On the rank of quadratic equations for curves of high degree

TL;DR

This paper investigates the generation of the ideal of high-degree, linearly normal algebraic curves by quadratic equations of low rank, improving known bounds for specific genus and degree conditions.

Contribution

It demonstrates that the ideal can be generated by quadratic equations of rank 3 under certain genus and degree conditions, refining previous results.

Findings

Quadratic equations of rank 3 generate the ideal for genus 0,1 with degree ≥ 2g+2.

Quadratic equations of rank 3 generate the ideal for genus ≥ 2 with degree ≥ 4g+4.

Improves bounds on the rank of quadratic generators for algebraic curve ideals.

Abstract

Let $\mathcal{C} \subset \mathbb{P}^r$ be a linearly normal curve of arithmetic genus $g$ and degree $d$. In \cite{SD}, B. Saint-Donat proved that the homogeneous ideal $I(\mathcal{C})$ of $\mathcal{C}$ is generated by quadratic equations of rank at most $4$ whenever $d \geq 2g+2$. Also, in \cite{EKS} Eisenbud, Koh and Stillman proved that $I(\mathcal{C})$ admits a determinantal presentation if $d \geq 4g+2$. In this paper, we will show that $I(\mathcal{C})$ can be generated by quadratic equations of rank $3$ if either $g=0,1$ and $d \geq 2g+2$ or else $g \geq 2$ and $d \geq 4g+4$.