6533b823fe1ef96bd127dfd9

RESEARCH PRODUCT

Humbert surfaces and the Kummer plane

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A Humbert surface is a hypersurface of the moduli space A 2 \mathcal A_2 of principally polarized abelian surfaces defined by an equation of the form a z 1 + b z 2 + c z 3 + d ( z 2 2 − z 1 z 3 ) + e = 0 az_1+bz_2+cz_3+d(z_2^2-z_1z_3)+e=0 with integers a , … , e 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.