On the Brauer group of Enriques surfaces

TL;DR

This paper investigates the Brauer group of complex Enriques surfaces, describing the pull-back of its unique nonzero element to the covering K3 surface and analyzing the moduli space where this element becomes trivial.

Contribution

It provides a detailed description of the Brauer group's nonzero element on Enriques surfaces and characterizes the moduli space regions where this element is trivial.

Findings

The Brauer group of an Enriques surface has a unique nonzero element.

The pull-back of this element to the covering K3 surface is explicitly described.

The locus where the Brauer element is trivial forms a countable union of hypersurfaces in the moduli space.

Abstract

Let S be a complex Enriques surface; it is the quotient of a K3 surface X by a fixed-point-free involution. The Brauer group Br(S) has a unique nonzero element. We describe its pull-back in Br(X), and show that the surfaces S for which it is trivial form a countable union of hypersurfaces in the moduli space of Enriques surfaces.