Torelli's theorem for high degree symmetric products of curves

TL;DR

This paper proves that for most genera, the symmetric products of two smooth projective curves uniquely determine the curves themselves, extending Martens' theorem to higher degrees.

Contribution

It extends Martens' theorem by showing that symmetric products of curves of genus g > 2 uniquely determine the curve up to isomorphism.

Findings

For g ≠ 2, symmetric products of curves are unique to the curve.

The case g=2 remains exceptional with non-uniqueness.

Provides a Torelli-type theorem for symmetric products.

Abstract

We show that two smooth projective curves C_1 and C_2 of genus g which have isomorphic symmetric products are isomorphic unless g=2. This extends a theorem of Martens.