On the Kodaira dimension of the moduli space of nodal curves

Abstract

We show that the compactification of the moduli space of $n-$nodal curves of genus g, i.e. ${\mathcal{N}}_{g,n}:={\mathcal{M}}_{g,2n}/G$, with $G:=({\mathbb{Z}}_2{)}^n⋊S_n$, is of general type for $g\ge 24$, for all $n\in \mathbb{N}$. While this is a fairly easy result, it requires completely different techniques to extend it to low genus $5\le g\le 23$. Here we need that the number of nodes varies in a band $n_{\min }(g)\le n\le n_{\max }(g)$, where $n_{\max }(g)$ is the largest integer smaller than (or in some cases equal to) $\frac{7}{2}(g-1)-3$. The lower bound $n_{\min }(g)$ is close to the bound found by Logan and Farkas for ${\mathcal{M}}_{g,2n}$ to be of general type (in many cases it is identical). This will be tabled in Theorem 1.1 which is the main result of this paper.