Linear systems cut out by quadrics on projections of varieties

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Let X be a variety embedded in P; where R is an algebraically closed field. Let L(X)(k) denote the linear system cut out on X by the hypersurfaces of degree k. Then L(X)(k) is complete for large enough k. The question arises: Given a particular embedding of X, how large does k have to be before L(X)(k) is complete? In the case C is a curve, the problem was studied by Castelnuovo [2], who showed, in particular, that if Cc P3, then L(C)(k) is complete if k> deg(C) 2 ( see Szpiro [ 12) for a modern proof due to Gruson) and more recently by Lazarfeld et al. [5], who have shown that if C a nondegenerate curve in P”, then L(C)(k) is complete for F> deg(C) n + 1. In this paper, we study the following special case. Let X be a variety in P” with ideal sheaf .7’. To say that L(X)(k) is complete is equivalent to saying that the sequence