

Andres Fernandez Herrero, Dario Weissmann, Xucheng Zhang

TL;DR

The paper proves that on a genus $g\ge2$ curve, the only nonempty open substack of rank-$r>0$ coherent sheaves with a universally closed adequate moduli space is the classical moduli of slope semistable bundles, encoded by $\mathscr{B}un_r^{d,ss}$ with its moduli space $M_r^d$. It develops a structural theory for sheaves with reductive automorphism groups, showing such sheaves are polysimple, and then uses reductions to bundles, Lange's conjecture, and Luna stratification to rule out exotic compactifications for positive rank under universal closedness. The paper further analyzes what happens when universal closedness is dropped: rank $2$ moduli spaces must be open immersions into the semistable locus (hence are schemes), while rank $3$ provides an explicit example of a non-schematic, separated, non-proper good moduli space arising from an open substack of simple semistable bundles. Overall, the work clarifies the rigidity of moduli for curves, linking reductive automorphisms, stability, and stratifications to show that exotic compactifications cannot occur under the compactness assumption, and it highlights the delicate phenomena that arise without it.

Abstract

We study moduli of coherent sheaves of some given degree and positive rank on a curve. We show that there is only one nonempty open condition on families of sheaves that yields a universally closed adequate moduli space, namely, the one that recovers the classical moduli of slope semistable vector bundles.

There are no exotic compact moduli of sheaves on a curve

TL;DR

The paper proves that on a genus curve, the only nonempty open substack of rank- coherent sheaves with a universally closed adequate moduli space is the classical moduli of slope semistable bundles, encoded by with its moduli space . It develops a structural theory for sheaves with reductive automorphism groups, showing such sheaves are polysimple, and then uses reductions to bundles, Lange's conjecture, and Luna stratification to rule out exotic compactifications for positive rank under universal closedness. The paper further analyzes what happens when universal closedness is dropped: rank moduli spaces must be open immersions into the semistable locus (hence are schemes), while rank provides an explicit example of a non-schematic, separated, non-proper good moduli space arising from an open substack of simple semistable bundles. Overall, the work clarifies the rigidity of moduli for curves, linking reductive automorphisms, stability, and stratifications to show that exotic compactifications cannot occur under the compactness assumption, and it highlights the delicate phenomena that arise without it.

Abstract

We study moduli of coherent sheaves of some given degree and positive rank on a curve. We show that there is only one nonempty open condition on families of sheaves that yields a universally closed adequate moduli space, namely, the one that recovers the classical moduli of slope semistable vector bundles.

Paper Structure

This paper contains 12 sections, 27 theorems, 19 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Sheaves with reductive automorphism groups
  3. Polysimple sheaves
  4. The case of a curve
  5. Universally closed moduli of sheaves on a curve
  6. Reduction to bundles
  7. Reduction to semistable bundles
  8. Luna's stratification for the moduli of bundles
  9. Substacks of the semistable locus
  10. Exotic moduli of bundles
  11. Moduli of rank 2 bundles
  12. Exotic moduli of rank 3 semistable bundles

Key Result

Theorem 1.1

Let $C$ be a smooth projective connected curve of genus at least $2$ over an algebraically closed field $k$. Fix a pair of integers $(d,r)$ such that $r>0$. Then, the stack of semistable vector bundles $\mathop{\mathscr{B}un}\nolimits_r^{d,ss} \subset \mathscr{C}oh_r^d$ is the only nonempty open sub

Theorems & Definitions (61)

  • Theorem 1.1: \ref{['prop: moduli spaces in coh(C)']} + \ref{['thm: proper moduli space is semistable locus']}
  • Remark 1.2
  • Remark 1.3
  • Corollary 1.4
  • Theorem 1.5: = \ref{['thm: main']}
  • Definition 2.1: Polysimple sheaf
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • ...and 51 more