Covering semigroups

Abstract

We introduce and study a semigroup structure on the set of irreducible components of the Hurwitz space of marked coverings of a complex projective curve with given Galois group of the coverings and fixed ramification type. As application, we give new conditions on the ramification type that are sufficient for irreducibility of the Hurwitz spaces, suggest some bounds on the number of irreducibility components under certain more general conditions, and show that the number of irreducible components coincides with the number of topological classes of the coverings if the number of brunch points is big enough.