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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Humbert surfaces and the Kummer plane


Authors: Christina Birkenhake and Hannes Wilhelm
Journal: Trans. Amer. Math. Soc. 355 (2003), 1819-1841
MSC (2000): Primary 14K10; Secondary 14H50
Published electronically: January 8, 2003
MathSciNet review: 1953527
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Abstract: A Humbert surface is a hypersurface of the moduli space $\mathcal A_2$ of principally polarized abelian surfaces defined by an equation of the form $az_1+bz_2+cz_3+d(z_2^2-z_1z_3)+e=0$ with integers $a,\ldots,e$. We give geometric characterizations of such Humbert surfaces in terms of the presence of certain curves on the associated Kummer plane. Intriguingly this shows that a certain plane configuration of lines and curves already carries all information about principally polarized abelian surfaces admitting a symmetric endomorphism with given discriminant.


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Additional Information

Christina Birkenhake
Affiliation: Christina Birkenhake, Universität Mainz, Staudingerweg 9, D-55099 Mainz, Germany
Email: birken@Mathematik.uni-mainz.de

Hannes Wilhelm
Affiliation: Hannes Wilhelm, 10 Studley Count, 4 Jamestown Way, London E14 2DA, England
Email: Hannes.Wilhelm@dresdner-bank.com

DOI: https://doi.org/10.1090/S0002-9947-03-03238-0
Received by editor(s): February 11, 2002
Published electronically: January 8, 2003
Additional Notes: Supported by DFG-contracts Bi 448/4-1 and Hu 337/5-1
Article copyright: © Copyright 2003 American Mathematical Society