The Geometric Bogomolov Conjecture for Curves of Small Genus

Abstract

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