×

The smooth surfaces on cubic hypersurface in \(\mathbb{P}^ 4\) with isolated singularities. (English) Zbl 0753.14032

Let \(S\) be a smooth surface in \(\mathbb{P}^ 4\) and \(V\) a hypersurface containing it. It is well known that if \(V\) is a quadric, then there exists another hypersurface \(V'\) such that the complete intersection of \(V\) and \(V'\) is either \(S\) or its union with a plane. In this paper, the author shows that, if \(V\) is a cubic with only isolated singularities, then there exists another \(V'\) such that the complete intersection of \(V\) and \(V'\) is either \(S\) or its union with a plane, a quadric surface, a cubic scroll or a Veronese surface.
The idea for the proof is to lift the analogous result for a general hyperplane section. The key tool for this is to observe that the number of singular points of \(V\) in \(S\) is bounded, which gives that the hyperplane section curve has almost maximal genus as a curve in a Del Pezzo cubic surface. Hence the results of Gruson and Peskine allow to conclude. The case of any cubic hypersurface \(V\) has been recently completed by L. Koelblen [J. Reine Angew. Math. 433, 113-141 (1992; see the following review)].
Reviewer: E.Arrondo (Madrid)

MSC:

14J25 Special surfaces
14J17 Singularities of surfaces or higher-dimensional varieties
14M10 Complete intersections

Citations:

Zbl 0753.14033
PDFBibTeX XMLCite
Full Text: DOI EuDML