## Experimental Mathematics

### The Geometric Bogomolov Conjecture for Curves of Small Genus

X. W. C. Faber

#### Abstract

The Bogomolov conjecture is a finiteness statement about algebraic points of small height on a smooth complete curve defined over a global field. We verify an effective form of the Bogomolov conjecture for all curves of genus at most $4$ over a function field of characteristic zero. We recover the known result for genus-$2$ curves and in many cases improve upon the known bound for genus-$3$ curves. For many curves of genus $4$ with bad reduction, the conjecture was previously unproved.

#### Article information

Source
Experiment. Math., Volume 18, Issue 3 (2009), 347-367.

Dates
First available in Project Euclid: 25 November 2009