A branched cover of the Riemann sphere is a non-constant holomorphic map $\phi: \Sigma \to \mathbb{C}P^1$ where $\Sigma$ is a compact Riemann surface. The Hurwitz space of branched coverings of the Riemann sphere is therefore equivalent to the space of all such holomorphic maps.

The Hurwitz space of branched coverings of fixed degree with $n$ branch points is a (non-branched) covering of the configuration space

$C_n = \{(z_1,...,z_n) \in (\mathbb{C}P^1)^n : z_i \neq z_j \ , \ \ if \ \ i \neq j \}$

Is there any similar relationship between holomorphic functions and generalized configuration spaces? By generalized configuration spaces I mean the following:

Choose an $n \times n$ matrix $A$ with elements $A_{ij}= 1$ or $0$ at will. Then a generalized configuration space is

$C^k_n = \{(z_1,...,z_n) \in (\mathbb{C}P^k)^n : z_i \neq z_j \ , \ \ if \ \ A_{ij} = 1 \}$

I do not know much algebraic geometry and so references and corrections would be much appreciated!

  • $\begingroup$ I am not sure exactly what you are asking, but Hurwitz spaces can be generalized to moduli spaces of covers of projective space with branch locus a fixed hyperplane arrangement (your $C^k_n$ for $A=Id$ is a parameter space for hyperplane arrangements in $\mathbb{C}P^k$ by duality). In some case they can be described explicitely; see the paper of Pardini, Abelian covers of algebraic varieties. $\endgroup$ – Simon Pepin Lehalleur Dec 20 '14 at 13:19
  • $\begingroup$ Hi Simon (if I may). That is very interesting but a little confusing. By $C^k_n$ for $A=Id$ do you actually mean this to correspond to the configuration space $z_i \neq z_j$ for all $i,j: i \neq j$ in which case $A$ is the matrix with unit entries everywhere except on the diagonal? If so, then a link to a paper/site discussing these spaces would be most useful. $\endgroup$ – David Dec 26 '14 at 12:57

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