Descent and étale-Brauer obstructions for 0-cycles

Abstract

We fill a gap in the theory of obstructions for 0-cycles by defining, in this context, an analogue of the classical descent set for rational points on varieties over number fields. This leads to, among other things, a definition of the étale-Brauer obstruction set for 0-cycles, which we show is contained in the Brauer-Manin set. 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.