The Geometric Bogomolov Conjecture for Curves of Small Genus

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.