An explicit solution to the weak Schottky problem

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