Irreducibility of Hurwitz spaces of coverings with one special fiber

Let Y be a smooth, projective complex curve of genus g ? 1. Let d be an integer ? 3, let e = {e₁, e₂,..., e_r} be a partition of d and let |e| = Σ_i=1^r(e_i − 1). In this paper we study the Hurwitz spaces which parametrize coverings of degree d of Y branched in n points of which n − 1 are points of simple ramification and one is a special point whose local monodromy has cyclic type e and furthermore the coverings have full monodromy group S_d. We prove the irreducibility of these Hurwitz spaces when n − 1 + |e| ? 2d, thus generalizing a result of Graber, Harris and Starr A note on Hurwitz schemes of covers of a positive genus curve, Preprint, math. AG/0205056].