Towards the Brauer-Manin obstruction on varieties fibred over the projective line
The paper under review proves that the Brauer-Manin obstruction is the only obstruction to weak approximation for \(0\)-cycles on certain varieties fibred over the projective line. This extends a result of \textit{D. Wei} [``On the equation \(N_{K/k}(\Xi)=P(t)\)'', Preprint, \url{arXiv:1202.4115}] which proves the analogous result for effective \(1\)-cycles -- that is, points -- on the same varieties.NEWLINENEWLINESpecifically, the varieties under consideration must satisfy the following conditions: NEWLINENEWLINE{\parindent=6mm \begin{itemize}\item[(a)] The (closed) fibres must be abelian-split, and \item[(b)] If \(X\) is the tensor product of the original surface with the algebraic closure of the function field of the base \(\mathbb{P}^1\), then the Picard group of \(X\) must be torsion free, and the Brauer group of \(X\) must be finite. \item[(c)] There is a third technical condition about the Brauer group of the original surface that must also be satisfied.NEWLINENEWLINE\end{itemize}}NEWLINEShould those conditions be satisfied, then the author proves the sufficiency of the Brauer-Manin obstruction to the Hasse principle and weak approximation for \(0\)-cycles from its sufficiency, respectively, for the Hasse principle and weak approximation for rational points.
scientific article; zbMATH DE number 6336182
scientific article