A Bogomolov type statement for function fields

TL;DR

This paper establishes a Bogomolov type statement for algebraic points over function fields, showing that points with sufficiently small height on a variety are contained in the base field.

Contribution

It proves a Bogomolov type result for varieties over algebraic closures of function fields, extending height bounds to characterize points in the base field.

Findings

Existence of a positive height bound c for points in V

Points with height less than c are contained in the base field k

The result applies to varieties over algebraic closures of function fields

Abstract

Let k be a an algebraically closed field of arbitrary characteristic, and we let h be the usual Weil height for the n-dimensional affine space corresponding to the function field k(t) (extended to its algebraic closure). We prove that for any affine variety V defined over the algebraic closure of k(t), there exists a positive real number c such that if P is an algebraic point of V and h(P)< c, then P has its coordinates in k.