On kth-order embeddings of K3 surfaces and Enriques surfaces

We give necessary and sufficient conditions for a big and nef line bundle L of any degree on a K3 surface or on an Enriques surface S to be k-very ample and k-spanned. Furthermore, we give necessary and sufficient conditions for a spanned and big line bundle on a K3 surface S to be birationally k-very ample and birationally k-spanned (our definition), and relate these concepts to the Clifford index and gonality of smooth curves in |L| and the existence of a particular type of rank 2 bundles on S.