Boundedness of the p-primary torsion of the Brauer groups of K3 surfaces

TL;DR

This paper proves that the transcendental Brauer group of a K3 surface over a finitely generated field is finite unless in positive characteristic with supersingularity, where it is p-torsion.

Contribution

It establishes the finiteness and p-torsion properties of the transcendental Brauer group for K3 surfaces over finitely generated fields.

Findings

Transcendental Brauer group is finite over characteristic zero fields.

In positive characteristic, the group is annihilated by p if the surface is supersingular.

The result clarifies the structure of Brauer groups in different characteristics.

Abstract

We prove that the transcendental Brauer group of a K3 surface X over a finitely generated field k is finite, unless k has positive characteristic p and X is supersingular, in which case it is annihilated by p.