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!