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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Varieties with a reducible hyperplane section whose two components are hypersurfaces


Authors: José Carlos Sierra and Andrea Luigi Tironi
Journal: Proc. Amer. Math. Soc. 135 (2007), 1263-1269
MSC (2000): Primary 14C20; Secondary 14N05.
Published electronically: November 13, 2006
MathSciNet review: 2276633
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Abstract: We classify smooth complex projective varieties $ X\subset\mathbb{P}^N$ of dimension $ n\geq 2$ admitting a divisor of the form $ A+B$ among their hyperplane sections, both $ A$ and $ B$ of codimension $ \leq 1$ in their respective linear spans. In this setting, one of the following holds: 1) $ X$ is either the Veronese surface in $ \mathbb{P}^5$ or its general projection to $ \mathbb{P}^4$, 2) $ n\leq 3$ and $ X\subset{\mathbb{P}}^{n+2}$ is contained in a quadric cone of rank $ 3$ or $ 4$, 3) $ n=2$ and $ X\subset\mathbb{P}^3$.


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Additional Information

José Carlos Sierra
Affiliation: Departamento de Álgebra, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain
Email: jcsierra@mat.ucm.es

Andrea Luigi Tironi
Affiliation: Dipartimento di Matematica “F. Enriques", Università degli Studi di Milano, Via C. Saldini 50, 20133 Milano, Italy
Email: atironi@mat.unimi.it

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08637-0
PII: S 0002-9939(06)08637-0
Keywords: Algebraic geometry, reducible hyperplane sections of varieties.
Received by editor(s): January 21, 2005
Received by editor(s) in revised form: December 6, 2005
Published electronically: November 13, 2006
Additional Notes: This work was done in the framework of the National Research Project “Geometry on Algebraic Varieties”, supported by the MIUR of the Italian Government (Cofin 2002).
Communicated by: Michael Stillman
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.