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Geometric helices on del Pezzo surfaces from tilting

Pierrick Bousseau

Abstract

We prove that all geometric helices in the derived category of coherent sheaves on a del Pezzo surface are related by a sequence of elementary operations: rotation, shifting, orthogonal reordering, derived dualization, tensoring by a line bundle, and tilting. As a consequence, any two non-commutative crepant resolutions of the affine cone over a del Pezzo surface are related by mutations. The proof relies on a geometric interpretation of tilting operations as cluster transformations acting on toric models of a log Calabi--Yau surface mirror to the del Pezzo surface.

Geometric helices on del Pezzo surfaces from tilting

Abstract

We prove that all geometric helices in the derived category of coherent sheaves on a del Pezzo surface are related by a sequence of elementary operations: rotation, shifting, orthogonal reordering, derived dualization, tensoring by a line bundle, and tilting. As a consequence, any two non-commutative crepant resolutions of the affine cone over a del Pezzo surface are related by mutations. The proof relies on a geometric interpretation of tilting operations as cluster transformations acting on toric models of a log Calabi--Yau surface mirror to the del Pezzo surface.
Paper Structure (27 sections, 46 theorems, 102 equations, 1 figure)

This paper contains 27 sections, 46 theorems, 102 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation and main result
  3. Structure of the proof
  4. Related works
  5. Mutations of exceptional collections
  6. Hille--Perling polygons and Hacking bundles
  7. Seiberg duality in physics
  8. Seeds, mutations, and log Calabi--Yau surfaces
  9. Seeds, cyclically ordered seeds, and mutations
  10. Seeds and log Calabi--Yau surfaces
  11. The intersection form
  12. Weyl groups
  13. Seeds of $q$-Painlevé type and T-polygons
  14. Affine Weyl groups and del Pezzo surfaces
  15. Del Pezzo surfaces, Euler form, and Weyl groups
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $Z$ be a del Pezzo surface. Any two geometric helices on $Z$ are related by a sequence of the following operations: rotation, shifting, orthogonal reordering, derived dualization, tensoring by a line bundle, and tilting.

Figures (1)

  • Figure 1: The cyclically oriented vectors $(r(F_i), d(F_i))$ for the very strong exceptional collection $\bE$ on $Z=\bP^1 \times \bP^1$ considered in Example \ref{['example_cyclic']}.

Theorems & Definitions (114)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 2.1
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • ...and 104 more