Weakly defective varieties

Chiantini, L.; Ciliberto, C.

doi:10.1090/S0002-9947-01-02810-0

Weakly defective varieties
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by L. Chiantini and C. Ciliberto PDF

Trans. Amer. Math. Soc. 354 (2002), 151-178 Request permission

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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