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 P 1 with Galois group isomorphic to Dn is in bijection with the set of possible numerical types. In this special case the numerical type is the equivalence class (for automorphisms of Dn) of the function which to each conjugacy class C in Dn associates the number of branch points whose local monodromy lies in the class C.