### Abstract

The purpose of this article is to show how the graded decomposition of complex simple Lie algebras g can be applied to studying adjoint varieties X and their secant varieties Sec X. Firstly quadratic equations defining adjoint varieties are explicitly given. Secondly it is shown that dim Sec X = 2 dim X for adjoint varieties X in two ways: one is based on Terracini's lemma, and the other is on some explicit description of Sec X in terms of an orbit of the adjoint action. Finally it is shown that the contact loci of the secant variety to its embedded tangent space have dimension two if X is adjoint.

Original language | English |
---|---|

Pages (from-to) | 45-57 |

Number of pages | 13 |

Journal | Indagationes Mathematicae |

Volume | 10 |

Issue number | 1 |

Publication status | Published - 1999 Mar 29 |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Indagationes Mathematicae*,

*10*(1), 45-57.

**Adjoint varieties and their secant varieties.** / Kaji, Hajime; Ohno, Masahiro; Yasukura, Osami.

Research output: Contribution to journal › Article

*Indagationes Mathematicae*, vol. 10, no. 1, pp. 45-57.

}

TY - JOUR

T1 - Adjoint varieties and their secant varieties

AU - Kaji, Hajime

AU - Ohno, Masahiro

AU - Yasukura, Osami

PY - 1999/3/29

Y1 - 1999/3/29

N2 - The purpose of this article is to show how the graded decomposition of complex simple Lie algebras g can be applied to studying adjoint varieties X and their secant varieties Sec X. Firstly quadratic equations defining adjoint varieties are explicitly given. Secondly it is shown that dim Sec X = 2 dim X for adjoint varieties X in two ways: one is based on Terracini's lemma, and the other is on some explicit description of Sec X in terms of an orbit of the adjoint action. Finally it is shown that the contact loci of the secant variety to its embedded tangent space have dimension two if X is adjoint.

AB - The purpose of this article is to show how the graded decomposition of complex simple Lie algebras g can be applied to studying adjoint varieties X and their secant varieties Sec X. Firstly quadratic equations defining adjoint varieties are explicitly given. Secondly it is shown that dim Sec X = 2 dim X for adjoint varieties X in two ways: one is based on Terracini's lemma, and the other is on some explicit description of Sec X in terms of an orbit of the adjoint action. Finally it is shown that the contact loci of the secant variety to its embedded tangent space have dimension two if X is adjoint.

UR - http://www.scopus.com/inward/record.url?scp=0033614166&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033614166&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0033614166

VL - 10

SP - 45

EP - 57

JO - Indagationes Mathematicae

JF - Indagationes Mathematicae

SN - 0019-3577

IS - 1

ER -