Differentiating Siegel modular forms, and the moving slope of ${\mathcal{A}}_g$

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 $g\le 4$, and gives an explicit upper bound for the moving slope of ${\mathcal{A}}_5$ and a conjectural upper bound for the moving slope of ${\mathcal{A}}_6$.