A Bogomolov Type Statement for Functions Fields
by Dragos Ghioca  

Vol. 9 No. 4 (2014) P.641~P.656

ABSTRACT

Let $ \ k \ $ be a an algebraically closed field of arbitrary characteristic, and we let $h:\mathbb{A}^n(\overline{k(t)})\longrightarrow \mathbb{R}_{\ge 0}$ 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 $V\subset \mathbb{A}^n$ defined over $\overline{k(t)}$, there exists a positive real number $\epsilon:=\epsilon(V)$ such that if $P\in V(\overline{k(t)})$ and $h(P)<\epsilon$, then $P\in V(k)$.

KEYWORDS
heights for function fields, the Bogomolov conjecture

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 11G50, 14G17, 11G10.

MILESTONES

Received: 2013-07-22
Revised : 2014-02-15
Accepted: 2014-02-27

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