A categorical Torelli theorem for quartic del Pezzo surfaces

TL;DR

This paper establishes a categorical Torelli theorem for quartic del Pezzo surfaces, showing they can be reconstructed from their Kuznetsov component, with implications for birationality and indecomposability.

Contribution

It proves that quartic del Pezzo surfaces are uniquely determined by their Kuznetsov components and confirms their semi-orthogonal indecomposability, extending Torelli-type results.

Findings

Surfaces are reconstructible from their Kuznetsov components.

Two minimal quartic del Pezzo surfaces are birational iff they are isomorphic.

Kuznetsov component of such surfaces is semi-orthogonally indecomposable.

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.