Rational points on some Fano cubic bundles

TL;DR

This paper studies specific Fano hypersurfaces in a product of projective spaces, providing lower bounds on the number of rational points of bounded height, challenging previous expectations about their distribution.

Contribution

It establishes new lower bounds for rational points on certain Fano hypersurfaces, contradicting prior conjectures about their distribution.

Findings

Lower bounds for $F$-rational points of bounded height

Contradiction of previous distribution expectations

Results apply to open subsets of the hypersurfaces

Abstract

We consider some families of smooth Fano hypersurfaces $X_{n+2}$ in ${\bf P}^{n+2} \times {\bf P}^3$ given by a homogeneous polynomial of bidegree $(1,3)$. For these varieties we obtain lower bounds for the number of $F$-rational points of bounded anticanonical height in arbitrary nonempty Zariski open subset $U \subset X_{n+2}$. These bounds contradict previous expectations about the distribution of $F$-rational points of bounded height on Fano varieties.