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

We consider the universal family End of superelliptic curves: each curve Σnd in the family is a d-fold covering of the unit disk, totally ramified over aset P of n distinct points; Σnd↪End→Cn is a fiber bundle, where Cn is the configuration space of n distinct points. We find that End 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 End, 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 Σnd, endowed with the monodromy action. While giving a geometric description of such monodromy of the above bundle, we introduce generalized 1d-twists, associated to each standard generator of the braid group, which reduce to standard Dehn twists for d = 2.