On Brauer groups of double covers of ruled surfaces

TL;DR

This paper provides a finite presentation of the 2-torsion in the Brauer group of certain double covers of ruled surfaces, with applications to understanding rational points on Enriques surfaces.

Contribution

It offers a new explicit description of the 2-torsion Brauer group for these surfaces, including generators and relations, and applies this to Enriques surfaces.

Findings

Finite presentation of 2-torsion Brauer group generators.

Explicit central simple algebra representatives for Brauer classes.

Application to rational points on Enriques surfaces.

Abstract

Let X be a smooth double cover of a geometrically ruled surface defined over a separably closed field of characteristic different from 2. The main result of this paper is a finite presentation of the 2-torsion in the Brauer group of X with generators given by central simple algebras over the function field of X and relations coming from the N\'eron-Severi group of X. In particular, the result gives a central simple algebra representative for the unique nontrivial Brauer class on any Enriques surface. An example demonstrating the applications to the study of rational points is given.