Galois covers of moduli of curves

Moduli spaces of pointed curves with some level structure are studied. We p rove that for so-called geometric level structures, the levels encountered in the boundary are smooth if the ambient variety is smooth, and in some ca ses we can describe them explicitly. The smoothness implies that the moduli space of pointed curves (over any field) admits a smooth finite Galois cov er. Finally, we prove that some of these moduli spaces are simply connected .