Reconstructing curves from their Hodge classes

Let $S$ be a smooth algebraic surface in $mathbb{P}^3(mathbb{C})$. A curve $C$ in $S$ has a cohomology class $eta_C in H^1 hspace{-3pt}left( Omega^1_S ight)$. Define $alpha(C)$ to be the equivalence class of $eta_C$ in the quotient of $H^1 hspace{-3pt}left( Omega^1_S ight)$ modulo the subspace generated by the class $eta_H$ of a plane section of $S$. In the paper "Reconstructing subvarieties from their periods" the authors Movasati and Sert"{o}z pose several interesting questions about the reconstruction of $C$ from the annihilator $I_{alpha(C)}$ of $alpha(C)$ in the polynomial ring $R=H^0_*(cO_{P^3})$. It contains the homogeneous ideal of $C$, but is much larger as $R/I_{alpha(C)}$ is artinian. We give sharp numerical conditions that guarantee $C$ is reconstructed by forms of low degree in $I_{alpha(C)}$. We also show it is not always the case that the class $alpha(C)$ is extit{perfect}, that is, that $I_{alpha(C)}$ could be bigger than the sum of the Jacobian ideal of $S$ and of the homogeneous ideals of curves $D$ in $S$ for which $I_{alpha(D)}=I_{alpha(C)}$