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

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For any pencil of conics or higher-dimensional quadrics over the rationals, with all degenerate fibres defined over the rationals, 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, which is a consequence of recent advances in additive combinatorics.