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

(Received June 30, 2015)
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.

Mathematics Subject Classification. Primary 14H42; Secondary 14C05.
Key words and phrases. Schottky problem, Hilbert scheme, Jacobian, theta duality, trisecant criterion.

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