Characters of fundamental groups of curve complements and orbifold pencils

  • Enrique Artal Bartolo
  • Jose Ignacio Cogolludo-Agustín
  • Anatoly Libgober
Conference paper
Part of the CRM Series book series (PSNS)

Abstract

The present work is a user’s guide to the results of [7], where a description of the space of characters of a quasi-projective variety was given in terms of global quotient orbifold pencils.

Below we consider the case of plane curve complements. In particular, an infinite family of curves exhibiting characters of any torsion and depth 3 will be discussed. Also, in the context of line arrangements, it will be shown how geometric tools, such as the existence of orbifold pencils, can replace the group theoretical computations via fundamental groups when studying characters of finite order, specially order two. Finally, we revisit an Alexander-equivalent Zariski pair considered in the literature and show how the existence of such pencils distinguishes both curves.

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References

  1. [1]
    A. Adem, J. Leida and Y. Ruan, “Orbifolds and Stringy Topology”, Cambridge University Press. 2007.Google Scholar
  2. [2]
    D. Arapura, Geometry of cohomology support loci for local systems I, J. of Alg. Geom. 6 (1997), 563–597.MATHMathSciNetGoogle Scholar
  3. [3]
    E. Artal, Sur les couples de Zariski, J. Algebraic Geom. 4 (1994), 223–247.Google Scholar
  4. [4]
    E. Artal, J. Carmona and J. I. Cogolludo, Essential coordinate components of characteristic varieties, Math. Proc. Cambridge Philos. Soc. 136 (2004), 287–299.CrossRefMATHMathSciNetGoogle Scholar
  5. [5]
    E. Artal, J. Carmona, J. I. Cogolludo and M.Á. Marco, Invariants of combinatorial line arrangements and Rybnikov’s example, Singularity theory and its applications, Adv. Stud. Pure Math., vol. 43, Math. Soc. Japan, Tokyo, 2006, pp. 1–34.Google Scholar
  6. [6]
    E. Artal and J. I. Cogolludo, On the connection between fundamental groups and pencils with multiple fibers, J. Singul. 2 (2010), 1–18.CrossRefMathSciNetGoogle Scholar
  7. [7]
    E. Artal, J. I. Cogolludo-Agustín and A. Libgober, Depth of cohomology support loci for quasi-projective varieties via orbifold pencils, J. Reine Angew. Math., to appear, also available at arXiv:1008.2018 [math.AG].Google Scholar
  8. [8]
    E. Artal, J. I. Cogolludo-Agustín and D. Matei, Characteristic varieties of quasi-projective manifolds and orbifolds, Preprint available at arXiv:1005.4761v2 [math.AG], 2010.Google Scholar
  9. [9]
    E. Artal, J. I. Cogolludo and H.O. Tokunaga, A survey on Zariski pairs, Algebraic geometry in East Asia—Hanoi 2005, Adv. Stud. Pure Math., vol. 50, Math. Soc. Japan, Tokyo, 2008, pp. 1–100.Google Scholar
  10. [10]
    G. Barthel, F. Hirzebruch and T. Höfer, “Geradenkonfigurationen und Algebraische Flächen”, Friedr. Vieweg & Sohn, Braunschweig, 1987.CrossRefGoogle Scholar
  11. [11]
    J. I. Cogolludo-Agustín and A. Libgober, Mordell-Weil groups of elliptic threefolds and the Alexander module of plane curves, Preprint available at arXiv:1008.2018v2 [math.AG], 2010.Google Scholar
  12. [12]
    J. I. Cogolludo-Agustín and M.Á. Marco Buzunáriz, The Max Noether fundamental theorem is combinatorial, Preprint available at arXiv:1002.2325v1 [math.AG], 2009.Google Scholar
  13. [13]
    J. I. Cogolludo and V. Florens, Twisted Alexander polynomials of plane algebraic curves, J. Lond. Math. Soc. (2) 76 (2007), 105–121.CrossRefMATHMathSciNetGoogle Scholar
  14. [14]
    D. C. Cohen and A. I. Suciu, Characteristic varieties of arrangements, Math. Proc. Cambridge Philos. Soc. 127 (1999), 33–53.CrossRefMATHMathSciNetGoogle Scholar
