The Kodaira classification of the moduli space of pointed curves in genus $3$

TL;DR

This paper completes the classification of the moduli space of genus 3 pointed curves, showing it is of general type for at least 15 marked points, by analyzing its singularities and canonical class.

Contribution

It proves that the moduli space ar{\u03bc}{}_{3,n} is of general type for n q 15, completing the Kodaira classification for genus 3.

Findings

ar{bc}{}_{3,n} is of general type for n q 15

Singularities impose no adjunction conditions for n q 1

Canonical class is big for n q 15

Abstract

We complete the Kodaira classification of the moduli spaces $\overline{\mathcal{M}}_{g,n}$ of curves with marked points in genus $g=3$, by proving that $\overline{\mathcal{M}}_{3,n}$ is of general type for $n \geq 15$. We prove that the singularities of $\overline{\mathcal{M}}_{3,n}$ impose no adjunction conditions for $n \geq 1$ and that the canonical class of $\overline{\mathcal{M}}_{3,n}$ is big for $n \geq 15$.