Advertisement

Functional Analysis and Its Applications

, Volume 46, Issue 2, pp 110–120 | Cite as

Real normalized differentials and Arbarello’s conjecture

  • I. M. Krichever
Article

Abstract

Using meromorphic differentials with real periods, we prove Arbarello’s conjecture that any compact complex cycle of dimension g - n in the moduli space M g of smooth algebraic curves of genus g must intersect the locus of curves having a Weierstrass point of order at most n.

Key words

moduli space of algebraic curves integrable system real normalized differential 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    E. Arbarello, “Weierstrass points and moduli of curves,” Compositio Math., 29 (1974), 325–342.MathSciNetMATHGoogle Scholar
  2. [2]
    E. Arbarello and G. Mondello, “Two remarks on the Weierstrass flag,” in: Compact Moduli spaces and Vector Bundles, Contemp. Math. (Proceedings) series, vol. 564, Amer. Math. Soc., Providence, RI, 2012, 137–144.CrossRefGoogle Scholar
  3. [3]
    Integrability: The Seiberg-Witten and Whitham Equations (eds. H. W. Braden and I. M. Krichever), Gordon and Breach Science Publishers, Amsterdam, 2000.MATHGoogle Scholar
  4. [4]
    S. Diaz, “Exceptional Weierstrass points and the divisor on moduli space that they define,” Mem. Amer. Math. Soc., 56:327 (1985).Google Scholar
  5. [5]
    A. Gorsky, I. Krichever, A. Marshakov, A. Mironov, and A. Morozov, “Integrability and Seiberg-Witten exact solution,” Phys. Lett. B, 355 (1995), 466–474.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    J. Harris and I. Morrison, Moduli of Curves, Graduate Texts in Math., vol. 187, Springer-Verlag, New York, 1998.Google Scholar
  7. [7]
    S. Grushevsky and I. Krichever, “The universalWhitham hierarchy and geometry of the moduli space of pointed Riemann surfaces,” in: Surveys in Differ. Geom., vol. 14, Int. Press, Somerville, MA, 2010, 111–129.Google Scholar
  8. [8]
    S. Grushevsky and I. Krichever, Foliations on the Moduli Space of Curves, Vanishing in Cohomology, and Calogero-Moser Curves, http://arxiv.org/abs/1108.4211.
  9. [9]
    I. M. Krichever, “Spectral theory of finite-zone nonstationary Schrödinger operators. A nonstationary Peierls model,” Funkts. Anal. Prilozhen., 20:3 (1986), 42–54; English transl.: Funct. Anal. Appl., 20:3 (1986), 203–214.MathSciNetGoogle Scholar
  10. [10]
    I. M. Krichever, “Method of an averaging for two-dimensional “integrable” equations,” Funkts. Anal. Prilozhen., 22:3 (1988), 37–52; English transl.: Funct. Anal. Appl., 22:3 (1988), 200–213.MathSciNetGoogle Scholar
  11. [11]
    I. Krichever, “The τ -function of the universal Whitham hierarchy, matrix models, and topological field theories,” Comm. Pure Appl. Math., 47:4 (1994), 437–475.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    I. Krichever and D. H. Phong, “On the integrable geometry of soliton equations and N = 2 supersymmetric gauge theories,” J. Differential Geom., 45:2 (1997), 349–389.MathSciNetMATHGoogle Scholar
  13. [13]
    I. Krichever and D. H. Phong, “Symplectic forms in the theory of solitons,” in: Surveys in Differ. Geom., vol. 4, Int. Press, Boston, MA, 1998, 239–313.Google Scholar
  14. [14]
    E. Looijenga, “On the tautological ring of Mg,” Invent. Math., 121:2 (1995), 411–419.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Columbia UniversityNew YorkUSA
  2. 2.Institute for Theoretical PhysicsKharkevich Institute for Problems of Information TransmissionMoscowRussia

Personalised recommendations