Let $\overline{\mathcal{M}}_{g,n}$ be the moduli space of pointed, stable, genus $g$ curves. Let $\overline{\mathcal{H}}_{g',r}$ be the hurwitz space of cyclic covers of degree $r$ of genus $g$ curves ramified along $n$ points. Under some easy assumptions on $g,g',n$ and $r$. There exist a natural "discriminant" (this is how I have seen people calling it in the literature) morphism $\delta:\overline{\mathcal{H}}_{g',r} \to \overline{\mathcal{M}}_{g,n}$, that basically forgets the "above" curve in the cover and keeps the base. This is a ramified cover of $\overline{\mathcal{M}}_{g,n}$, what is the branch locus of $\delta$? Also a first answer for the space of $r$Prym curves (i.e. étale degree $r$ covers) and $\overline{\mathcal{M}}_g$ would be a nice starting point. A wild guess is that the components of the boundary s.t. the generic member is a nodal curve that decomposes in a genus one curve plus a genus $g1$ should be contained in the branch locus.
