A complete curve of genus 10 in the moduli space of curves of genus 3 and related questions

Abstract

We show the existence of a complete trigonal curve of geometric genus 10 in the moduli space $\mathcal{M}_3$ of smooth complex projective curves of genus $3$, and of a complete curve of geometric genus 102 through a general point of $\mathcal{M}_3$. We also show that the moduli space $\mathcal{A}_3$ of principally polarized complex abelian threefolds contains a complete curve whose normalization is a hyperelliptic curve of genus 4.