On the Rank of Quadratic Equations for Curves of High Degree

Abstract

Let C subset of 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 >= 2g+ 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 >= 4g + 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 >= 2g + 2 or else g >= 2 and d >= 4g + 4.