Rational surfaces in P^4 containing a plane curve

Details Rational surfaces in P^4 containing a plane curve Rational surfaces in P^4 containing a plane curve

1995-05-23

arxiv.org/abs/alg-geom/9505021v1

Mathematics

Algebraic Geometry

25 pages, LaTeX2e

Scientific paper

The families of smooth rational surfaces in $\PP^4$ have been classified in degree $\le 10$. All known rational surfaces in $\PP^4$ can be represented as blow-ups of the plane $\PP^2$. 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 $S_5$. We also discuss in how far this method can be used to study other rational surfaces in $\PP^4$.

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