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Let $G$ be a fixed finite group. I'm interested in the structure of the set $\mathcal{H}_{r,g,h,G}$ of tuples $(C,f,\delta)$, where $C$ is a smooth projective genus $g\geq 2$ curve, $\delta:G\to\mbox{Aut}(C)$ is an injective group homomorphism, $f:C\to C'$ is a finite Galois morphism with $r$ ramification points and Galois group $\delta(G)$, and $C'$ is a smooth projective curve of genus $h\geq 1$. When $h=0$, then this is just the usual Hurwitz scheme (or at least one of the usual ones) of Galois morphisms to $\mathbb{P}^1$ with fixed Galois group, and can be seen as the coarse moduli scheme of a certain functor (basically the above construction but relativized).

However, it seems to me that for $h>0$, not much is known. I have seen certain functors defined but where the $C'$ above is a fixed curve, not one that moves. I, on the other hand, am interested in not fixing the $C'$. Is there any literature on this? Can this set be seen as a coarse moduli scheme for a functor?

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  • $\begingroup$ Is $C'$ the quotient of $C$ by the action of $G$? If not, what's the connection between $G,\delta$ and $f$? $\endgroup$
    – benblumsmith
    Commented Jun 3, 2016 at 2:03
  • $\begingroup$ Sorry, yes! I will edit it right away. $\endgroup$
    – rfauffar
    Commented Jun 3, 2016 at 2:43
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    $\begingroup$ If you do not fix $C'$, then how do you define the classical Hurwitz schemes where $C'$ is a smooth, projective genus $0$ curve? Anyway, for topological properties, it will not make much difference whether or not you fix $C'$. Some of the modern references are by Fulton, particularly in positive characteristic. The generalization to arbitrary $C'$ of Hurwitz's irreducibility theorem (in characteristic 0) has been "rediscovered" many times, perhaps first in the thesis of Richard Hamilton, but then also by Gabai-Kazez and others (at one time, Graber, Harris, and I needed this as well). $\endgroup$
    – Jason Starr
    Commented Jun 3, 2016 at 4:47
  • $\begingroup$ Dear Jason, thanks for your comment. I think you're right, that the construction should be analogous to the classical case. I will take a look at the papers you describe. $\endgroup$
    – rfauffar
    Commented Jun 3, 2016 at 12:05
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    $\begingroup$ I just saw that what I am calling "Hurwitz's irreducibility theorem" is actually due jointly to L"uroth, Clebsch, and Hurwitz. $\endgroup$
    – Jason Starr
    Commented Jun 4, 2016 at 21:53

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Apart from Jason Starr's excellent references in the comments above, I also found

José Bertin, Compactification des schémas de Hurwitz, C.R. Acad. Sci. Paris I, vol. 322 (1996), 1063-1066.

and 

S. Wewers, Construction of Hurwitz spaces, Dissertation, Universität Essen, 1998.

that basically show that the space parametrizing $G$-covers with specified genus of the quotient and specified ramification type (i.e. the conjugacy classes of the stabilizers of the action) has a natural structure of quasi-projective variety. Since the ramification type data is finite, then this shows that the space I ask about above also has the structure of a quasi-projective variety.

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