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 \(X\). 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.
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Notes
A Magma [3] script verifying these claims and all other computational claims in this section can be found with the arXiv distribution of this article.
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Acknowledgments
Bianca Viray would like to thank Dan Abramovich, Asher Auel, Jean-Louis Colliot-Thélène, and Bjorn Poonen for helpful conversations.
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B. Viray was partially supported by NSF Grant DMS-1002933.
