Rational surfaces in P^4 containing a plane curve

Abstract

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 $\PP^4$.

arXiv comment25 pages, LaTeX2e

References in corpus (2)

• Surfaces of Degree 10 in the Projective Fourspace via Linear Systems and Linkage
• Embeddings of curves and surfaces

Cited by in corpus (1)

• Smooth rational surfaces of $d=11$ and $\pi =8$ in ${\mathbb{P}}^5$