A note on Griffiths infinitesimal invariant for curves

Given a generic curve of genus $g⩾4$ and a smooth point $L\in W_{g-1}^{1}©$, 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.