A note on Griffiths infinitesimal invariant for curves

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.