On the rank of quadratic equations for curves of high degree

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 $d \geq 4g+4$.