van Geemen–van der Geer–Donagi Schottky characterization via second-order theta divisors

Let $(P,\Xi)$ be a principally polarized abelian variety of dimension $p\geq 4$, with $\Xi$ a symmetric theta divisor. Let $\Gamma=H^0(P,2\Xi)$, and let $\Gamma_{00}\subset\Gamma$ consist of sections vanishing to order at least $4$ at the origin. Write $V_{00}$ for the base locus of the linear system $|2\Xi|_{00}$ associated with $\Gamma_{00}$. For a smooth curve $C$ of genus $g$, write $(JC,\Theta)$ for its Jacobian, and let ${\cal J}_g$ denote the closure of the Jacobian locus in the moduli space of principally polarized abelian varieties of dimension $g$. van Geemen–van der Geer–Donagi conjecture.

1. If $(P,\Xi)=(JC,\Theta)$ is the Jacobian of a smooth curve $C$ of genus $g$, then $V_{00}$ is set-theoretically equal to the reduced surface

$$C-C=\{\mathcal O_C(s-t):s,t\in C\}.$$

2. If $(P,\Xi)$ is not in ${\cal J}_g$, then $V_{00}=\{0\}$ set-theoretically.

This gives a proposed characterization of Jacobians among principally polarized abelian varieties through the base locus of second-order theta divisors with multiplicity at least four at the origin. The source attributes the proposal to van Geemen and van der Geer and its more precise formulation to Donagi; its resolution is not specified here.