Normalization of the 1-stratum of the moduli space of stable curves

Abstract

Let ${\mathcal{M}}_g(\Gamma )$ be the stack of stable curves of genus $g$ with a given dual graph $\Gamma$ and let ${\overline{\mathcal{M}}}_g(\Gamma )$ be its closure in ${\overline{\mathcal{M}}}_g$. We consider the normalization of ${\overline{\mathcal{M}}}_g(\Gamma )$ in order to classify the residual orbifolds of the normalizations of the irreducible components of the locus in ${\overline{\mathcal{M}}}_g$ corresponding to curves with at least $3g-4$ nodes.