### Abstract

The reflexivity, the (semi-)ordinariness, the dimension of dual varieties and the structure of Gauss maps are discussed for Segre varieties, where a Segre variety is the image of the product of two or more projective spaces under Segre embedding. A generalization is given to a theorem of A. Hefez and A. Thorup on Segre varieties of two projective spaces. In particular, a new proof is given to a theorem of F. Knop, G. Menzel, I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky that states a necessary and sufficient condition for Segre varieties to have codimension one duals. On the other hand, a negative answer is given to a problem raised by S. Kleiman and R. Piene as follows: For a projective variety of dimension at least two, do the Gauss map and the natural projection from the conormal variety to the dual variety have the same inseparable degree?

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

Pages (from-to) | 221-229 |

Number of pages | 9 |

Journal | Geometriae Dedicata |

Volume | 99 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2003 Jun |

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### Keywords

- Dual variety
- Gauss map
- Reflexivity
- Segre variety
- Symmetric matrix

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Geometriae Dedicata*,

*99*(1), 221-229. https://doi.org/10.1023/A:1024968503486

**On the Duals of Segre Varieties.** / Kaji, Hajime.

Research output: Contribution to journal › Article

*Geometriae Dedicata*, vol. 99, no. 1, pp. 221-229. https://doi.org/10.1023/A:1024968503486

}

TY - JOUR

T1 - On the Duals of Segre Varieties

AU - Kaji, Hajime

PY - 2003/6

Y1 - 2003/6

N2 - The reflexivity, the (semi-)ordinariness, the dimension of dual varieties and the structure of Gauss maps are discussed for Segre varieties, where a Segre variety is the image of the product of two or more projective spaces under Segre embedding. A generalization is given to a theorem of A. Hefez and A. Thorup on Segre varieties of two projective spaces. In particular, a new proof is given to a theorem of F. Knop, G. Menzel, I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky that states a necessary and sufficient condition for Segre varieties to have codimension one duals. On the other hand, a negative answer is given to a problem raised by S. Kleiman and R. Piene as follows: For a projective variety of dimension at least two, do the Gauss map and the natural projection from the conormal variety to the dual variety have the same inseparable degree?

AB - The reflexivity, the (semi-)ordinariness, the dimension of dual varieties and the structure of Gauss maps are discussed for Segre varieties, where a Segre variety is the image of the product of two or more projective spaces under Segre embedding. A generalization is given to a theorem of A. Hefez and A. Thorup on Segre varieties of two projective spaces. In particular, a new proof is given to a theorem of F. Knop, G. Menzel, I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky that states a necessary and sufficient condition for Segre varieties to have codimension one duals. On the other hand, a negative answer is given to a problem raised by S. Kleiman and R. Piene as follows: For a projective variety of dimension at least two, do the Gauss map and the natural projection from the conormal variety to the dual variety have the same inseparable degree?

KW - Dual variety

KW - Gauss map

KW - Reflexivity

KW - Segre variety

KW - Symmetric matrix

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U2 - 10.1023/A:1024968503486

DO - 10.1023/A:1024968503486

M3 - Article

AN - SCOPUS:0042710730

VL - 99

SP - 221

EP - 229

JO - Geometriae Dedicata

JF - Geometriae Dedicata

SN - 0046-5755

IS - 1

ER -