Mathematische Zeitschrift

, Volume 267, Issue 3–4, pp 549–582 | Cite as

Geometry of curves with exceptional secant planes: linear series along the general curve

Article

Abstract

We study linear series on a general curve of genus g, whose images are exceptional with regard to their secant planes. Working in the framework of an extension of Brill–Noether theory to pairs of linear series, we prove that a general curve has no linear series with exceptional secant planes, in a very precise sense, whenever the total number of series is finite. Next, we partially solve the problem of computing the number of linear series with exceptional secant planes in a one-parameter family in terms of tautological classes associated with the family, by evaluating our hypothetical formula along judiciously-chosen test families. As an application, we compute the number of linear series with exceptional secant planes on a general curve equipped with a one-dimensional family of linear series. We pay special attention to the extremal case of d-secant (d − 2)-planes to (2d − 1)-dimensional series, which appears in the study of Hilbert schemes of points on surfaces. In that case, our formula may be rewritten in terms of hypergeometric series, which allows us both to prove that it is nonzero and to deduce its asymptotics in d.

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References

  1. 1.
    Arbarello E., Cornalba M., Griffiths P., Harris J.: Geometry of Algebraic Curves, Grundlehren der Math. Wiss., vol. 267. Springer, New York (1985)Google Scholar
  2. 2.
    Cotterill, E.: Geometry of curves with exceptional secant planes. arXiv:0706.2049Google Scholar
  3. 3.
    Cotterill, E.: Geometry of curves with exceptional secant planes: effective divisors on \({\overline{\mathcal{M}}_g}\) (in preparation)Google Scholar
  4. 4.
    Deutsch E.: Dyck path enumeration. Discret. Math. 204(1–3), 167–202 (1999)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Diaz S., Harris J.: Geometry of the Severi variety. Trans. Am. Math. Soc. 309(1), 1–34 (1988)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Du, R., Yin, J.: Counting labelled trees with given indegree sequence. http://arXiv.org/pdf/0712.4032v1
  7. 7.
    Eisenbud D., Harris J.: Divisors on general curves and cuspidal rational curves. Invent. Math. 74(3), 371–418 (1983)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Eisenbud D., Harris J.: A simpler proof of the Gieseker–Petri theorem on special divisors. Invent. Math. 74(2), 269–280 (1983)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Eisenbud D., Harris J.: Limit linear series: basic theory. Invent. Math. 85(2), 337–371 (1986)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Eisenbud D., Harris J.: Irreducibility and monodromy of some families of linear series. Ann. Sci. École Norm. Sup. (4) 20(1), 65–87 (1987)MathSciNetMATHGoogle Scholar
  11. 11.
    Farkas G.: Higher ramification and varieties of secant divisors on the generic curve. J. Lond. Math. Soc. (2) 78(2), 418–440 (2008)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Gasper G., Rahman M.: Basic Hypergeometric Series, Encyclopedia of Mathematics and its Applications, vol. 96. Cambridge University Press, New York (2004)Google Scholar
  13. 13.
    Harris J., Morrison I.: Slopes of effective divisors on the moduli space of curves. Invent. Math. 99(2), 321–355 (1990)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Harris J., Tu L.: Chern numbers of kernel and cokernel bundles. Invent. Math. 75, 467–475 (1984)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Khosla, D.: Moduli of curves with linear series. Harvard doctoral thesis (2005)Google Scholar
  16. 16.
    Koepf W.: Hypergeometric Summation: An Algorithmic Apporach to Summation and Special Function Identities. Vieweg, Braunschweig/Wiesbaden (1998)Google Scholar
  17. 17.
    Le Barz P.: Sur les espaces multisécants aux courbes algébriques. Manuscr. Math. 119, 433–452 (2006)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Le Barz P.: Sur une formule de Castelnuovo pur les espaces multisécants. Bollettino U.M.I. 10-B, 381–387 (2007)MathSciNetGoogle Scholar
  19. 19.
    Lehn M.: Chern classes of tautological sheaves on Hilbert schemes of points on surfaces. Invent. Math. 136(1), 157–207 (1999)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Macdonald I.G.: Some enumerative formulae for algebraic curves. Proc. Camb. Philos. Soc. 54, 399–416 (1958)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Petkovšek M., Wilf H., Zeilberger D.: A = B. A. K. Peters, Wellesley (1996)MATHGoogle Scholar
  22. 22.
    Ran Z.: Curvilinear enumerative geometry. Acta. Math. 155(1–2), 81–101 (1985)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Ran Z.: Geometry on nodal curves. Compos. Math. 141(5), 1191–1212 (2005)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Ran, Z.: Geometry and intersection theory on Hilbert schemes of families of nodal curves. arXiv:0803.4512Google Scholar
  25. 25.
    Stanley R.: Enumerative Combinatorics, vol. 2. Cambridge Studies in Advanced Math. 49. Cambridge University Press, Cambridge (1997)Google Scholar
  26. 26.
    Shin, H., Zeng, J.: Proof of two combinatorial results arising in algebraic geometry. arXiv:0805.0067Google Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsQueen’s UniversityKingstonCanada

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