Introduction: We work over an algebraically closed field k of characteristic zero, except in section (3) where the characteristic is arbitry. By a surface we will mean a smooth projective surface and a curve will be any effective divisor on a surface. We recall that in [A], the speciality of a rational surface X in ℙn is defined to be the number q(1)=h1(O×(H)), where H is a hyperplane section of X. We say that X is special or non-special in accordance with q(1)>0 or q(1)=0.

In [A], a complete classification of non-special rational surfaces in ℙ4 was given, showing that the linearly normal ones form, for each degree 3≤d≤9, a single irreducible family. Recently in [E-P] it was shown that there are only a finite number of irreducible components of the Hilbert scheme of ℙ4 containing rational surfaces; in particular the degrees of such surfaces is bounded. The results which we present here are a contribution to the eventual determination of all such components and contributes to the classification of surfaces in ℙ4 of small degree [A], [A-R], [R], [Ro], and varieties with small invariants [l1, l2, l3].

We will be concerned with rational surfaces of speciality one in ℙ4. By [O1, O2, O3] these have degree eight or more and a simple argument shows that their degree is at most eleven (prop.(1.1)).