Manuscripta Mathematica

, Volume 141, Issue 1–2, pp 111–124 | Cite as

The Scorza correspondence in genus 3



In this note we prove the genus 3 case of a conjecture of Farkas and Verra on the limit of the Scorza correspondence for curves with a theta-null. Specifically, we show that the limit of the Scorza correspondence for a hyperelliptic genus 3 curve C is the union of the curve \({\{x, \sigma(x) \mid x \in C\}}\) (where σ is the hyperelliptic involution), and twice the diagonal. Our proof uses the geometry of the subsystem Γ00 of the linear system |2Θ|, and Riemann identities for theta constants.


Modulus Space Modular Form Theta Function Abelian Variety Hyperelliptic Curve 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Diaz S.: A bound on the dimensions of complete subvarieties of \({{\mathcal M}_{g}}\). Duke Math. J. 51(2), 405–408 (1984)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Dolgachev, I.: Classical Algebraic Geometry: A Modern View. Cambridge University Press, Cambridge (2012), to appearGoogle Scholar
  3. 3.
    Farkas G.: The birational type of the moduli space of even spin curves. Adv. Math. 223(2), 433–443 (2010)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Farkas, G., Verra, A.: The intermediate type of certain moduli spaces of curves. arXiv:0910.3905, preprint (2009)Google Scholar
  5. 5.
    Grushevsky S., Krichever I.: The universal Whitham hierarchy and the geometry of the moduli space of pointed Riemann surfaces. Geometry of Riemann surfaces and their moduli spaces, Surv. Differ. Geom. 14, 111–129 (2009)MathSciNetGoogle Scholar
  6. 6.
    Grushevsky S., Salvati Manni R.: The vanishing of two-point functions for three-loop superstring scattering amplitudes. Commun. Math. Phys. 294(2), 343–352 (2010)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Harris J., Morrison I.: Moduli of Curves, Volume 187 of Graduate Texts in Mathematics. Springer, New York (1998)Google Scholar
  8. 8.
    Igusa, J.-I.: Theta Functions. Springer, New York, Die Grundlehren der mathematischen Wissenschaften, Band 194 (1972)Google Scholar
  9. 9.
    Igusa, J.-I.: Schottky’s invariant and quadratic forms. In: E.B. Christoffel (Aachen/Monschau, 1979), pp. 352–362. Birkhäuser, Basel (1981)Google Scholar
  10. 10.
    Izadi E.: Fonctions thêta du second ordre sur la Jacobienne d’une courbe lisse. Math. Ann. 289(2), 189–202 (1991)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Mumford D.: On the equations defining abelian varieties. I. Invent. Math. 1, 287–354 (1966)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Mumford D.: On the equations defining abelian varieties. II. Invent. Math. 3, 75–135 (1967)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Mumford D.: On the equations defining abelian varieties. III. Invent. Mat. 3, 215–244 (1967)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Mumford, D.: Tata lectures on theta. II. Modern Birkhäuser Classics. Birkhäuser Boston Inc., Boston, MA, Jacobian theta functions and differential equations, With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman and H. Umemura, Reprint of the 1984 original (2007)Google Scholar
  15. 15.
    Poor C.: The hyperelliptic locus. Duke Math. J. 76(3), 809–884 (1994)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Runge B.: On Siegel modular forms. I. J. Reine Angew. Math. 436, 57–85 (1993)MathSciNetMATHGoogle Scholar
  17. 17.
    Salvati Manni R.: Modular varieties with level 2 theta structure. Am. J. Math. 116(6), 1489–1511 (1994)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Salvati Manni R.: On the projective varieties associated with some subrings of the ring of Thetanullwerte. Nagoya Math. J. 133, 71–83 (1994)MathSciNetMATHGoogle Scholar
  19. 19.
    Salvati Manni R., Top J.: Cusp forms of weight 2 for the group Γ 2(4,8). Am. J. Math. 115(2), 455–486 (1993)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    van Geemen B., van der Geer G.: Kummer varieties and the moduli spaces of abelian varieties. Am. J. Math. 108(3), 615–641 (1986)MATHCrossRefGoogle Scholar
  21. 21.
    Welters G.: The surface CC on Jacobi varieties and 2nd order theta functions. Acta Math. 157(1–2), 1–22 (1986)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Zaal C.: Explicit complete curves in the moduli space of curves of genus three. Geom. Dedicata 56(2), 185–196 (1995)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Zaal C.: A complete surface in M 6 in characteristic >2. Compositio Math. 119(2), 209–212 (1999)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Mathematics DepartmentStony Brook UniversityStony BrookUSA
  2. 2.Dipartimento di MatematicaUniversità “La Sapienza”RomaItaly

Personalised recommendations