Schottky via the punctual Hilbert scheme

TL;DR

This paper demonstrates how a smooth projective curve can be reconstructed from its polarized Jacobian using specific loci in the Hilbert scheme, extending classical theorems with scheme-theoretic methods.

Contribution

It introduces a novel scheme-theoretic approach to reconstructing curves from their Jacobians via Hilbert schemes, extending classical geometric criteria.

Findings

Reconstruction of curves from Jacobians using Hilbert scheme loci

Application of Gunning--Welters trisecant criterion

Extension of the Castelnuovo--Schottky theorem scheme-theoretically

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.