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Transactions of the American Mathematical Society

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Weakly defective varieties


Authors: L. Chiantini and C. Ciliberto
Journal: Trans. Amer. Math. Soc. 354 (2002), 151-178
MSC (2000): Primary 14E25
DOI: https://doi.org/10.1090/S0002-9947-01-02810-0
Published electronically: July 13, 2001
MathSciNet review: 1859030
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Abstract:

A projective variety $X$ is `$k$-weakly defective' when its intersection with a general $(k+1)$-tangent hyperplane has no isolated singularities at the $k+1$ points of tangency. If $X$ is $k$-defective, i.e. if the $k$-secant variety of $X$ has dimension smaller than expected, then $X$ is also $k$-weakly defective. The converse does not hold in general. A classification of weakly defective varieties seems to be a basic step in the study of defective varieties of higher dimension. We start this classification here, describing all weakly defective irreducible surfaces. Our method also provides a new proof of the classical Terracini's classification of $k$-defective surfaces.


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Additional Information

L. Chiantini
Affiliation: Department of Mathematics, University of Siena, Via del Capitano 15, 53100 Siena, Italy
Email: chiantini@unisi.it

C. Ciliberto
Affiliation: Department of Mathematics, University of Rome II, Viale della Ricerca Scientifica, 16132 Rome, Italy
Email: cilibert@axp.mat.uniroma2.it

DOI: https://doi.org/10.1090/S0002-9947-01-02810-0
Received by editor(s): March 1, 2000
Published electronically: July 13, 2001
Article copyright: © Copyright 2001 American Mathematical Society