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 [10] had given a birational classification of such surfaces. If g = 0 or 1 then X has been classified, see [8].
If g = 2 ≠ hl,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, h2,0(X) = 0 and
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X0001110X/resource/name/S0008414X0001110X_eqn1.gif?pub-status=live)
see [4] and [12, p. 390].