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 Saint-Donat (CR Acad Sci Paris Ser A 274: 324–327, 1972), 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 $d \ge 2g+2$. Also, in Eisenbud et al. (Amer J Math 110: 513–539, 1988) Eisenbud, Koh and Stillman proved that $I({\mathcal {C}})$ admits a determinantal presentation if $d \ge 4g+2$. 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 $d \ge 2g+2$ or else $g \ge 2$ and $d \ge 4g+4$.