Astuce de Salberger et z\'ero-cycles sur certaines fibrations

TL;DR

This paper demonstrates that for specific fibrations over curves or projective space, the Brauer-Manin obstruction fully explains failures of the Hasse principle and weak approximation for zero-cycles, under minimal fiber hypotheses.

Contribution

It establishes that the Brauer-Manin obstruction is the only obstruction for zero-cycles on certain fibrations, with hypotheses only on fibers over a generalized Hilbertian set.

Findings

Brauer-Manin obstruction is the only obstruction for zero-cycles.

Results apply to fibrations over curves and projective space.

Minimal fiber hypotheses are sufficient for the main theorems.

Abstract

We prove that the Brauer-Manin obstruction is the only obstruction to the Hasse principle and to the weak approximation for zero-cycles on certain fibrations over a smooth curve or over the projective space. The principal novelty is that the arithmetic hypotheses are supposed only on the fibres over a generalized Hilbertian subset.