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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Weakly defective varieties
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by L. Chiantini and C. Ciliberto PDF
Trans. Amer. Math. Soc. 354 (2002), 151-178 Request permission


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
  • MR Author ID: 194958
  • ORCID: 0000-0001-5776-1335
  • Email:
  • C. Ciliberto
  • Affiliation: Department of Mathematics, University of Rome II, Viale della Ricerca Scientifica, 16132 Rome, Italy
  • MR Author ID: 49480
  • Email:
  • Received by editor(s): March 1, 2000
  • Published electronically: July 13, 2001
  • © Copyright 2001 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 354 (2002), 151-178
  • MSC (2000): Primary 14E25
  • DOI:
  • MathSciNet review: 1859030