Derived categories of families of sextic del Pezzo surfaces

TL;DR

This paper develops a semiorthogonal decomposition for the derived categories of flat families of sextic del Pezzo surfaces, revealing their structure via twisted derived categories of finite schemes and employing homological projective duality.

Contribution

It introduces a natural semiorthogonal decomposition for derived categories of sextic del Pezzo surface families, with explicit computations and a modular interpretation of associated schemes.

Findings

Decomposition components correspond to twisted derived categories of schemes of degrees 1, 3, and 2.

Explicit computations for standard families using homological projective duality.

Modular interpretation of the schemes involved.

Abstract

We construct a natural semiorthogonal decomposition for the derived category of an arbitrary flat family of sextic del Pezzo surfaces with at worst du Val singularities. This decomposition has three components equivalent to twisted derived categories of finite flat schemes of degrees 1, 3, and 2 over the base of the family. We provide a modular interpretation for these schemes and compute them explicitly in a number of standard families. For two such families the computation is based on a symmetric version of homological projective duality for $\mathbb{P}^2 \times \mathbb{P}^2$ and $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$, which we explain in an appendix.