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

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Abstract

We consider the universal family Edn of superelliptic curves: each curve Σdn in the family is a d-fold covering of the unit disk, totally ramified over aset P of n distinct points; \(\Sigma _n^d \hookrightarrow E_n^d \to {{\rm{C}}_n}\) is a fiber bundle, where Cn is the configuration space of n distinct points.

We find that Edn 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 Edn , 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 Σdn , 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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Authors and Affiliations

  1. Dipartimento di Matematica, University of Pisa, Largo Bruno Pontecorvo 5, Pisa, 56126, Italy

    Filippo Callegaro & Mario Salvetti

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  1. Filippo Callegaro
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  2. Mario Salvetti
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Correspondence to Filippo Callegaro.

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Callegaro, F., Salvetti, M. Families of superelliptic curves, complex braid groups and generalized Dehn twists. Isr. J. Math. 238, 945–1000 (2020). https://doi.org/10.1007/s11856-020-2040-x

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  • DOI: https://doi.org/10.1007/s11856-020-2040-x

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