Homogeneous projective varieties with degenerate secants

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Abstract

The secant variety of a projective variety X in ℙ, denoted by SecX, is denned to be the closure of the union of lines in ℙ passing through at least two points of X, and the secant deficiency of X is defined by δ := 2 dim X + 1 -dim Sec X. We list the homogeneous projective varieties X with δ > 0 under the assumption that X arise from irreducible representations of complex simple algebraic groups. It turns out that there is no homogeneous, non-degenerate, projective variety X with SecX 5 P and δ > 8, and the E6-variety is the only homogeneous projective variety with largest secant deficiency δ = 8. This gives a negative answer to a problem posed by R. Lazarsfeld and A. Van de Ven if we restrict ourselves to homogeneous projective varieties.

Original languageEnglish
Pages (from-to)533-545
Number of pages13
JournalTransactions of the American Mathematical Society
Volume351
Issue number2
Publication statusPublished - 1999 Dec 1

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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