Schottky via the punctual Hilbert scheme

@article{Gulbrandsen2014SchottkyVT,
  title={Schottky via the punctual Hilbert scheme},
  author={Martin G. Gulbrandsen and Mart'i Lahoz},
  journal={arXiv: Algebraic Geometry},
  year={2014}
}
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. 

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