Let L be a very ample line bundle on a smooth, connected, projective, ruled not rational surface X. We have considered the problem of classifying biholomorphically smooth, connected, projected, ruled, non rational surfaces X with smooth hyperplane section C such that the genus g = g(C) is less than or equal to six and dim where is the map associated to . L. Roth in  had given a birational classification of such surfaces. If g = 0 or 1 then X has been classified, see .
If g = 2 ≠ h
l,0(X) by [12, Lemma (2.2.2) ] it follows that X is a rational surface. Thus we can assume g ≦ 3.
Since X is ruled, h
2,0(X) = 0 and
see  and [12, p. 390].