A note on Griffiths infinitesimal invariant for curves

TL;DR

This paper investigates the infinitesimal invariant of the Abel-Jacobi normal function for certain line bundles on generic curves, demonstrating its ability to uniquely determine the curve and line bundle, and revisiting Griffiths' Torelli theorem for genus 4.

Contribution

It shows that the infinitesimal invariant associated to specific line bundles on generic curves can reconstruct the curve and line bundle pair, extending Griffiths' Torelli theorem.

Findings

Infinitesimal invariant reconstructs (C,L) for generic curves of genus ≥4.

Reproduces Griffiths' Torelli theorem for genus 4.

Provides a new perspective on the relationship between normal functions and curve invariants.

Abstract

Given a generic curve of genus $g\geqslant4$ and a smooth point $L\in W_{g-1}^{1}(C)$, whose linear system is base-point free, we consider the Abel-Jacobi normal function associated to $L^{\otimes 2}\otimes \omega_{C}^{-1}$, when $(C,L)$ varies in moduli. We prove that its infinitesimal invariant reconstruct the couple $(C,L)$. When $g=4$, we obtain the generic Torelli theorem proved by Griffiths.