Irreducible components of Hurwitz spaces parameterizing Galois coverings of curves of positive genus

Given a smooth, projective curve Y of genus g>=1 and a finite group G, let H^G_n(Y) be the Hurwitz space which parameterizes the G-equivalence classes of G-coverings of Y branched in n points. This space is a finite e'tale covering of Y^{(n)}\setminus \Delta, where \Delta is the big diagonal. In this paper we calculate explicitly the monodromy of this covering. This is an extension to curves of positive genus of a well known result in the case of Y = P^1, and may be used for determining the irreducible components of H^G_n(Y) in particular cases.