Moduli of $G$-covers of curves: geometry and singularities

In a recent paper Chiodo and Farkas described the singular locus and the locus of non-canonical singularities of the moduli space of level curves. In this work we generalize their results to the moduli space $\overline{\mathcal R}_{g,G}$ofcurveswitha$G$-coverforanyfinitegroup$G$. We show that non-canonical singularities are of two types: $T$-curves, that is singularities lifted from the moduli space ${\bar{\mathcal{M}}}_g$ of stable curves, and $J$-curves, that is new singularities entirely characterized by the dual graph of the cover. Finally, we prove that in the case $G=S_3$, the $J$-locus is empty, which is the first fundamental step in evaluating the Kodaira dimension of ${\bar{\mathcal{R}}}_{g,S_3}$.