On the Rank of Quadratic Equations for Curves of High Degree

Let C⊂ 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(C) of C is generated by quadratic equations of rank at most 4 whenever d≥ 2 g+ 2. Also, in Eisenbud et al. (Amer J Math 110: 513–539, 1988) Eisenbud, Koh and Stillman proved that I(C) admits a determinantal presentation if d≥ 4 g+ 2. In this paper, we will show that I(C) can be generated by quadratic equations of rank 3 if either g= 0 , 1 and d≥ 2 g+ 2 or else g≥ 2 and d≥ 4 g+ 4.