On the fibration method for zero-cycles and rational points

TL;DR

This paper proves that certain conjectures about zero-cycles and rational points on algebraic varieties are compatible with fibrations into rationally connected varieties, extending their validity to new classes of varieties over number fields.

Contribution

It demonstrates the compatibility of conjectures on zero-cycles and rational points with fibrations into rationally connected varieties, including families of homogeneous spaces of linear groups.

Findings

Conjectures hold for total spaces of fibrations into rationally connected varieties.

Compatibility established for families of homogeneous spaces of linear groups.

Conditional results on rational points based on conjectures about polynomial values.

Abstract

Conjectures on the existence of zero-cycles on arbitrary smooth projective varieties over number fields were proposed by Colliot-Th\'el\`ene, Sansuc, Kato and Saito in the 1980's. We prove that these conjectures are compatible with fibrations, for fibrations into rationally connected varieties over a curve. In particular, they hold for the total space of families of homogeneous spaces of linear groups with connected geometric stabilisers. We prove the analogous result for rational points, conditionally on a conjecture on locally split values of polynomials which a recent work of Matthiesen establishes in the case of linear polynomials over the rationals.