Complete Families of Curves in the Moduli Space of Genus g Curves

Abstract

Let ${\mathcal{M}}_g$ be the moduli space of smooth curves of genus $g$. The image of a non-constant morphism from a curve $T$ to ${\mathcal{M}}_g$ is a curve in ${\mathcal{M}}_g$. By work of González Díez and Harvey, for every integer $g\ge 3$, there exists a complete curve in ${\mathcal{M}}_g$. Here we generalize the construction to produce new complete curves in ${\mathcal{M}}_g$. We also find a formula for the genus of each curve $T$ using Galois theory for function fields.