Humbert surfaces and the Kummer plane

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.