Generators for the defining ideal of certain rational surfaces

Let Z be a set of s distinct points in the projective plane and let I be the defining ideal of Z. Let X be the surface obtained from P^2 by blowing up the points of Z. In this paper we shall begin a detailed study of the defining ideals of certain projective embeddings of X. We shall be particularly interested in the degrees of the elements in a minimal generating set for these ideals and, more generally, in the graded Betti numbers in a minimal free resolution of these ideals. Such questions have been considered by many authors in the past.