Skip to main content
SpringerLink
Log in

Theta divisors with curve summands and the Schottky problem

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We prove the following converse of Riemann’s Theorem: let \((A,\Theta )\) be an indecomposable principally polarized abelian variety whose theta divisor can be written as a sum of a curve and a codimension two subvariety \(\Theta =C+Y\). Then C is smooth, A is the Jacobian of C, and Y is a translate of \(W_{g-2}(C)\). As applications, we determine all theta divisors that are dominated by a product of curves and characterize Jacobians by the existence of a d-dimensional subvariety with curve summand whose twisted ideal sheaf is a generic vanishing sheaf.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. A priori \(n\ge \dim (X)\), but by [21, Lem. 6.1], we may actually assume \(n=\dim (X)\).

  2. In fact, Pareschi and Popa treat the more general case of an equidimensional closed reduced subscheme \(Z\subseteq A\), but for our purposes the case of subvarieties will be sufficient.

References

  1. Arbarello, E., Cornalba, M., Griffiths, P.A., Harris, J.: Geometry of Algebraic Curves I. Springer, New York (1985)

    Book  MATH  Google Scholar 

  2. Birkenhake, C., Lange, H.: Complex Abelian Varieties, 2nd edn. Springer, New York (2004)

  3. Clemens, C.H., Griffiths, P.A.: The intermediate Jacobian of the cubic threefold. Ann. Math. 95, 281-356 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  4. Debarre, O.: Minimal cohomology classes and Jacobians. J. Algebraic Geom. 4(2), 321-335 (1995)

    MathSciNet  MATH  Google Scholar 

  5. Debarre, O.: Tores et variétés abéliennes complex, Cours Spécialisés 6. Société Mathématique de France, EDP Sciences, Paris (1999)

    Google Scholar 

  6. Deligne, P.: La conjecture de Weil pour les surfaces K3. Invent. Math. 15, 206-226 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ein, L., Lazarsfeld, R.: Singularities of theta divisors and the birational geometry of irregular varieties. J. Am. Math. Soc. 10, 243-258 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gunning, R.C.: Some curves in abelian varieties. Invent. Math. 66, 377-389 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  9. Grushevsky, S.: The Schottky problem. In: Current Developments in Algebraic Geometry, MSRI Publications, vol. 59, pp. 129-164. Cambridge University Press, Cambridge (2012)

  10. Hoyt, W.L.: On products and algebraic families of Jacobian varieties. Ann. Math. 77, 415-423 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  11. Huybrechts, D.: Fourier-Mukai Transforms in Algebraic Geometry. Oxford Mathematical Monographs, Oxford (2006)

    Book  MATH  Google Scholar 

  12. Krichever, I.: Characterizing Jacobians via trisecants of the Kummer variety. Ann. Math. 172, 485-516 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Little, J.: Correction to: On Lie’s approach to the study of translation manifolds. http://mathcs.holycross.edu/ little/Corrs.html (on his personal webpage)

  14. Little, J.: On Lie’s approach to the study of translation manifolds. J. Differ. Geom. 26, 253-272 (1987)

    MathSciNet  MATH  Google Scholar 

  15. Matsusaka, T.: On a characterization of a Jacobian variety. Mem. Coll. Sci. Kyoto Ser. A Math. 32, 1-19 (1959)

    MathSciNet  MATH  Google Scholar 

  16. Martens, H.H.: An extended Torelli theorem. Am. J. Math. 87, 257-261 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  17. Pareschi, G., Popa, M.: Generic vanishing and minimal cohomology classes on abelian varieties. Math. Ann. 340, 209-222 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ran, Z.: A characterization of five-dimensional Jacobian varieties. Invent. Math. 67, 395-422 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ran, Z.: On a theorem of Martens. Rend. Sem. Mat. Univers. Politec. Torino 44, 287-291 (1986)

    MathSciNet  MATH  Google Scholar 

  20. Ran, Z.: On subvarieties of abelian varieties. Invent. Math. 62, 459-479 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  21. Schoen, C.: Varieties dominated by product varieties. Int. J. Math. 7, 541-571 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  22. Serre, J.-P.: Letter to Grothendieck, March 31. In: Grothendieck-Serre Correspondence, p. 2004. AMS, Providence (1964)

  23. Weil, A.: Zum Beweis des Torellischen Satzes. Nachr. Akad. Wiss. Göttingen, 33-53 (1957)

  24. Welters, G.E.: A characterization of non-hyperelliptic Jacobi varieties. Invent. Math. 74, 437-440 (1983)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

I would like to thank my advisor D. Huybrechts for constant support, encouragement and discussions about the DPC problem. Thanks go also to C. Schnell for his lectures on generic vanishing theory, held in Bonn during the winter semester 2013/14, where I learned about GV-sheaves and Ein–Lazarsfeld’s result [7]. I am grateful to J. Fresan, D. Kotschick, L. Lombardi and M. Popa for useful comments. Special thanks to the anonymous referee for helpful comments and corrections. The author is member of the BIGS and the SFB/TR 45 and supported by an IMPRS Scholarship of the Max Planck Society.

Author information

Authors and Affiliations

  1. Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111, Bonn, Germany

    Stefan Schreieder

  2. Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Stefan Schreieder

Authors
  1. Stefan Schreieder
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stefan Schreieder.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Schreieder, S. Theta divisors with curve summands and the Schottky problem. Math. Ann. 365, 1017–1039 (2016). https://doi.org/10.1007/s00208-015-1287-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-015-1287-8

Mathematics Subject Classification

Search

Navigation