Tohoku Mathematical Journal

Schottky via the punctual Hilbert scheme

Martin G. Gulbrandsen and Martí Lahoz

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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.

Article information

Tohoku Math. J. (2), Volume 69, Number 4 (2017), 611-619.

First available in Project Euclid: 2 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14H42: Theta functions; Schottky problem [See also 14K25, 32G20]
Secondary: 14C05: Parametrization (Chow and Hilbert schemes)

Schottky problem Hilbert scheme Jacobian theta duality trisecant criterion


Gulbrandsen, Martin G.; Lahoz, Martí. Schottky via the punctual Hilbert scheme. Tohoku Math. J. (2) 69 (2017), no. 4, 611--619. doi:10.2748/tmj/1512183632.

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