Remarques sur un article r'ecent de B. Poonen

TL;DR

This paper discusses smooth threefolds over number fields that lack rational points and are not obstructed by the Brauer-Manin obstruction, showing they possess zero-cycles of degree one, thus contributing to understanding rational points and zero-cycles.

Contribution

It demonstrates that the specific threefolds constructed by Poonen have zero-cycles of degree one, providing new insights into their rational points and obstructions.

Findings

These threefolds have zero-cycles of degree 1.

They lack rational points despite no Brauer-Manin obstruction.

The results impact the study of rational points and zero-cycles.

Abstract

B. Poonen recently produced smooth threefolds over a number field which do not have a rational point but have no Brauer-Manin obstruction even after descent to a finite 'etale cover. In this note I show that the varieties he produces have zero-cycles of degree 1. ----- B. Poonen a r'ecemment exhib'e des exemples de vari'et'es projectives et lisses de dimension 3 sur un corps de nombres qui n'ont pas de point rationnel et pour lesquelles il n'y a pas d'obstruction de Brauer-Manin apr`es rev^etement fini 'etale. Dans cette note, je montre que les vari'et'es qu'il construit poss`edent des z'ero-cycles de degr'e 1.