Rational points on pencils of conics and quadrics with many degenerate fibres

Pages 381-402 from Volume 180 (2014), Issue 1 by Tim D. Browning, Lilian Matthiesen, Alexei N. Skorobogatov

Abstract

For any pencil of conics or higher-dimensional quadrics over $\mathbb{Q}$, with all degenerate fibres defined over $\mathbb{Q}$, we show that the Brauer–Manin obstruction controls weak approximation. The proof is based on the Hasse principle and weak approximation for some special intersections of quadrics over $\mathbb{Q}$, which is a consequence of recent advances in additive combinatorics.

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Keywords
, , , ,
DOI
https://doi.org/10.4007/annals.2014.180.1.8
MR
3194818
Zbl
06316073
Milestones
Received: 1 October 2012
Revised: 4 August 2013
Accepted: 23 January 2014
Published online: 1 July 2014

Authors

Tim D. Browning

School of Mathematics, University of Bristol, Bristol BS8, 1TW, Bristol U.K.

Lilian Matthiesen

Institut de Mathématiques de Jussieu --- Paris Rive Gauche, 75205 Paris Cedex 13, France

Alexei N. Skorobogatov

Imperial College London, London SW7 2AZ, United Kingdom and
Institute for the Information Transmission Problems, Moscow, Russia