A note on Torelli-type theorems for Gorenstein curves

TL;DR

This paper extends the theory of adjoint forms to Gorenstein curves using generalized divisors, proving an infinitesimal Torelli-type theorem for vector bundles and providing counterexamples for reducible cases.

Contribution

It introduces a novel adaptation of adjoint form theory to Gorenstein curves and establishes an infinitesimal Torelli theorem for vector bundles on these curves.

Findings

Proves an infinitesimal Torelli-type theorem for Gorenstein curves

Constructs counterexamples for reducible Gorenstein curves

Adapts Hartshorne's generalized divisors to the Torelli context

Abstract

Using the notion of generalized divisors introduced by Hartshorne, we adapt the theory of adjoint forms to the case of Gorenstein curves. We show an infinitesimal Torelli-type theorem for vector bundles on Gorenstein curves. We also construct explicit counterexamples to the infinitesimal Torelli claim in the case of a reduced reducible Gorenstein curve.