Geometric Bogomolov conjecture for semiabelian varieties

TL;DR

This paper proves the geometric Bogomolov conjecture for semiabelian varieties over function fields, characterizing subvarieties with dense small points and revealing new phenomena about height zero points.

Contribution

It establishes the conjecture for semiabelian varieties over function fields and uncovers a new phenomenon regarding special subvarieties and height zero points.

Findings

Subvarieties with dense small points are torsion translates of constant varieties after stabilizer quotient.

A special subvariety may lack a Zariski dense set of points of height 0.

The result extends the understanding of the distribution of small points in semiabelian varieties.

Abstract

We establish the geometric Bogomolov conjecture for semiabelian varieties over function fields. We show a closed subvariety contains Zariski dense sets of small points, if and only if, after modulo its stabilizer, it is a torsion translate of a constant variety. A new phenomenon is that a special subvariety may not have a Zariski dense set of points of height 0.