The base-locus characterization of Jacobians via the Gauss map

Let $(A,\Theta)$ be a principally polarized abelian variety, and for a fixed $p\in\mathbb P^{g-1}$ let $Z_p\subset |2\Theta|$ be the linear system generated by the divisors

$$\Theta_x\cup\Theta_{-x}$$

where $x$ lies in $\Theta^{\rm sing}$ or in the fiber $\mathcal G^{-1}(p)$ of the Gauss map. Since $x\in\Theta$, the origin belongs to every such divisor and hence to the base locus of $Z_p$. Base-locus characterization. If $(A,\Theta)$ is not a Jacobian, then for some $p\in\mathbb P^{g-1}$ the base locus of $Z_p$ is zero-dimensional in a neighborhood of the origin. This is a reformulation of a conjecture attributed in the source to [BD]; the assertion gives a geometric criterion distinguishing Jacobians from non-Jacobian principally polarized abelian varieties, but its resolution is not specified in the supplied text.