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Varieties with quadratic entry locus, II

Part of: Special varieties Surfaces and higher-dimensional varieties Projective and enumerative geometry

Published online by Cambridge University Press:  01 July 2008

Paltin Ionescu  and
Francesco Russo
Paltin Ionescu
Affiliation:
University of Bucharest and Institute of Mathematics of the Romanian Academy
Francesco Russo
Affiliation:
Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Viale A. Doria 6, 95125 Catania, Italy (email: frusso@dmi.unict.it)
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Abstract

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We continue the study, begun in [F. Russo, Varieties with quadratic entry locus. I, Preprint (2006), math. AG/0701889] , of secant defective manifolds having ‘simple entry loci’. We prove that such manifolds are rational and describe them in terms of tangential projections. Using also the work of [P. Ionescu and F. Russo, Conic-connected manifolds, Preprint (2006), math. AG/0701885], their classification is reduced to the case of Fano manifolds of high index, whose Picard group is generated by the hyperplane section class. Conjecturally, the former should be linear sections of rational homogeneous manifolds. We also provide evidence that the classification of linearly normal dual defective manifolds with Picard group generated by the hyperplane section should follow along the same lines.

Keywords

secant defectivequadratic entry locus manifoldtangential projectionconic-connecteddual defective

MSC classification

Secondary: 14N05: Projective techniques 14M99: None of the above, but in this section 14J45: Fano varieties
Type
Research Article
Information
Compositio Mathematica , Volume 144 , Issue 4 , July 2008 , pp. 949 - 962
Copyright
Copyright © Foundation Compositio Mathematica 2008

References

The first author is partially supported by the Italian Programme ‘Incentivazione alla mobilità di studiosi stranieri e italiani residenti all’estero’. The second author is partially supported by CNPq (Centro Nacional de Pesquisa), Grants 300961/2003-0, 308745/2006-0 and 474475/2006-9, and by PRONEX/FAPERJ–Algebra Comutativa e Geometria Algebrica.