Factoriality of certain hypersurfaces of \(\mathbb P^4\) with ordinary double points. (English) Zbl 1068.14010
Popov, Vladimir L. (ed.), Algebraic transformation groups and algebraic varieties. Proceedings of the conference on interesting algebraic varieties arising in algebraic transformation group theory, Vienna, Austria, October 22–26, 2001. Berlin: Springer (ISBN 3-540-20838-0/hbk). Encyclopaedia of Mathematical Sciences 132. Invariant Theory and Algebraic Transformation Groups 3, 1-7 (2004).
The authors develop an approach of F. Severi [Rom. Acc. L. Rend. (5) \(15_2\), 691–696 (1906; JFM 37.0131.01)] who proved that any surface contained in a smooth hypersurface of \({\mathbb P}^4\) is a complete intersection.
In the paper under review a slightly more general situation is considered. So let \(V\subset {\mathbb P}^4\) be a reduced and irreducible hypersurface of degree at least 3 whose singularities are ordinary double points only. Then under some restrictions on the number and geometry of the nodes of \(V\) the authors prove that Severi’s theorem remains valid. In particular, \(V\) is \(\mathbb Q\)-factorial. As a consequence new examples of \(\mathbb Q\)-factorial projective varieties are obtained [Y. Miyaoka and T. Peternell, “Geometry of higher dimensional algebraic varieties” (1997; Zbl 0865.14018)]. In addition, the case of smooth surfaces in \(V\) is analyzed in detail.
For the entire collection see [Zbl 1051.14003].
In the paper under review a slightly more general situation is considered. So let \(V\subset {\mathbb P}^4\) be a reduced and irreducible hypersurface of degree at least 3 whose singularities are ordinary double points only. Then under some restrictions on the number and geometry of the nodes of \(V\) the authors prove that Severi’s theorem remains valid. In particular, \(V\) is \(\mathbb Q\)-factorial. As a consequence new examples of \(\mathbb Q\)-factorial projective varieties are obtained [Y. Miyaoka and T. Peternell, “Geometry of higher dimensional algebraic varieties” (1997; Zbl 0865.14018)]. In addition, the case of smooth surfaces in \(V\) is analyzed in detail.
For the entire collection see [Zbl 1051.14003].
Reviewer: Aleksandr G. Aleksandrov (Moskva)
MSC:
| 14C22 | Picard groups |
| 14J17 | Singularities of surfaces or higher-dimensional varieties |
| 14M05 | Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) |
| 14J70 | Hypersurfaces and algebraic geometry |
| 14M10 | Complete intersections |
