Humbert surfaces and the Kummer plane

Birkenhake, Christina; Wilhelm, Hannes

doi:10.1090/S0002-9947-03-03238-0

Humbert surfaces and the Kummer plane
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by Christina Birkenhake and Hannes Wilhelm PDF

Trans. Amer. Math. Soc. 355 (2003), 1819-1841 Request permission

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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