On Hurwitz spaces of coverings with one special fiber

Let X ! X0 f ! Y be a covering of smooth, projective complex curves such that is a degree 2 \ue9tale covering and f is a degree d covering, with monodromy group Sd, branched in nC 1 points one of which is a spe- cial point whose local monodromy has cycle type given by the partition eD.e1;:::; er/ of d. We study such coverings whose monodromy group is either W.Bd/ or w N.W.Bd//.G1/w 1 for some w2 W.Bd/, where W.Bd/ is the Weyl group of type Bd, G1 is the subgroup of W.Bd/ generated by reflections with respect to the long roots "i " j and N.W.Bd//.G1/ is the normalizer of G1. We prove that in both cases the corresponding Hurwitz spaces are not connected and hence are not irreducible. In fact, we show that if nCjej 2d, wherejejD Pr iD1.ei 1/, they have 2 2g 1 connected components