Descent and \'etale-Brauer obstructions for 0-cycles

TL;DR

This paper introduces an analogue of the descent set and the étale-Brauer obstruction for 0-cycles on varieties over number fields, extending tools from rational point arithmetic to 0-cycles and applying them to various algebraic surfaces and torsors.

Contribution

It defines the étale-Brauer obstruction set for 0-cycles, shows its relation to the Brauer-Manin set, and adapts techniques from rational point arithmetic to the study of 0-cycles.

Findings

The étale-Brauer obstruction set for 0-cycles is contained in the Brauer-Manin set.

Tools from rational point arithmetic are successfully transferred to 0-cycles.

Applications include new insights into the arithmetic of Enriques surfaces and torsors.

Abstract

For 0-cycles on a variety over a number field, we define an analogue of the classical descent set for rational points. This leads to, among other things, a definition of the \'etale-Brauer obstruction set for 0-cycles, which we show is contained in the Brauer-Manin set and is compatible with Suslin's singular homology of degree 0. We then transfer some tools and techniques used to study the arithmetic of rational points into the setting of 0-cycles. For example, we extend the strategy developed by Y. Liang, relating the arithmetic of rational points over finite extensions of the base field to that of 0-cycles, to torsors. We give applications of our results to study the arithmetic behaviour of 0-cycles for Enriques surfaces, torsors given by (twisted) Kummer varieties, universal torsors, and torsors under tori.