Remarques sur un article r'ecent de B. Poonen

Details Remarques sur un article r'ecent de B. Poonen Remarques sur un article r'ecent de B. Poonen

2008-09-08

arxiv.org/abs/0809.1337v1

Mathematics

Number Theory

8 pages

Scientific paper

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.

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