The intersection of quadrics and defining equations of a projective curve

Abstract

Let $C$ be a complete nonsingular curve over an algebrically closed field $K$ and $L$ a very ample invertible sheaf on $C$. We denote by $\phi_{L}$: $C\rightarrow P(H^{0}(L))$, the projective embedding of $C$ by means of the vector space $H^{0}(C, L)$. There are two purposes in this paper. One is to the question: What is the intersection of quadrics through $\phi_{L}(C)$? The other is to answer the question: What degrees are the minimal generators of the associated homogeneous ideal?