The intersection of quadrics and defining equations of a projective curve (Q1375802)

Let \(C\) be a complete nonsingular curve over an algebraically closed field \(K\) and \(L\) a very ample invertible sheaf on \(C\). We denote by \(\varphi_L: C\to \mathbb{P} (H^0 (L))\), 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 \(\varphi_L (C)\)? The other is to answer the question: What degrees are the minimal generators of the associated homogeneous ideal? Our purpose is to answer for the case of \(g\geq 4\) (mainly \(g=4)\).