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International Mathematics Research Notices (2008) Vol. 2008 : article ID rnm128, 19 pages, doi:10.1093/imrn/rnm128 published on January 29, 2008
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Copyright © The Author 2008. Published by Oxford University Press.

Some Intersections in the Poincaré Bundle and the Universal Theta Divisor on Formula

Samuel Grushevsky1,

David Lehavi2

1 Mathematics Department, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544, USA
2 Mathematics Department, University of Michigan, 2074 East Hall, 530 Church St, Ann Arbor, USA

Correspondence: Correspondence to be sent to: sam{at}math.princeton.edu

We compute all the top intersection numbers of divisors on the total space of the Poincaré bundle restricted to B x C (where B is an abelian variety, and C sub B is any test curve). We use these computations to find the class of the universal theta divisor and m-theta divisor inside the universal corank 1 semiabelian variety — the boundary of the partial toroidal compactification of the moduli space of abelian varieties. We give two computational examples: we compute the boundary coefficient of the Andreotti–Mayer divisor (computed by Mumford but in a much harder and ad hoc way), and the analog of this for the universal m-theta divisor.

communicated by Prof. Enrico Arbarello



References

  1. Alexeev V. Complete moduli in the presence of semiabelian group action. Annals of Mathematics (2002) 155:611–708.[Web of Science]
  2. Alexeev V., Nakamura I. On Mumford's construction of degenerating abelian varieties. Tohoku Mathematical Journal (1999) 51(no. 2):399–420.[CrossRef][Web of Science]
  3. Andreotti A., Mayer A. L. On period relations for abelian integrals on algebraic curves. Annali della Scuola Normale Superiore di Pisa (1967) 21(no. 3):189–238.
  4. Birkenhake Ch, Lange H. Complex Abelian Varieties. (2004) 302, 2nd ed. Berlin: Springer. Grundlehren der mathematischen Wissenschaften.
  5. Faltings G., Chai C.-L. Degeneration of Abelian Varieties. (1990) Berlin: Springer. Ergebnisse der Mathematik und ihrer Grenzgebiete 22.
  6. Fulton W. Intersection Theory. (1998) 2nd ed. Berlin: Springer. Ergebnisse der Mathematik und ihrer Grenzgebiete 3.
  7. Hulek K. "Degenerations of abelian varieties." Lecture notes of Pragmatic Summer School. (2000) 7. available from http://www-ifm.math.uni-hannover.de/~hulek.
  8. Hulek K., Weintraub S. The principal degenerations of abelian surfaces and their polarisations. Mathematische Annalen (1990) 286:281–307.[CrossRef][Web of Science]
  9. Hulek K., Kahn C., Weintraub S. Moduli Spaces of Abelian Surfaces: Compactification, Degenerations, and Theta Functions. (1993) Berlin: de Gruyter. Expositions in Mathematics 12.
  10. Milne J. Abelian varieties. Version 1.1. Ann Arbor, MI: Self-Published, 1998. http://jmilne.org/math/index.html (accessed October 19, 2007).
  11. Mumford D. On the Kodaira Dimension of the Siegel Modular Variety. (1983) Berlin: Springer. 348–75. Lecture Notes in Mathematics 997.
  12. Olsson M. Canonical compactifications of moduli spaces for abelian varieties. University of California - Berkeley. http://math.berkeley.edu/~molsson.
  13. Shepherd-Barron N. Perfect forms and the moduli space of abelian varieties. Inventiones Mathematicae (2006) 163:25–45.[CrossRef][Web of Science]
  14. Tai Y.-S. On the Kodaira dimension of the moduli space of abelian varieties. Inventiones Mathematicae (1982) 68:425–39.[CrossRef][Web of Science]
  15. Yoshikawa K.-I. Discriminant of theta divisors and Quillen metrics. Journal of Differential Geometry (1999) 52:73–115.[Web of Science]

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This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
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Right arrow Email this article to a friend
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Right arrow Articles by Grushevsky, S.
Right arrow Articles by Lehavi, D.
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