Equations defining a space curve

Let C ⊂ P be a smooth irreducible complete algebraic curve of degree d embedded in a projective 3-space over an algebraically closed field k. It is called n-regular if H(IC(n− 1)) = H(IC(n− 2)) = 0, where IC is the ideal sheaf of C. Among all this implies that the homogeneous ideal IC = ⊕ l H (IC(l)) ⊂ k[x0, x1, x2, x3] is generated in degree ≤ n ([7], Lecture 14) and C ⊂ P is (scheme-theoretically) an intersection of surfaces of degree n. A line ⊂ P is called an s-secant line if deg( ∩ C) ≥ s. If C is an intersection of surfaces of degree n, then it has no (n + 1)-secant lines. We say that the number of s-secant lines is Picard finite if the number of isomorphism classes of the line bundles OC( ∩ C), moving all s-secant lines, is finite. A line bundle ξ on C is called a g s if deg ξ = s and h (ξ) ≥ 3. In this article, applying a vanishing of Raynaud type to a certain family of vector bundles on C, we shall show the following: