TY - JOUR

T1 - Homogeneous projective varieties with degenerate secants

AU - Kaji, Hajime

N1 - Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 1999

Y1 - 1999

N2 - 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.

AB - 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.

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U2 - 10.1090/s0002-9947-99-02378-8

DO - 10.1090/s0002-9947-99-02378-8

M3 - Article

AN - SCOPUS:22444456435

VL - 351

SP - 533

EP - 545

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 2

ER -