The Prym characterization by the minimal number of conditions

Let $(A,\Theta)$ be a principally polarized abelian variety of dimension $g$, and let $\Gamma$ be a theta-general, uniform collection of points on $(A,\Theta)$. Suppose that $\Gamma$ imposes generically $n<|\Gamma|$ conditions on $2\Theta$-linear series, and that the divisor class associated to $\Gamma$ is $[2\Theta]$.

The Prym characterization by the minimal number of conditions. Then $n\geq 2g$, and equality characterizes Prym varieties. This is proposed as a higher Castelnuovo–Schottky problem. The motivation is Welters' characterization of Prym varieties through Abel–Prym curves of maximal genus representing twice the minimal curve class; the conjectural equality case is not resolved in the supplied text.