On Brauer groups of double covers of ruled surfaces

Brendan Creutz, Bianca Viray

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 Néron-Severi group of X. The path to this result naturally involves a study of the 2-torsion Brauer classes of a smooth double cover of the projective line, yielding results of independent interest. Arithmetic applications are given for both curves and surfaces.

Keywords: Brauer group, Rational points.

AMS Subject Classification: Primary 14F22; secondary 14G05.