The degree of the Jacobian locus and the Schottky problem

Abstract

We show that the degree of the images of the moduli space of (principally polarized) abelian varieties A_g and of the moduli space of curves M_g in the projective space under the theta constant embedding are equal to the top self-intersection numbers of one half the first Hodge class on them. This allows us to obtain an explicit formula for the degree of A_g, and an explicit upper bound for the degree of M_g. Knowing the degree of A_g allows us to effectively determine the subvariety itself, i.e. to effectively obtain all polynomial equations satisfied by theta constants. Furthermore, combining the bound on the degree of M_g with effective Nullstellensatz allows us to rewrite the Kadomtsev-Petvsiashvili (KP) partial differential equation as a system of algebraic equations for theta constants, and thus obtain an effective algebraic solution to the Schottky problem.