Reconstructing curves from their Hodge classes

TL;DR

This paper investigates how to reconstruct algebraic curves on surfaces from their Hodge classes, providing conditions under which the curves can be uniquely recovered from associated algebraic data.

Contribution

It establishes sharp numerical criteria for reconstructing curves from their Hodge class annihilators and explores the limitations of this reconstruction process.

Findings

Reconstruction is possible under certain numerical conditions.

The class  is not always perfect, affecting reconstruction.

Provides bounds and conditions for low-degree form reconstruction.

Abstract

Let $S$ be a smooth algebraic surface in $\mathbb{P}^3(\mathbb{C})$. A curve $C$ in $S$ has a cohomology class $\eta_C \in H^1 \hspace{-3pt}\left( \Omega^1_S \right)$. 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 $I_{\alpha(D)}=I_{\alpha(C)}$.