The degree of the Jacobian locus and the Schottky problem

2004/02/29 by Samuel Grushevsky, Grushevsky, Samuel
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #math.AG

paper · pdf · doi:10.48550/arxiv.math/0403009

arxiv created 2004/02/29 · openalex publication_date 2004/02/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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