The Castelnuovo-Schottky theorem of Pareschi-Popa characterizes Jacobians, among indecomposable principally polarized abelian varieties $(A,\Theta )$ of dimension $g$, by the existence of $g+2$ points $\Gamma \subset A$ in special position with respect to $2\Theta$, but general with respect to $\Theta$, and furthermore states that such collections of points must be contained in an Abel-Jacobi curve. Building on the ideas in the original paper, we give here a self contained, scheme theoretic proof of the theorem, extending it to finite, possibly...