Let S be a surface and L be a line bundle on S. For any zero-dimensional closed subschemes x of S, there is natural map from global sections of L to the global sections of L restricting to x (which is a (r+1)-dimension vector space. ). For any positive integers r, the line bundle L on a surface S is called r-very ample if for any length r+1 zero-dimensional closed subschemes x of S, this map is surjective. For example, very ampleness implies 1-very ampleness. If L and K are both very ample then tensoring l copies of L and k copies of K is (k+l)-very ample.

My question is, for every positive integer r, can you find an algebraic K3 surface with Picard number 1 such that the primitive ample line bundle is r-very ample? Moreover, can you find infinitely many line bundles of this kind whose self-intersection numbers are all different?