Finiteness of Brauer groups of K3 surfaces in characteristic 2

TL;DR

This paper proves the finiteness of the cokernel of the Brauer group map for K3 surfaces over fields of characteristic 2, extending known results to this special case using recent advances in algebraic geometry.

Contribution

It establishes the finiteness of the Brauer group cokernel for K3 surfaces in characteristic 2, utilizing recent Kuga-Satake and Tate conjecture results.

Findings

Finiteness of the Brauer group cokernel in characteristic 2.

Extension of previous results from characteristic not 2.

Application of recent Kuga-Satake and Tate conjecture results.

Abstract

For a K3 surface over a field of characteristic 2 which is finitely generated over its prime subfield, we prove that the cokernel of the natural map from the Brauer group of the base field to that of the K3 surface is finite modulo the 2-primary torsion subgroup. In characteristic different from 2, such results were previously proved by A. N. Skorobogatov and Y. G. Zarhin. We basically follow their methods with an extra care in the case of superspecial K3 surfaces using the recent results of W. Kim and K. Madapusi Pera on the Kuga-Satake construction and the Tate conjecture for K3 surfaces in characteristic 2.