Reduction of Brauer classes on K3 surfaces, rationality and derived equivalence

TL;DR

This paper studies how Brauer classes on K3 surfaces over number fields behave under reduction, revealing implications for rationality and derived equivalence of algebraic varieties.

Contribution

It demonstrates that Brauer classes on very general polarized K3 surfaces become trivial at many places, impacting rationality and derived equivalence upon reduction.

Findings

Brauer classes trivialize at a positive density of places

Certain cubic fourfolds become rational after reduction

Twisted derived equivalent K3 surfaces become derived equivalent after reduction

Abstract

We consider the reduction of Brauer classes on surfaces over number fields, with a view toward applications to rationality and derived equivalence. We show that a Brauer class on a very general polarized K3 surface over a number field becomes trivial upon reduction for a set of places of positive natural density. As a consequence, there are cubic fourfolds which become rational upon reduction for a positive proportion of places, and there are twisted derived equivalent K3 surfaces which become derived equivalent upon reduction for a positive proportion of places.