Effectivity of Brauer-Manin obstructions. (English) Zbl 1142.14013
The main object of the paper under review is a smooth, projective, geometrically irreducible variety \(X\) defined over a number field \(k\). The authors are interested in computing the Brauer–Manin obstruction to the Hasse principle and weak approximation. They present an efficient general procedure for computing the obstruction related to the algebraic part of the Brauer group (i.e. to the kernel of the map (\({\text{{Br}}}(X)\to {\text{{Br}}}(X\times _k\bar k)\)) under the assumptions that the geometric Picard group \({\text{{Pic}}}(X\times _k\bar k)\) is finitely generated, torsion-free, and explicitly given by generators (a collection of codimension one geometric cycles) and relations with an explicit Galois action. The authors use one of the earliest counter-examples to the Hasse principle (the Cassels–Guy diagonal cubic surface) to illustrate each step of the proposed algorithm.
Reviewer: Boris Kunyavskii (Ramat Gan)
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