In this question all the varieties are over $\mathbb{C}$. Classic Hurwitz spaces $\mathcal{H}_{g,r}$ are moduli spaces of simple branched coverings $f \colon X \to \mathbb{P}^1$ of degree $d$, where $X$ is a smooth curve ov genus $g$ and there are exactly $r$ branch points. Simple means that every point in $\mathbb{P}^1$ has at least $d-1$ preimages. My question is whether similar moduli spaces were considered for finite branched coverings $X \to \mathbb{P}^k$, for $k > 1$? One of course has to define a corresponding notion of "simple" and classify the branching.
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$\begingroup$ There are many different ways you could try to extend Hurwitz schemes to higher dimensional projective spaces. Do you want the branch locus in projective space to be irreducible? $\endgroup$– Jason StarrCommented May 28, 2015 at 14:36
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$\begingroup$ I think that it makes sense to consider the case where the branch locus is a divisor of normal crossing for example. $\endgroup$– shamovicCommented May 29, 2015 at 9:43
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