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.
Chiantini, L., Ciliberto, C. (2002). Weakly defective varieties. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 354(1), 151-178 [10.1090/S0002-9947-01-02810-0].
Weakly defective varieties
CHIANTINI, LUCA;
2002-01-01
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.File | Dimensione | Formato | |
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https://hdl.handle.net/11365/22981
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