MODULI OF G-COVERS OF CURVES: GEOMETRY AND SINGULARITIES

We analyze the singular locus and the locus of non-canonical singularities of the moduli space barR_{G,g} of curves with a G-cover for any finite group G. We show that non-canonical singularities are of two types: T-curves, that is singularities lifted from the moduli space barM_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=S3, the J-locus is empty, which is the first fundamental step in evaluating the Kodaira dimension of R_{S3,g}.