The Brauer group of torsors and its arithmetic applications.
The main object of the paper under review is a torsor \(f\colon Y\to X\) under an algebraic torus \(T\). In the first part the authors are interested in the study of the cokernel of the natural map \(f^*:\text{Br}(X)\to \text{Br}(Y)\). The main result of this part, Theorem 1.7, describes this cokernel in the case where \(k\) is a field of characteristic zero such that \(H^3(k,\overline k^*)=0\) and \(X\) is a smooth and geometrically integral \(k\)-variety such that \(\overline k[X]^*=\overline k^*\) and \(\text{Pic}(\overline X)\) is a free abelian group of finite type. In the case of the universal torsor (i.e. when the natural map \(\widehat T\to \text{Pic}(\overline X)\) is an isomorphism) this description is particularly simple: the map \(\text{Br}(X)\to \text{Br}(Y)/\text{Br}(k)\) can be identified with the map \(\text{Br}(X)\to \text{Br}(\overline X)^{\Gamma}\), where \(\Gamma =\text{Gal} (\overline k/k)\). The second part of the paper is devoted to the case where \(k\) is a number field. The authors continue the search for new classes of varieties in which the Brauer-Manin obstruction to the Hasse principle and weak approximation is the only one. More precisely, they consider the case where a smooth projective variety \(X\) admits a dominant morphism \(\pi\colon X\to \mathbb P^1\) with geometrically integral generic fibre and ask whether assuming the Brauer-Manin obstruction is the only one for the smooth fibres of \(\pi\) one can deduce the same for \(X\). Several earlier works showed that this approach is successful when the fibration has a few ``bad'' fibres. In the paper under review several new instances of this principle have been found. Explicit examples of applications include conic bundles over the plane and some varieties of norm type.
scientific article; zbMATH DE number 2093464
scientific article