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 $\eta_C \in H^1 \hspace{-3pt}\left( \Omega^1_S \right)$. Define $\alpha(C)$ to be the equivalence class of $\eta_C$ in the quotient of $H^1 \hspace{-3pt}\left( \Omega^1_S \right)$ 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 $R=H^0_*(\mathcal{O}_{\mathbb{P}^3})$. 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 \textit{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 $I_{\alpha(D)}=I_{\alpha(C)}$.