The Hasse principle for random Fano hypersurfaces

TL;DR

This paper proves that almost all Fano hypersurfaces of dimension at least 3 over the rationals satisfy the Hasse principle, confirming a conjecture for most cases except cubic surfaces.

Contribution

It establishes that the Hasse principle holds for almost all high-dimensional Fano hypersurfaces over the rationals, except for cubic surfaces, advancing understanding of rational points.

Findings

Almost all Fano hypersurfaces of dimension ≥ 3 satisfy the Hasse principle.

The result confirms a conjecture of Poonen and Voloch for these cases.

Cubic surfaces remain an exception to the proven cases.

Abstract

It is known that the Brauer--Manin obstruction to the Hasse principle is vacuous for smooth Fano hypersurfaces of dimension at least $3$ over any number field. Moreover, for such varieties it follows from a general conjecture of Colliot-Th\'el\`ene that the Brauer--Manin obstruction to the Hasse principle should be the only one, so that the Hasse principle is expected to hold. Working over the field of rational numbers and ordering Fano hypersurfaces of fixed degree and dimension by height, we prove that almost every such hypersurface satisfies the Hasse principle provided that the dimension is at least $3$. This proves a conjecture of Poonen and Voloch in every case except for cubic surfaces.