An explicit solution to the weak Schottky problem

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.