Families of superelliptic curves, complex braid groups and generalized Dehn twists

We consider the universal family E n d of superelliptic curves: each curve Σ n d in the family is a d -fold covering of the unit disk, totally ramified over aset P of n distinct points; Σ n d ↪ E n d → C n is a fiber bundle, where C n is the configuration space of n distinct points. We find that E n d is the classifying space for the complex braid group of type B( d, d, n ) and we compute a big part of the integral homology of E n d , including a complete calculation of the stable groups over finite fields by means of Poincaré series. The computation of the main part of the above homology reduces to the computation of the homology of the classical braid group with coefficients in the first homology group of Σ n d , endowed with the monodromy action. While giving a geometric description of such monodromy of the above bundle, we introduce generalized 1 d -twists, associated to each standard generator of the braid group, which reduce to standard Dehn twists for d = 2.