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Zero-cycles of degree 1 on Poonen threefolds. (Zéro-cycles de degré 1 sur les solides de Poonen.) (French) Zbl 1205.11075
The main object of the paper under review is a class of threefolds constructed by B. Poonen [Ann. Math. 171, No. 3, 2157–2169 (2010; Zbl 1284.11096)]. These varieties are defined over a number field \(k\) and do not satisfy the Hasse principle. These counter-examples are not accounted for not only by the Brauer–Manin obstruction (as in the example by A. N. Skorobogatov [Invent. Math. 135, No. 2, 399–424 (1999; Zbl 0951.14013)]) but by any its variant related to a finite étale covering either.
The main result of the paper (Theorem 3.1) states the existence of a zero-cycle of degree 1 on every Poonen’s threefold. This thus confirms an earlier conjecture of the author [Algebraic \(K\)-theory. Proceedings of an AMS-IMS-SIAM summer research conference, Seattle, WA, USA, July 13-24, 1997. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 67, 1–12 (1999; Zbl 0981.14003)] that the Brauer–Manin obstruction to the existence of zero-cycles of degree 1 on a smooth, geometrically integral, projective variety is the only one. (Note that the existence of a zero-cycle of degree 1 on Skorobogatov’s surface is not known.)
The proof is based on a variant of the fibration method [cf. D. Harari, Duke Math. J. 75, No. 1, 221–260 (1994; Zbl 0847.14001)] adjusted to the zero-cycle setting.

MSC:
11G35 Varieties over global fields
14F22 Brauer groups of schemes
14G05 Rational points
14G25 Global ground fields in algebraic geometry
14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties
14J30 \(3\)-folds
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