Enriques involutions and Brauer classes

TL;DR

This paper establishes a correspondence between order 2 elements in the Brauer group of complex Kummer surfaces and Enriques quotients, revealing a deep link between algebraic and geometric structures.

Contribution

It proves that all order 2 Brauer classes on a complex Kummer surface descend to Enriques quotients and characterizes the relationship between these classes and Enriques quotients.

Findings

Every order 2 Brauer class descends to an Enriques quotient.

In generic cases, a bijection exists between Enriques quotients and Brauer classes of order 2.

For some Picard rank 20 K3 surfaces, fibers of the map from Enriques quotients to Brauer classes have uniform order.

Abstract

We prove that every element of order 2 in the Brauer group of a complex Kummer surface X descends to an Enriques quotient of X. In 'generic' cases this gives a bijection between the set Enr(X) of Enriques quotients of X up to isomorphism and the set of Brauer classes of X of order 2. For some K3 surfaces of Picard rank 20 we prove that the fibres of the map from Enr(X) to Br(X)[2] above the non-zero points have the same order.