On curves on K3 surfaces,II

We study the Clifford index c of a smooth irreducible curve X in the linear series |2H| on a special K3 surface S of degree 2n in ({{mathbb P}}^{n+1}), with hyperplane section H, and we look for the complete and base point free linear series of S whose restrictions to X compute c. In a more general context, we discuss the features of such series, for an assigned curve on a K3 surface; this discussion is of some independent interest.