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arXiv · 1805.11968

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

Filippo Callegaro·Mario Salvetti
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Abstract

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 a set $P$ of $n$ distinct points; $Σ_n^d\hookrightarrow E_n^d\to C_n$ 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 Poincarè 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 $1\over d$-twists, associated to each standard generator of the braid group, which reduce to standard Dehn twists for $d=2.$

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Filippo Callegaro, Mario Salvetti. 2018-08-24. Families of superelliptic curves, complex braid groups and generalized Dehn twists. https://arxiv.org/abs/1805.11968

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