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

TL;DR

This paper establishes a correspondence between the irreducible components of dihedral Galois covers of the projective line and their numerical types, providing a classification framework for these covers.

Contribution

It proves that the set of irreducible components of dihedral covers is in bijection with their numerical types, clarifying the structure of these covers.

Findings

Irreducible components correspond to numerical types.

Numerical types are classified by conjugacy classes in D_n.

The classification aids in understanding dihedral Galois covers.

Abstract

We show in this paper that the set of irreducible components of the family of Galois coverings of P^1_C with Galois group isomorphic to 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 D_n) of the function which to each conjugacy class \mathcal{C} in D_n associates the number of branch points whose local monodromy lies in the class \mathcal{C}.