Principe local-global pour les z\'ero-cycles sur certaines fibrations au-dessus de l'espace projectif

TL;DR

This paper investigates the local-global principle for zero-cycles on varieties fibered over projective space, demonstrating that the Brauer-Manin obstruction is the only barrier to the Hasse principle and weak approximation in specific cases.

Contribution

It establishes the sufficiency of the Brauer-Manin obstruction for zero-cycles on certain fibrations over projective space, extending previous results to new classes of varieties.

Findings

Brauer-Manin obstruction is the only obstruction for zero-cycles on these fibrations.

Proves the Hasse principle and weak approximation hold under this obstruction.

Results apply to Severi-Brauer and Châtelet-surface bundles over projective space.

Abstract

We study the local-global principle for zero-cycles of degree 1 on certain varieties fibered over the projective space. Among other applications, we prove that the Brauer-Manin obstruction is the only obstruction to the Hasse principle and weak approximation for zero-cycles of degree 1 on Severi-Brauer-variety bundles or Ch\^atelet-surface bundles over the projective space.