Rational surfaces in ℙ4 containing a plane curve

The families of smooth rational surfaces in ℙ4 have been classified in degree ⩽ 10. All known rational surfaces in ℙ4 can be represented as blow-ups of the plane P2. The fine classification of these surfaces consists of giving explicit open and closed conditions which determine the configurations of points corresponding to all surfaces in a given family. Using a restriction argument originally due independently to Alexander and Bauer we achieve the fine classification in two cases, namely non-special rational surfaces of degree 9 and special rational surfaces of degree 8. The first case completes the fine classification of all non-special rational surfaces. In the second case we obtain a description of the moduli space as the quotient of a rational variety by the symmetric group S5. We also discuss in how far this method can be used to study other rational surfaces in ℙ4.