The non-existence of stable Schottky forms

Let $A_g^S$ be the Satake compactification of the moduli space $A_g$ of principally polarized abelian $g$-folds and $M_g^S$ the closure of the image of the moduli space $M_g$ of genus $g$ curves in $A_g$ under the Jacobian morphism. Then $A_g^S$ lies in the boundary of $A_{g+m}^S$ for any $m$. We prove that $M_{g+m}^S$ and $A_g^S$ do not meet transversely in $A_{g+m}^S$, but rather that their intersection contains the $m$th order infinitesimal neighbourhood of $M_g^S$ in $A_g^S$. We deduce that there is no non-trivial stable Siegel modular form that vanishes on $M_g$ for every $g$. In particular, given two inequivalent positive even unimodular quadratic forms $P$ and $Q$, there is a curve whose period matrix distinguishes between the theta series of $P$ and $Q$.