## Tohoku Mathematical Journal 2017 December SECOND SERIES VOL. 69, NO. 4

 Tohoku Math. J. 69 (2017), 611-619

Title SCHOTTKY VIA THE PUNCTUAL HILBERT SCHEME

Author Martin G. Gulbrandsen and Martí Lahoz

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.