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Bridgeland Stability of Sheaves on del Pezzo Surface of Picard Rank Three

Yuki Mizuno, Tomoki Yoshida

TL;DR

This work analyzes Bridgeland stability for sheaves on a del Pezzo surface of Picard rank three (the blow-up of P^2 at two general points). It develops the divisorial stability framework, proves that destabilizing subobjects of line bundles reduce to twists of the structure sheaf, and establishes that O(E)|_E is stable for all divisorial stability conditions along any (-1)-curve E. The approach combines wall-crossing geometry (notably left hyperbola walls), Bertram’s nested-wall lemmas, and an induction on subobject rank to extend rank-one results to higher ranks. These contributions extend established results for lower Picard ranks and set the stage for moduli-space projectivity questions in the Picard rank three setting.

Abstract

This article discusses the Bridgeland stability of some sheaves on the blow-up of $\mathbb{P}^{2}$ at two general points. We have determined the destabilizing objects of the line bundles and have shown that $\mathscr{O}(E)|_{E}$ is Bridgeland stable for any $(-1)$-curve $E$ and any divisorial Bridgeland stability condition.

Bridgeland Stability of Sheaves on del Pezzo Surface of Picard Rank Three

TL;DR

This work analyzes Bridgeland stability for sheaves on a del Pezzo surface of Picard rank three (the blow-up of P^2 at two general points). It develops the divisorial stability framework, proves that destabilizing subobjects of line bundles reduce to twists of the structure sheaf, and establishes that O(E)|_E is stable for all divisorial stability conditions along any (-1)-curve E. The approach combines wall-crossing geometry (notably left hyperbola walls), Bertram’s nested-wall lemmas, and an induction on subobject rank to extend rank-one results to higher ranks. These contributions extend established results for lower Picard ranks and set the stage for moduli-space projectivity questions in the Picard rank three setting.

Abstract

This article discusses the Bridgeland stability of some sheaves on the blow-up of at two general points. We have determined the destabilizing objects of the line bundles and have shown that is Bridgeland stable for any -curve and any divisorial Bridgeland stability condition.

Paper Structure

This paper contains 23 sections, 23 theorems, 98 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Bridgeland Stability Conditions
  3. Main Theorem
  4. Related and future works
  5. Preliminaries on Bridgeland Stability Conditions
  6. Bridgeland Stability Conditions
  7. Stability conditions on Surfaces
  8. Walls in the Spaces of the Bridgeland Stability Conditions
  9. Some Geometries of del Pezzo Surfaces
  10. Divisorial Stability Conditions on Surface of Picard Rank Three
  11. Spaces of Bridgeland Stability Consitions
  12. Bridgeland Stability of Line Bundles
  13. Rank 1 case
  14. Settings and properties
  15. The case of $z<0$
  16. ...and 8 more sections

Key Result

Theorem B

The Conjecture: stability conjecture is true for the del Pezzo surface of Picard rank $3$. Moreover, $C$ is either $E_1$, $E_2$, or $E$, which are the $(-1)$-curves.

Figures (4)

  • Figure 1: The case $x+y+z=0$
  • Figure 2: Right Hyperbola
  • Figure 3: Left Hyperbola
  • Figure 4: Walls $\mathcal{W}^{0}_{H, G}(\mathscr{O}(-E_i), \mathscr{O})$

Theorems & Definitions (54)

  • Conjecture A: arcara_miles_2016_bridgeland_stability_of_line_bundles_on_surfaces
  • Theorem B: See \ref{['Theorem: stability of line bundle']} and \ref{['proposition: stability of shifted sheaf']}
  • Proposition C: See \ref{['proposition: stability of torsion sheaf']}
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • ...and 44 more