Reconstructing curves from their Hodge classes

Abstract

Let $S$ be a smooth algebraic surface in ${\mathbb{P}}^3(\mathbb{C})$. A curve $C$ in $S$ has a cohomology class ${\eta }_C\in H^1\left({\Omega }_S^1\right)$. Define $\alpha (C)$ to be the equivalence class of ${\eta }_C$ in the quotient of $H^1\left({\Omega }_S^1\right)$ 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 Sertöz pose several interesting questions about the reconstruction of $C$ from the annihilator $I_{\alpha (C)}$ of $\alpha (C)$ in the polynomial ring $R=H_{\ast }^0({\mathcal{O}}_{{\mathbb{P}}^3})$. 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 $I_{\alpha (D)}=I_{\alpha (C)}$.