Irreducibility of the space of dihedral covers of the projective line of a given numerical type

Abstract

We show in this paper that the set of irreducible components of the family of Galois coverings of ${\mathbb{P}}_{\mathbb{C}}^1$ with Galois group isomorphic to ${\mathrm{D}}_n$ is in bijection with the set of possible numerical types. In this special case the numerical type is the equivalence class (for automorphisms of ${\mathrm{D}}_n$) of the function which to each conjugacy class $\mathcal{C}$ in ${\mathrm{D}}_n$ associates the number of branch points whose local monodromy lies in the class $\mathcal{C}$.