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Wall in the stability space of the gluing stability conditions on Hirzebruch surfaces

Yusuke Ohmiya

TL;DR

The paper analyzes how walls in the stability manifold interact with the geometric chamber for Hirzebruch surfaces by employing the gluing stability construction along a fixed semiorthogonal decomposition. It identifies the wall defined by zero gluing perversity and computes its intersection with the divisorial stability boundary, distinguishing four glued types and the corresponding moduli of skyscraper-sheaf objects. Through explicit central charges and Grothendieck-group calculations, it characterizes when the skyscraper sheaves are stable, semistable, or unstable, and describes the resulting coarse or fine moduli spaces (often projective lines, points, or the surface itself). The results illuminate the wall-crossing behavior of skyscraper-moduli under gluing, connect to divisorial stability structures, and yield concrete geometric descriptions of moduli spaces in terms of the Hirzebruch surface data. The methods combine Bridgeland stability, semiorthogonal gluing, and Beilinson-type exceptional collections to produce exact, computable criteria with potential implications for moduli theory on ruled surfaces and wall-crossing phenomena.

Abstract

This paper investigates the wall structure of the space of stability conditions on Hirzebruch surfaces. Using the gluing construction of \cite{CP} and \cite{Uch} with respect to a fixed semiorthogonal decomposition, we focus on two main objectives: observing the intersection of the geometric chamber $U(S) \subset \mathrm{Stab}(Σ_e)$ with the walls of the resulting subspace, and determining the moduli space of $σ$-semistable objects on this subspace.

Wall in the stability space of the gluing stability conditions on Hirzebruch surfaces

TL;DR

The paper analyzes how walls in the stability manifold interact with the geometric chamber for Hirzebruch surfaces by employing the gluing stability construction along a fixed semiorthogonal decomposition. It identifies the wall defined by zero gluing perversity and computes its intersection with the divisorial stability boundary, distinguishing four glued types and the corresponding moduli of skyscraper-sheaf objects. Through explicit central charges and Grothendieck-group calculations, it characterizes when the skyscraper sheaves are stable, semistable, or unstable, and describes the resulting coarse or fine moduli spaces (often projective lines, points, or the surface itself). The results illuminate the wall-crossing behavior of skyscraper-moduli under gluing, connect to divisorial stability structures, and yield concrete geometric descriptions of moduli spaces in terms of the Hirzebruch surface data. The methods combine Bridgeland stability, semiorthogonal gluing, and Beilinson-type exceptional collections to produce exact, computable criteria with potential implications for moduli theory on ruled surfaces and wall-crossing phenomena.

Abstract

This paper investigates the wall structure of the space of stability conditions on Hirzebruch surfaces. Using the gluing construction of \cite{CP} and \cite{Uch} with respect to a fixed semiorthogonal decomposition, we focus on two main objectives: observing the intersection of the geometric chamber with the walls of the resulting subspace, and determining the moduli space of -semistable objects on this subspace.
Paper Structure (26 sections, 37 theorems, 149 equations, 2 figures)

This paper contains 26 sections, 37 theorems, 149 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Bridgeland's works
  3. Geometric chambers and walls
  4. Gluing construction and perversity
  5. Main Result
  6. Moduli space of $\sigma$-stable objects
  7. Intersection of geometric chamber with wall
  8. Outline of this paper
  9. Notation and Conventions
  10. Acknowledgement
  11. Preliminaries
  12. Stability conditions
  13. Gluing construction
  14. Inducing stability conditions
  15. Exceptional collection on triangulated categories
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $\sigma = \sigma_{gl,m}$ be a gluing stability condition of glued type $m$ (Definition typeofgluingstab). Then, we have the following: (Case $m=1$) (Case $m=2,3)$ (Case $m=4$) $\\$ In this case, which necessarily implies ${\rm per}(\sigma)=0$ and $per_i(\sigma)\ne 0$ (see Lemma phaseoflambda1Oxandrho2Ox, Lemma lemmasetofMsigma), $\mathbb{P}^1$ is the coarse moduli space of S-equivalence cla

Figures (2)

  • Figure 1: The region $\overline{S_{div}}$ for glued type $m=1,2$ at a fixed point $(x,y) \in \mathbb{R}^2$, and the wall $\mathcal{W}_{0,m}$ contained within the boundary $\partial_z$.
  • Figure 2: The region $\overline{S_{div}}$ for glued type $m=3$ at a fixed point $(x,y) \in \mathbb{R}^2$. The wall $\mathcal{W}_{0,3}$ intersects the boundary, and touches the vertex of $\overline{S_{div}}$.

Theorems & Definitions (83)

  • Theorem 1.1: Theorem \ref{['Mainthm1']}
  • Theorem 1.2: Theorem \ref{['thm.boundaryofW01']}, Theorem \ref{['thm.boundaryofW02']}, Theorem \ref{['thm.m=3']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7: Br07 Theorem 7.1
  • Definition 2.8: CP § 2
  • ...and 73 more