  We study the cone of moving divisors on the moduli space ${\mathcal A}_g$ of
principally polarized abelian varieties. Partly motivated by the generalized
Rankin-Cohen bracket, we construct a non-linear holomorphic differential
operator that sends Siegel modular forms to Siegel modular forms, and we apply
it to produce new modular forms. Our construction recovers the known divisors
of minimal moving slope on ${\mathcal A}_g$ for $g\leq 4$, and gives an
explicit upper bound for the moving slope of ${\mathcal A}_5$ and a conjectural
upper bound for the moving slope of ${\mathcal A}_6$.
