Completeness and non-speciality of linear series on space curves and deficiency of the corresponding linear series on normalization (Q1076746)

Let \(X\subset {\mathbb{P}}^ r\) be a reduced, irreducible curve with normalization C. Let \(A=\oplus_{n\geq 0}A_ n\) be a saturated homogeneous ideal of \(k[x_ 0,...,x_ r]\) containing strictly the homogeneous ideal of X. Let D(A) be the positive divisor of C determined by A. Fix an integer N such that for all \(n\geq N\), \(h^ 1({\mathbb{P}}^ r,{\mathcal J}_ X(n))=h^ 1(X,{\mathcal O}_ X(n))=0\). Here the author proves that for every \(n\geq N\), the natural map \(A_ n\to H^ 0(C,{\mathcal O}_ C(n)\otimes {\mathcal O}_ C(-D(A))\) is surjective. Thanks to Castelnuovo and Gruson-Lazarsfeld-Peskine, there are good bounds on N.