  We show that a smooth projective curve of genus $g$ can be reconstructed from
its polarized Jacobian $(X, \Theta)$ as a certain locus in the Hilbert scheme
$\mathrm{Hilb}^d(X)$, for $d=3$ and for $d=g+2$, defined by geometric
conditions in terms of the polarization $\Theta$. The result is an application
of the Gunning--Welters trisecant criterion and the Castelnuovo--Schottky
theorem by Pareschi--Popa and Grushevsky, and its scheme theoretic extension by
the authors.
