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

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.