The degree of the Jacobian locus and the Schottky problem

TL;DR

This paper establishes a link between the degrees of moduli spaces of abelian varieties and curves with their Hodge classes, leading to explicit formulas and bounds that facilitate solving the Schottky problem algebraically.

Contribution

It provides explicit formulas for degrees of moduli spaces and an effective algebraic approach to the Schottky problem using these degrees.

Findings

Derived explicit degree formulas for A_g and M_g.

Established an upper bound for the degree of M_g.

Reformulated the Schottky problem as a system of algebraic equations.

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.