A Generalized Torelli Theorem

TL;DR

This paper extends the classical Torelli theorem, demonstrating that a curve can be uniquely determined by its Jacobian and certain symmetric power images for a broader range of degrees.

Contribution

It generalizes the Torelli theorem by replacing the symmetric power degree with any integer between 1 and g-1, broadening the theorem's applicability.

Findings

The Torelli theorem holds for all degrees d in 1 to g-1.

The pair (J(C), W^d) determines the curve C for these degrees.

The result unifies and extends previous cases of the theorem.

Abstract

Given a smooth projective curve $C$ of genus $g$ over the complex numbers, Torelli's thoerem asserts that the pair $(J(C),W^{g-1})$ determines $C$, where $W^{g-1}$ is an image of the $g-1$st symmetric power of $C$ inside the Jacobian under an Abel-Jacobi map. We show that the theorem holds with $g-1$ replaced by an integer $d$ in the range $1\le d\le g-1$.