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Moduli spaces of stable sheaves over quasi-polarized surfaces, and the relative Strange Duality morphism

Svetlana Makarova

TL;DR

The paper develops a relative moduli theory for slope-stable sheaves over the stack of quasi-polarized surfaces, proving algebraicity and the existence of a relative good moduli space $\\mathcal{M} \ o \\mathcal{K}$ whose fibers are the moduli schemes of stable sheaves for the restricted polarization. It then constructs theta line bundles in families and extends the Marian–Oprea Strange Duality construction to the relative setting, producing a morphism $D: W^\vee \to V$ in the presence of a universal K3 family. Central to the approach are the openness of the stability locus, the descent properties of good morphisms, and the use of the AHHL criterion to glue relative moduli spaces. The work enables a coherent extension of Verlinde-type dualities to the locus of quasi-polarized K3 surfaces, including degree-two elliptic cases, and provides a robust framework for future Fourier–Mukai and descent techniques in families.

Abstract

The main result of the present paper is a construction of relative moduli spaces of stable sheaves over the stack of quasipolarized projective surfaces. For this, we use the theory of good moduli spaces, whose study was initiated by Alper. As a corollary, we extend the relative Strange Duality morphism to the locus of quasipolarized K3 surfaces.

Moduli spaces of stable sheaves over quasi-polarized surfaces, and the relative Strange Duality morphism

TL;DR

The paper develops a relative moduli theory for slope-stable sheaves over the stack of quasi-polarized surfaces, proving algebraicity and the existence of a relative good moduli space whose fibers are the moduli schemes of stable sheaves for the restricted polarization. It then constructs theta line bundles in families and extends the Marian–Oprea Strange Duality construction to the relative setting, producing a morphism in the presence of a universal K3 family. Central to the approach are the openness of the stability locus, the descent properties of good morphisms, and the use of the AHHL criterion to glue relative moduli spaces. The work enables a coherent extension of Verlinde-type dualities to the locus of quasi-polarized K3 surfaces, including degree-two elliptic cases, and provides a robust framework for future Fourier–Mukai and descent techniques in families.

Abstract

The main result of the present paper is a construction of relative moduli spaces of stable sheaves over the stack of quasipolarized projective surfaces. For this, we use the theory of good moduli spaces, whose study was initiated by Alper. As a corollary, we extend the relative Strange Duality morphism to the locus of quasipolarized K3 surfaces.

Paper Structure

This paper contains 10 sections, 17 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Relative moduli spaces of stable sheaves with respect to a quasi-polarization
  3. Constructibility and generization
  4. Technicalities to prove openness
  5. Good morphisms
  6. The good morphism is fiberwise a scheme
  7. Proof of the main theorem
  8. The Strange Duality morphism
  9. Defining theta line bundles
  10. Constructing the Strange Duality morphism

Key Result

Theorem 1.1

Let $\mathcal{K}$ be the moduli stack of quasi-polarized projective surfaces, and let $\mathcal{X}$ be the universal surface with the universal quasi-polarization $\mathcal{H}$. Fix a Chern character $v$ over $\mathcal{X}$. Assume that, pointwise over $\mathcal{K}$, slope stability is equivalent to

Theorems & Definitions (41)

  • Theorem 1.1: Theorems \ref{['Thrm:good_mm']} and \ref{['Thrm:construct_Mc']}
  • Theorem 1.2: Equation \ref{['Eq:SD_mm']}
  • Remark 1.3
  • Theorem : cf. Theorem \ref{['Thrm:construct_Mc']}
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4
  • proof
  • ...and 31 more