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Bridgeland walls destabilizing one-dimensional space sheaves

Daniel Bernal, Cristian Martinez

TL;DR

This paper analyzes Bridgeland stability for 1-dimensional sheaves on $\mathbb{P}^3$ via the Bayer–Macrì–Toda double-tilt framework, focusing on the $(_{\beta},\alpha,s)$-slice and the twisted Chern character $v=(-R,0,D,0)$. It derives general numerical wall conditions, proves the existence of actual walls, and provides sharp bounds for the largest numerical wall, yielding first Gieseker-chamber bounds for a threefold when $R=0$. An algorithmic approach is used to study $D=3,4$ with $\beta\neq0$, identifying explicit walls that bound the Gieseker chamber and connecting destabilizing objects to pushforwards from $\mathbb{P}^2$ and to the structure of $\mathcal{M}_{dt}$, the moduli of 1-dimensional Gieseker semistable space sheaves. The results offer a concrete description of wall-crossing on threefolds and bridge the numerical wall data with the geometry of the corresponding moduli spaces. They also clarify how the maximal numerical wall controls the transition to the Gieseker moduli, and provide computational tools for exploring unstable configurations in low-degree cases.

Abstract

Following the setup proposed by Jardim-Maciocia-Martinez in the case of the projective space, we study some numerical and actual Bridgeland walls for the (twisted) Chern character $v=(-R,0,D,0)$ in certain half-plane of stability conditions, where walls are nested and finite. We give bounds for the largest numerical wall that may appear. When $R=0$, these bounds in particular produce the first known bounds for the Gieseker chamber in the case of a threefold. We also study the cases $R=0$ and $D=3,4$ in detail using a small algorithm in Python.

Bridgeland walls destabilizing one-dimensional space sheaves

TL;DR

This paper analyzes Bridgeland stability for 1-dimensional sheaves on via the Bayer–Macrì–Toda double-tilt framework, focusing on the -slice and the twisted Chern character . It derives general numerical wall conditions, proves the existence of actual walls, and provides sharp bounds for the largest numerical wall, yielding first Gieseker-chamber bounds for a threefold when . An algorithmic approach is used to study with , identifying explicit walls that bound the Gieseker chamber and connecting destabilizing objects to pushforwards from and to the structure of , the moduli of 1-dimensional Gieseker semistable space sheaves. The results offer a concrete description of wall-crossing on threefolds and bridge the numerical wall data with the geometry of the corresponding moduli spaces. They also clarify how the maximal numerical wall controls the transition to the Gieseker moduli, and provide computational tools for exploring unstable configurations in low-degree cases.

Abstract

Following the setup proposed by Jardim-Maciocia-Martinez in the case of the projective space, we study some numerical and actual Bridgeland walls for the (twisted) Chern character in certain half-plane of stability conditions, where walls are nested and finite. We give bounds for the largest numerical wall that may appear. When , these bounds in particular produce the first known bounds for the Gieseker chamber in the case of a threefold. We also study the cases and in detail using a small algorithm in Python.

Paper Structure

This paper contains 14 sections, 19 theorems, 51 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. The double-tilt construction on threefolds
  5. The preferred slice and numerical constraints
  6. Moduli spaces of 1-dimensional space sheaves
  7. General walls and numerical conditions
  8. Numerical walls for beta not zero
  9. Bounding the Gieseker wall
  10. Low degree examples
  11. Degree 3
  12. Degree 4
  13. Code
  14. Numerical computations

Key Result

Theorem 1.1

Let $X=\mathbb{P}^3$ and $\mathop{\mathrm{ch}}\nolimits^\beta= (0,0,D,0)$. Then

Figures (2)

  • Figure 1: The actual walls $\alpha_0=1,\sqrt{7}$ for $v=(0,0,3,0)$ in orange, and the quiver region shadowed in light orange.
  • Figure 2: The general figure of walls in $\beta=0$. The red curve represents the killing wall and the blue curve represents the maximal numerical wall, after which the Bridgeland moduli space is the Gieseker moduli space, and in the green region the destabilizing objects have rank $0$.

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Example 3.3
  • Remark 3.4
  • Proposition 4.1
  • ...and 30 more