Families of superelliptic curves, complex braid groups and generalized Dehn twists
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FAMILIES OF SUPERELLIPTIC CURVES, COMPLEX BRAID GROUPS AND GENERALIZED DEHN TWISTS BY
Filippo Callegaro and Mario Salvetti Dipartimento di Matematica, University of Pisa Largo Bruno Pontecorvo 5, Pisa 56126, Italy e-mail: [email protected], [email protected]
ABSTRACT d of superelliptic curves: each curve We consider the universal family En Σdn in the family is a d-fold covering of the unit disk, totally ramified over d → C is a fiber bundle, where C a set P of n distinct points; Σdn → En n n is the configuration space of n distinct points. d is the classifying space for the complex braid group We find that En of type B(d, d, n) and we compute a big part of the integral homology d , including a complete calculation of the stable groups over finite of En 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 Σdn , endowed with the monodromy action. While giving a geometric description of such monodromy of the above bundle, we introduce generalized 1 -twists, associated to each standard generator of the braid group, which d reduce to standard Dehn twists for d = 2.
Received January 25, 2019 and in revised form July 31, 2019
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F. CALLEGARO AND M. SALVETTI
Isr. J. Math.
Contents
1. Introduction . . . . . . . . . . . . . . . . . . 2. General setting . . . . . . . . . . . . . . . . 3. Generalized twists . . . . . . . . . . . . . . 4. Motivations and homology of complex braid 5. Homology of some Artin groups . . . . . . . 6. Exact sequences . . . . . . . . . . . . . . . 7. A first bound for torsion order . . . . . . . 8. No pk+1 -torsion . . . . . . . . . . . . . . . . 9. Natural maps between families . . . . . . . 10. Stabilization and computations . . . . . . References . . . . . . . . . . . . . . . . . . . . .
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1. Introduction This paper is a natural continuation of [CS19]. We are interested in the homology of the families of superelliptic curves Edn := {(P, z, y) ∈ Cn ×D × C|y d = (z − x1 ) · · · (z − xn )} where D is the open unit disk in C, Cn is the configuration space of n distinct unordered points in D and P = {x1 , . . . , xn } ∈ Cn . For a fixed P ∈ Cn , the curve Σdn = {(z, y) ∈ D × C|y d = (z − x1 ) · · · (z − xn )} in the family Edn is a d-fold covering of the disk D, totally ramified over P, and there is a fibration π : Edn → Cn which takes Σdn onto its set of ramification points. One can see Edn as a universal family over the Hurwitz space H n,d (for precise definitions see [Ful69], [EVW16]). In [CS19] we worked out the case d = 2 of hyperelliptic families. In such a case the classical geometrical representation of the braid group as automorphisms of an elliptic curve play a key role.
Vol. TBD, 2020
FAMILIES OF SUPERELLIPTIC CURVES
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