Schottky via the punctual Hilbert scheme

Abstract

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.