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 $$d \ge 4g+4$$ .