Divisors of small genus on algebraic surfaces and projective embeddings

Abstract. In this article we give a numerical criterion, valid in all characteristics, for the very ampleness of a line bundle H on a curve C (possibly reducible and non reduced) lying on a smooth algebraic surface. We show by the way that this criterion essentially implies the results of Bombieri and Ekedahl on pluricanonical embeddings of surfaces of general type. We use our numerical criterion to obtain the ne classication of non special rational surfaces in P 4 (with one exception dealt with in a forthcoming article). We also describe in full detail the embeddings when C is a curve of genus 2 and H has degree 5. Content