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Irreducible Components of Hurwitz Spaces of Coverings with Two Special Fibers

Abstract

In this paper we prove new results of irreducibility for Hurwitz spaces of coverings whose monodromy group is a Weyl group of type B d and whose local monodromies are all reflections except two.

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References

  1. 1.

    Berstein I., Edmonds A.L.: On the classification of generic branched coverings of surfaces. Illinois J. Math. 28, 64–82 (1984)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    R. Biggers and M. Fried, Irreducibility of moduli spaces of cyclic unramified covers of genus g curves, Trans. Amer. Math. Soc. 295 (no. 1) (1986), 59–70.

    Google Scholar 

  3. 3.

    Birman J.S.: On braid groups. Comm. Pure Appl. Math. 22, 41–72 (1969)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    N. Bourbaki, Groupes et algebres de Lie, Chap. 4–6, Éléments de Mathématique, 34 (1968), Hermann, Paris.

  5. 5.

    Carter R.W.: Conjugacy classes in the Weyl group. Compos. Math. 25, 1–59 (1972)

    MATH  Google Scholar 

  6. 6.

    F. Catanese, M. Lönne and F. Perroni, Irreducibility of the space of dihedral covers of the projective line of a given numerical type, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 22 (2011), 291–309.

  7. 7.

    Chen D.: Covers of elliptic curves and the moduli space of stable curves. J. Reine Angew. Math. 649, 167–205 (2010)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Donagi R.: Decomposition of spectral covers. Astérisque 218, 145–175 (1993)

    MathSciNet  Google Scholar 

  9. 9.

    Fadell E., Neuwirth L.: Configuration spaces. Math. Scand. 10, 111–118 (1962)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    M. Fried and H. Volklein, The inverse Galois problem and rational points on moduli spaces, Math. Ann. 290 (no. 4) (1991), 771–800.

  11. 11.

    W. Fulton, Hurwitz schemes and irreducibility of moduli of algebraic curves, Ann. of Math. 90 (no. 2) (1969), 542–575.

  12. 12.

    T. Graber, J. Harris and J. Starr, A note on Hurwitz schemes of covers of a positive genus curve, preprint, arXiv: math. AG/0205056 (2002).

  13. 13.

    T. Graber, J. Harris and J. Starr, Families of rationally connected varieties, J. Amer. Math. Soc. 16 (no. 1) (2003), 57–67.

  14. 14.

    Hurwitz A.: Ueber Riemann’schen Flächen mit gegebenen Verzweigungspunkten, Math. Ann. 39, 1–61 (1891)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    V. Kanev, Irreducibility of Hurwitz spaces, Preprint n. 241 Dip. di Matematica ed Applicazioni, Universit‘a degli Studi di Palermo; arXiv: math. AG/0509154 (2004).

  16. 16.

    V. Kanev, Hurwitz spaces of Galois coverings of IP 1 whose Galois groups are Weyl groups, J. Algebra 305 (no. 1) (2006), 442–456.

  17. 17.

    P. Kluitmann, Hurwitz action and finite quotients of braid groups, in: Braids (Santa Cruz, CA, 1986), Contemp. Math. 78, Amer. Math. Soc., Providence, RI, (1988), 299–325.

  18. 18.

    S. M. Natanzon, Topology of 2-dimensional coverings and meromorphic functions on real and complex algebraic curves, Selecta Math. (N. S.) 12 (no. 3) (1993), 251–291.

  19. 19.

    R. Scognamillo, Prym - Tjurin varieties and the Hitchin map, Math. Ann. 303 (no. 1) (1995), 47–62.

  20. 20.

    Scott G.P.: Braid groups and the group of homeomorphisms of a surface. Math. Proc. Cambridge Philos. Soc. 68, 605–617 (1970)

    MATH  Article  Google Scholar 

  21. 21.

    F. Severi, Vorlesungen uber algebraische Geometrie, Teubuer, Leibzig, 1921.

  22. 22.

    F. Vetro, Irreducibility of Hurwitz spaces of coverings with one special fiber, Indag. Math. (N. S.) 17 (2006), 115–127.

  23. 23.

    F. Vetro, Irreducibility of Hurwitz spaces of coverings with monodromy groups Weyl groups of type W(B d ), Boll. Unione Mat. Ital. 10 (no. 2) (2007), 405–431.

  24. 24.

    F. Vetro,Connected components of Hurwitz spaces of coverings with one special fiber and monodromy groups contained in a Weyl group of type B d , Boll. Unione Mat. Ital. 1 (no. 1) (2008), 87–103.

  25. 25.

    Vetro F.: Irreducibility of Hurwitz spaces of coverings with one special fiber and monodromy group a Weyl group of type Dd, Manuscripta Math. 125, 353–368 (2008)

    Google Scholar 

  26. 26.

    F. Vetro, On Hurwitz spaces of coverings with one special fiber, Pacific J. Math. 240 (no. 2) (2009), 383–398.

  27. 27.

    Vetro F.: On the irreducibility of Hurwitz spaces of coverings with two special fibers. Georgian Math. J. 19, 375–389 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    F. Vetro, Hurwitz spaces of coverings with two special fibers and monodromy group a Weyl group of type B d , Pacific J. Math. 255 (no. 1) (2012), 241–255.

  29. 29.

    B.Wajnryb, Orbits of Hurwitz action for coverings of a sphere with two special fibers, Indag. Math. (N. S.), 7 (no. 4) (1996), 549–558.

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Correspondence to Francesca Vetro.

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To my mother

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Vetro, F. Irreducible Components of Hurwitz Spaces of Coverings with Two Special Fibers. Mediterr. J. Math. 10, 1151–1170 (2013). https://doi.org/10.1007/s00009-013-0256-2

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Mathematics Subject Classification (2010)

  • Primary 14H30
  • Secondary 14H10

Keywords

  • Hurwitz spaces
  • special fibers
  • branched coverings
  • Weyl group of type B d
  • monodromy
  • braid moves