An explicit solution to the weak Schottky problem

TL;DR

The paper provides an explicit set of polynomial equations in theta constants that characterize Jacobian varieties within the moduli space of abelian varieties, advancing the understanding of the Schottky problem.

Contribution

It introduces a new explicit construction of polynomials in theta constants that contain Jacobians as a component, using Schottky-Jung proportionality and Riemann's quartic relations.

Findings

Polynomials in theta constants define Jacobian locus as an irreducible component.

Explicit formulas derived from Schottky-Jung proportionality.

Connection to classical Riemann quartic relations.

Abstract

We give an explicit weak solution to the Schottky problem, in the spirit of Riemann and Schottky. For any genus $g$, we write down a collection of polynomials in genus $g$ theta constants, such that their common zero locus contains the locus of Jacobians of genus $g$ curves as an irreducible component. These polynomials arise by applying a specific Schottky-Jung proportionality to an explicit collection of quartic identities for theta constants in genus $g-1$, which are suitable linear combinations of Riemann's quartic relations.