Let $S$ be a smooth algebraic surface in $\mathbb{P}^3$. A curve $C$ in $S$ has a cohomology class $[C] \in H^1\left( \Omega^1_S \right)$. The Hodge class $\alpha(C)$ of $C$ is the equivalence class of $[C]$ in the quotient of $H^1\left( \Omega^1_S \right)$ modulo the subspace generated by the class $[H]$ of a plane section of $S$: the Hodge class depends on the embedding of $S$ in $\mathbb{P}^3$, and can be seen as a linear form on the primitive cohomology $H^1\left( \Omega^1_S \right)^{\perp_H}$.

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 its Hodge class.

I will report on work in progress, joint with Maria Gioia Cifani and Pietro Pirola, in which we give an answer to some of these questions related to the notion of a {\em perfect class}: the Hodge class of a curve $C$ is perfect if its annihilator is a sum of ideals of curves $C_i$ whose Hodge class is a nonzero rational multiple of that of $C$.
We show that the Hodge class of a smooth rational quartic on a surface of degree $4$ is not perfect, and that the Hodge classn of an arithmetically Cohen-Macaulay curve is always perfect.