Applications of the fibration method for zero-cycles to the Brauer-Manin obstruction to the existence of zero-cycles on certain varieties

TL;DR

This paper explores the use of the fibration method to analyze the Brauer-Manin obstruction for zero-cycles on specific varieties over number fields, extending previous results to cases with non-finite geometric Brauer groups.

Contribution

It generalizes existing results by removing the finiteness assumption on the geometric Brauer group of the generic fibre in fibrations over the projective line.

Findings

Established conditions under which the Brauer-Manin obstruction explains the absence of zero-cycles.

Extended the fibration method to varieties with non-finite geometric Brauer groups.

Provided new insights into the relationship between obstructions for fibres and total varieties.

Abstract

We study the Brauer-Manin obstruction to the existence of zero-cycles of degree $d$ on certain classes of varieties over number fields. We generalise existing results in the literature and prove some results about fibrations over the projective line, where the geometric Brauer group of the generic fibre is not assumed to be finite. The idea is to assume that the Brauer-Manin obstruction to the Hasse principle is the only one for certain fibres and then deduce analogous results for zero-cycles.