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A categorical Torelli theorem for quartic del Pezzo surfaces

Alexey Elagin

Abstract

We solve categorical Torelli problem for quartic del Pezzo surfaces. That is, we prove that a del Pezzo surface of degree $4$ can be canonically reconstructed from its Kuznetsov component, which is the orthogonal subcategory to the structure sheaf in the derived category of the surface. Our methods work in equivariant setting and over arbitrary perfect fields. Using recent theory of atomic semi-orthogonal decompositions arXiv:2512.05064, we conclude that two minimal quartic del Pezzo surfaces are birational if and only if they are isomorphic. We also verify that the Kuznetsov component of a minimal quartic del Pezzo surface is semi-orthogonally indecomposable, confirming a conjecture by Auel and Bernardara.

A categorical Torelli theorem for quartic del Pezzo surfaces

Abstract

We solve categorical Torelli problem for quartic del Pezzo surfaces. That is, we prove that a del Pezzo surface of degree can be canonically reconstructed from its Kuznetsov component, which is the orthogonal subcategory to the structure sheaf in the derived category of the surface. Our methods work in equivariant setting and over arbitrary perfect fields. Using recent theory of atomic semi-orthogonal decompositions arXiv:2512.05064, we conclude that two minimal quartic del Pezzo surfaces are birational if and only if they are isomorphic. We also verify that the Kuznetsov component of a minimal quartic del Pezzo surface is semi-orthogonally indecomposable, confirming a conjecture by Auel and Bernardara.

Paper Structure

This paper contains 14 sections, 30 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Conventions
  4. Preliminaries
  5. Coxeter groups of type $B_5$ and $D_5$
  6. Weighted projective lines and their autoequivalences
  7. Quartic del Pezzo surfaces and their automorphisms
  8. Orthogonal to the structure sheaf on quartic del Pezzo surfaces
  9. Separable functors
  10. Main results
  11. Algebraically closed field case
  12. Arbitrary perfect field case
  13. An application to birationality
  14. Indecomposability of the orthogonal subcategory

Key Result

Theorem A

Let $\Bbbk$ be an algebraically closed field of $\mathop{\mathrm{\mathrm{char}}}\nolimits\ne 2$ and $G$ be a group. Then the natural functor from the category of $G$-del Pezzo surfaces of degree $4$ over $\Bbbk$ and $G$-isomorphisms to the category of triangulated $\Bbbk$-linear categories with a $G

Theorems & Definitions (62)

  • Theorem A: Theorem \ref{['th_mainG']}
  • Theorem B: Theorem \ref{['theorem_main-perfect']}
  • Theorem C: Theorem \ref{['th_birational-perfect']}, see also ElaginSchneiderShinder
  • Theorem D: Theorems \ref{['th_indecomposability']}, \ref{['th_indecomposability2']}
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 52 more