On Brauer groups of double covers of ruled surfaces

Let be a smooth double cover of a geometrically ruled surface defined over a separably closed field of characteristic different from . The main result of this paper is a finite presentation of the -torsion in the Brauer group of with generators given by central simple algebras over the function field of and relations coming from the N,ron-Severi group of . 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.