Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields

TL;DR

This paper explores the birational properties of del Pezzo surfaces over arbitrary fields using derived categories, providing criteria for rationality and explicit semiorthogonal decompositions linked to algebraic invariants.

Contribution

It introduces a derived categorical criterion for the rationality of del Pezzo surfaces over any field and constructs explicit semiorthogonal decompositions for degrees ≥5.

Findings

Derived category decomposes into zero-dimensional components for rational surfaces.

Explicit semiorthogonal decompositions for degree ≥5 del Pezzo surfaces.

Retrieval of surface index from Brauer and Chern classes.

Abstract

We study the birational properties of geometrically rational surfaces from a derived categorical point of view. In particular, we give a criterion for the rationality of a del Pezzo surface over an arbitrary field, namely, that its derived category decomposes into zero-dimensional components. For del Pezzo surfaces of degree at least 5, we construct explicit semiorthogonal decompositions by subcategories of modules over semisimple algebras arising as endomorphism algebras of vector bundles and we show how to retrieve information about the index of the surface from Brauer classes and Chern classes associated to these vector bundles.