### 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 E_{6}-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 language | English |
---|---|

Pages (from-to) | 533-545 |

Number of pages | 13 |

Journal | Transactions of the American Mathematical Society |

Volume | 351 |

Issue number | 2 |

Publication status | Published - 1999 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Transactions of the American Mathematical Society*,

*351*(2), 533-545.

**Homogeneous projective varieties with degenerate secants.** / Kaji, Hajime.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 351, no. 2, pp. 533-545.

}

TY - JOUR

T1 - Homogeneous projective varieties with degenerate secants

AU - Kaji, Hajime

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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UR - http://www.scopus.com/inward/citedby.url?scp=22444456435&partnerID=8YFLogxK

M3 - Article

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 -