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Chamber structure of polarizations and the moduli of stable sheaves on a ruled surface

Kōta Yoshioka

TL;DR

The paper extends Qin's chamber structure to rank $r\ge 3$ on a non-rational ruled surface and develops a wall-crossing framework for stability, yielding a cohomological description of moduli via equivariant techniques. It establishes existence and factoriality results for μ-stable sheaves and derives explicit determinant-line-bundle descriptions that give precise Picard groups of the Gieseker–Maruyama compactifications in terms of Jacobian products and a small free abelian factor; these descriptions depend on the genus and the rank decomposition of the first Chern class along the ruling. By combining chamber analysis, Harder–Narasimhan data, and equivariant intersection theory, the work connects stability chambers to the topology and line bundle structure of moduli spaces, enabling explicit calculations of Betti/Hodge-type information and Picard groups in new non-rational settings. The results provide concrete, rank- and genus-dependent formulas for Picard groups and offer a pathway to Betti-number computations through chamber decompositions on non-rational ruled surfaces.

Abstract

In this paper we shall generalize the chamber structure of polarizations defined by Qin, and as an application we shall compute the Picard groups of moduli spaces of stable sheaves on a non-rational ruled surface.

Chamber structure of polarizations and the moduli of stable sheaves on a ruled surface

TL;DR

The paper extends Qin's chamber structure to rank on a non-rational ruled surface and develops a wall-crossing framework for stability, yielding a cohomological description of moduli via equivariant techniques. It establishes existence and factoriality results for μ-stable sheaves and derives explicit determinant-line-bundle descriptions that give precise Picard groups of the Gieseker–Maruyama compactifications in terms of Jacobian products and a small free abelian factor; these descriptions depend on the genus and the rank decomposition of the first Chern class along the ruling. By combining chamber analysis, Harder–Narasimhan data, and equivariant intersection theory, the work connects stability chambers to the topology and line bundle structure of moduli spaces, enabling explicit calculations of Betti/Hodge-type information and Picard groups in new non-rational settings. The results provide concrete, rank- and genus-dependent formulas for Picard groups and offer a pathway to Betti-number computations through chamber decompositions on non-rational ruled surfaces.

Abstract

In this paper we shall generalize the chamber structure of polarizations defined by Qin, and as an application we shall compute the Picard groups of moduli spaces of stable sheaves on a non-rational ruled surface.

Paper Structure

This paper contains 5 sections, 14 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. chamber structure
  3. Equivariant cohomology of $M_H^{\gamma}$
  4. The existence of stable sheaves.
  5. The Picard group of $\overline{M}_H(r;c_1,c_2)$

Key Result

Lemma 2.1

Let $E$ be a torsion free sheaf which is defined by an extension $0 \to F_1 \to E \to F_2 \to 0$. Then $\Delta(E)=\frac{\operatorname{rk}(F_1)}{\operatorname{rk}(E)}\Delta(F_1)+\frac{\operatorname{rk}(F_2)}{\operatorname{rk}(E)}\Delta(F_2)- \frac{\operatorname{rk}(F_1)\operatorname{rk}(F_2)}{2\opera

Theorems & Definitions (20)

  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.1
  • Definition 2.1
  • Lemma 2.3
  • Definition 2.2
  • Proposition 2.4
  • Lemma 3.1
  • Theorem 3.2
  • Corollary 3.3
  • ...and 10 more