Bad reduction of the Brauer-Manin obstruction

TL;DR

This paper investigates how the Brauer group of smooth varieties over p-adic fields relates to the geometry of their special fibers, impacting the understanding of rational points and the Brauer-Manin obstruction.

Contribution

It establishes a connection between the Brauer group and special fiber geometry, providing criteria for evaluation maps and applications to the Brauer-Manin obstruction over number fields.

Findings

The Brauer group is determined by the singularity type of the reduction.

Evaluation maps can be constant or surjective depending on the geometry.

Surjectivity of evaluation maps can imply the vanishing of the Brauer-Manin obstruction.

Abstract

We relate the Brauer group of a smooth variety over a p-adic field to the geometry of the special fibre of a regular model, using the purity theorem in \'etale cohomology. As an illustration, we describe how the Brauer group of a smooth del Pezzo surface is determined by the singularity type of its reduction. We then relate the evaluation of an element of the Brauer group to the existence of points on certain torsors over the special fibre; we use this to describe situations when the evaluation is constant, and situations when the evaluation is surjective. In the latter case, we describe how this surjectivity can be used to prove vanishing of the Brauer-Manin obstruction on varieties over number fields.