Towards the Brauer-Manin obstruction on varieties fibred over the projective line
type: Item
instance of: scholarly article
title: Towards the Brauer-Manin obstruction on varieties fibred over the projective line
zbMATH Open document ID: 1372.11066
author: Yongqi Liang
published in: Journal of Algebra
publication date: 2014-08-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/1210.4190
reviewed by: David McKinnon
Mathematics Subject Classification ID: 11G35
Mathematics Subject Classification ID: 14G25
Mathematics Subject Classification ID: 14D10
zbMATH DE Number: 6336182
zbMATH Keywords: zero-cycles
zbMATH Keywords: Hasse principle
zbMATH Keywords: weak approximation
zbMATH Keywords: Brauer-Manin obstruction
MaRDI profile type: Publication
OpenAlex ID: W2962977045
arXiv ID: 1210.4190
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DOI: 10.1016/J.JALGEBRA.2014.04.023
review text: 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.
description: scientific article; zbMATH DE number 6336182