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Brauer-Manin obstruction for zero-cycles on certain varieties. (English) Zbl 1525.11070

Let \(X\) be a smooth projective variety over a number field \(K\). The Brauer-Manin obstruction for \(X\) consists of a set \(X(\mathbb{A}_K)^{\operatorname{Br}(X)}\) sitting in between \(X(K)\) and \(X(\mathbb{A}_K)\), which when empty may explain the failure of the local-global principle for \(X\). Instead of studying rational points, one may consider the analogous construction for zero-cycles of a given degree. Let \(\operatorname{Hyp}(d)\) denote the assertion that the Brauer-Manin set for degree \(d\) cycles is nonempty, and let \((*)\) denote the assertion that the ordinary Brauer-Manin set is empty. Clearly \((*)\) implies \(\operatorname{Hyp}(d)\); the main result of this paper is that \(\operatorname{Hyp}(d)\) implies \((*)\) in the following cases:
(i) \(d\) is an odd integer and \(X\) is a del Pezzo surface of degree 4 or a Châtelet surface,
(ii) \(d\) is coprime to \(3\) and \(X\) is a cubic surface,
(iii) \(X\) is a rationally connected variety satisfying certain additional conditions at some places, \(X\) has a zero-cycle of degree \(n\), and \(d\) is coprime to \(n\).
The proofs of these results use a variety of techniques but are not particularly long.
Finally, the paper gives numerous explicit examples of varieties for which the Brauer-Manin obstruction exists for zero-cycles of various degrees, as well as some counterexamples to show the strength of the results. These include del Pezzo surface, K3 surfaces, a threefold, and curves.

MSC:

11G25 Varieties over finite and local fields
11G35 Varieties over global fields
14F22 Brauer groups of schemes
14J28 \(K3\) surfaces and Enriques surfaces
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