# branch locus of the discriminant map $\overline{\mathcal{H}}_{g',r} \to \overline{\mathcal{M}}_{g,n}$

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 $g-1$ should be contained in the branch locus.

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For $n=0$ the branch locus is $\delta_{irr}$. This is described in many places, for instance here: arxiv.org/abs/1205.0201 – Dan Petersen Oct 24 '13 at 9:35
There seems to be a difference whether you consider the stack or the coarse moduli space. For instance, in the "twin" paper arxiv.org/pdf/1205.0661.pdf (page 13) the authors claim that for the coarse moduli space the discriminant morphism is ramified along some of the divisors I mentioned. Can you explain this? – IMeasy Oct 24 '13 at 11:31
You're right, I was thinking of the stack. On the level of course spaces you have to think also about automorphisms. On most divisors the generic curve has no automorphism, so this won't give you any extra ramification, but on $\delta_1$ the generic curve has $2$ automorphisms. If this generically defined automorphism does not lift to the level curve then you get ramification on the level of coarse spaces. – Dan Petersen Oct 24 '13 at 11:46
You can see this (informally) by computing degrees: on the level of stacks the map is étale, but then the map to the course space has "degree 1/2" downstairs (that is, on the divisor $\delta_1$) and is generically an isomorphism upstairs so has "degree 1". – Dan Petersen Oct 24 '13 at 11:47
Oh no, genus one is special. As coarse spaces you have $\overline M_{1,1} \cong \mathbb P^1$ so there's no way you can have only one branch point of a ramified cover!! You can see this by thinking about automorphisms too. Branching always happens in codimension one, but in higher genus the loci of smooth curves with automorphisms have bigger codimension. This is why the only branching comes from the divisor $\delta_1$. For the projection from a modular curve you should expect branching over $\infty$ and over the points with $j$-invariants $0$ and $1728$. – Dan Petersen Oct 31 '13 at 9:36