  15. [15]
    A. Dimca, Pencils of plane curves and characteristic varieties Preprint available at math.AG/0606442, 2006.Google Scholar
  16. [16]
    A. Dimca, S. Papadima and A.I. Suciu, Formality, Alexander invariants, and a question of Serre, Preprint available at arXiv:math/0512480v3 [math.AT], 2005.Google Scholar
  17. [17]
    D. Eisenbud, “Commutative Algebra. With a View Toward Algebraic Geometry”, Graduate Texts in Mathematics, Vol. 150. Springer-Verlag, New York, 1995.Google Scholar
  18. [18]
    M. Falk, Arrangements and cohomology, Ann. Comb. 1 (1997), 135–157.CrossRefMATHMathSciNetGoogle Scholar
  19. [19]
    R. Friedman and J. W. Morgan, “Smooth Four-manifolds and Complex Surfaces”, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 27, Springer-Verlag, Berlin, 1994.Google Scholar
  20. [20]
    E. Hironaka, Abelian coverings of the complex projective plane branched along configurations of real lines, Mem. Amer. Math. Soc. 105 (1993), vi+85.MathSciNetGoogle Scholar
  21. [21]
    W. Magnus, A. Karrass and D. Solitar, Combinatorial group theory, second ed., Dover Publications Inc., Mineola, NY, 2004, Presentations of groups in terms of generators and relations.MATHGoogle Scholar
  22. [22]
    A. Libgober, On the homology of finite abelian coverings, Topology Appl. 43 (1992), 157–166.CrossRefMATHMathSciNetGoogle Scholar
  23. [23]
    A. Libgober Characteristic varieties of algebraic curves, Applications of algebraic geometry to coding theory, physics and computation (Eilat, 2001), Kluwer Acad. Publ., Dordrecht, 2001, pp. 215–254.CrossRefGoogle Scholar
  24. [24]
    A. Libgober, Non vanishing loci of Hodge numbers of local systems, Manuscripta Math. 128 (2009), 1–31.CrossRefMATHMathSciNetGoogle Scholar
  25. [25]
    A. Libgober and S. Yuzvinsky, Cohomology of local systems, Arrangements—Tokyo 1998, Kinokuniya, Tokyo, 2000, pp. 169–184.Google Scholar
  26. [26]
    M. Oka, Alexander polynomial of sextics, J. Knot Theory Ramifications 12 (2003), 619–636.CrossRefMATHMathSciNetGoogle Scholar
  27. [27]
    M. Sakuma, Homology of abelian coverings of links and spatial graphs, Canad. J. Math. 47 (1995), 201–224.CrossRefMATHMathSciNetGoogle Scholar
  28. [28]
    P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), 401–487.CrossRefMATHMathSciNetGoogle Scholar
  29. [29]
    C. Simpson, Subspaces of moduli spaces of rank one local systems, Ann. Sci. Scuola Norm. Sup. (4) 26 (1993), 361–401.MATHGoogle Scholar
  30. [30]
    A.I. Suciu, Fundamental groups of line arrangements: enumerative aspects, Advances in algebraic geometry motivated by physics (Lowell, MA, 2000), Contemp. Math., vol. 276, Amer. Math. Soc., 2001, pp. 43–79.Google Scholar
  31. [31]
    A.I. Suciu, Translated tori in the characteristic varieties of complex hyperplane arrangements, Topology Appl. 118 (2002), no. 1-2, 209–223, Arrangements in Boston: a Conference on Hyperplane Arrangements (1999).CrossRefMATHMathSciNetGoogle Scholar
  32. [32]
    H. O. Tokunaga, Some examples of Zariski pairs arising from certain K 3 surfaces, Math. Z. 227 (1998), no. 3, 465–477.CrossRefMATHMathSciNetGoogle Scholar
  33. [33]
    O. Zariski, On the problem of existence of algebraic functions of two variables possessing a given branch curve, Amer. J. of Math. 51 (1929).Google Scholar

Copyright information

© Scuola Normale Superiore Pisa 2012

Authors and Affiliations

  • Enrique Artal Bartolo
  • Jose Ignacio Cogolludo-Agustín
  • Anatoly Libgober

There are no affiliations available

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