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

TL;DR

This paper demonstrates that for certain families of conics and quadrics over the rationals, the Brauer-Manin obstruction fully explains the failure of weak approximation, leveraging recent advances in additive combinatorics.

Contribution

It establishes the Brauer-Manin obstruction as the sole reason for weak approximation failure in pencils of conics and quadrics with all degenerate fibres rational, using new results on intersections of quadrics.

Findings

Brauer-Manin obstruction controls weak approximation in the studied cases

Weak approximation holds for specific intersections of quadrics due to recent additive combinatorics results

The approach links the Hasse principle with geometric properties of degenerate fibres

Abstract

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.