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


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.


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