### Abstract

It is known that if a projective variety X in P ^{N} is reflexive with respect to the projective dual, then the Gauss map of X defined by embedded tangent spaces is separable, and moreover that the converse is not true in general. We prove that the converse holds under the assumption that X is of dimension two. Explaining the subtleness of the problem, we present an example of smooth projective surfaces in arbitrary positive characteristic, which gives a negative answer to a question raised by S. Kleiman and R. Piene on the inseparability of the Gauss map.

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

Pages (from-to) | 699-703 |

Number of pages | 5 |

Journal | Mathematische Zeitschrift |

Volume | 256 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2007 Aug |

### Fingerprint

### Keywords

- Gauss map
- Reflexive
- Separable

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Mathematische Zeitschrift*,

*256*(4), 699-703. https://doi.org/10.1007/s00209-006-0085-0

**The separability of the Gauss map and the reflexivity for a projective surface.** / Fukasawa, Satoru; Kaji, Hajime.

Research output: Contribution to journal › Article

*Mathematische Zeitschrift*, vol. 256, no. 4, pp. 699-703. https://doi.org/10.1007/s00209-006-0085-0

}

TY - JOUR

T1 - The separability of the Gauss map and the reflexivity for a projective surface

AU - Fukasawa, Satoru

AU - Kaji, Hajime

PY - 2007/8

Y1 - 2007/8

N2 - It is known that if a projective variety X in P N is reflexive with respect to the projective dual, then the Gauss map of X defined by embedded tangent spaces is separable, and moreover that the converse is not true in general. We prove that the converse holds under the assumption that X is of dimension two. Explaining the subtleness of the problem, we present an example of smooth projective surfaces in arbitrary positive characteristic, which gives a negative answer to a question raised by S. Kleiman and R. Piene on the inseparability of the Gauss map.

AB - It is known that if a projective variety X in P N is reflexive with respect to the projective dual, then the Gauss map of X defined by embedded tangent spaces is separable, and moreover that the converse is not true in general. We prove that the converse holds under the assumption that X is of dimension two. Explaining the subtleness of the problem, we present an example of smooth projective surfaces in arbitrary positive characteristic, which gives a negative answer to a question raised by S. Kleiman and R. Piene on the inseparability of the Gauss map.

KW - Gauss map

KW - Reflexive

KW - Separable

UR - http://www.scopus.com/inward/record.url?scp=34249058526&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34249058526&partnerID=8YFLogxK

U2 - 10.1007/s00209-006-0085-0

DO - 10.1007/s00209-006-0085-0

M3 - Article

VL - 256

SP - 699

EP - 703

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

IS - 4

ER -