A note on rational surfaces in

In this paper we show that a smooth rational surface in ruled in cubics or quartics (i.e. possessing a base point free pencil of rational curves of degree three or four in ) has degree d≤12. As a corollary it is shown that if is the image of a blow-up of embedded by a linear system with simple base points” (see Theorem 10 for definition), then, again, deg(S)≤12.