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

TL;DR

This paper studies the universal family of superelliptic curves, linking it to complex braid groups, and computes significant parts of its homology, revealing new geometric and algebraic structures including generalized Dehn twists.

Contribution

It establishes that the universal superelliptic family classifies the complex braid group of type B(d,d,n) and computes its integral homology, including stable groups over finite fields.

Findings

E_n^d is the classifying space for the complex braid group of type B(d,d,n)

Computed the integral homology of E_n^d, including stable groups over finite fields

Introduced generalized 1/d-twists extending standard Dehn twists

Abstract

We consider the universal family $E_n^d$ of superelliptic curves: each curve $\Sigma_n^d$ in the family is a $d$-fold covering of the unit disk, totally ramified over a set $P$ of $n$ distinct points; $\Sigma_n^d\hookrightarrow E_n^d\to C_n$ is a fibre 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 Poincar\`e 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 $\Sigma_n^d,$ endowed with the monodromy action. While giving a geometric description of such monodromy of the above bundle, we introduce generalized $1\over d$-twists, associated to each standard generator of the braid group, which reduce to standard Dehn twists for $d=2.$