Potential density for some families of homogeneous spaces

TL;DR

This paper proves that potential density of rational points in a family of homogeneous varieties over a number field extends from the base to the total space, linking it to a conjecture on nef tangent bundles.

Contribution

It establishes a connection between potential density in families of homogeneous spaces and a conjecture on nef tangent bundles, advancing understanding of rational points.

Findings

Potential density extends from base to total space in certain families.

A conjecture on nef tangent bundles implies potential density for specific varieties.

Results apply to families over number fields with known potential density in the base.

Abstract

For a smooth, projective family of homogeneous varieties defined over a number field, we show that if potential density holds for the rational points of the base, then it also holds for the total space. A conjecture of Campana and Peternell, known in dimension at most 4 and for certain higher dimensional cases, would then imply potential density for the rational points of smooth projective varieties over number fields whose tangent bundle is nef.