### Abstract

We study the relationship between the generic smoothness of the Gauss map and the reflexivity (with respect to the projective dual) for a projective variety defined over an algebraically closed field. The problem we discuss here is whether it is possible for a projective variety X in ℙ^{N} to re-embed into some projective space ℙ^{M} so as to be non-reflexive with generically smooth Gauss map. Our result is that the answer is affirmative under the assumption that X has dimension at least 3 and the differential of the Gauss map of X in ℙ^{N} is identically zero; hence the projective variety X re-embedded in ℙ^{M} yields a negative answer to Kleiman-Piene's question: Does the generic smoothness of the Gauss map imply reflexivity for a projective variety? A Fermat hypersurface in ℙ^{N} with suitable degree in positive characteristic is known to satisfy the assumption above. We give some new, other examples of X in ℙ^{N} satisfying the assumption.

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

Pages (from-to) | 1412-1417 |

Number of pages | 6 |

Journal | Mathematische Nachrichten |

Volume | 281 |

Issue number | 10 |

DOIs | |

Publication status | Published - 2008 Oct |

### Fingerprint

### Keywords

- Gauss map
- Generic smoothness
- Reflexivity

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Mathematische Nachrichten*,

*281*(10), 1412-1417. https://doi.org/10.1002/mana.200610688

**Existence of a non-reflexive embedding with birational Gauss map for a projective variety.** / Fukasawa, Satoru; Kaji, Hajime.

Research output: Contribution to journal › Article

*Mathematische Nachrichten*, vol. 281, no. 10, pp. 1412-1417. https://doi.org/10.1002/mana.200610688

}

TY - JOUR

T1 - Existence of a non-reflexive embedding with birational Gauss map for a projective variety

AU - Fukasawa, Satoru

AU - Kaji, Hajime

PY - 2008/10

Y1 - 2008/10

N2 - We study the relationship between the generic smoothness of the Gauss map and the reflexivity (with respect to the projective dual) for a projective variety defined over an algebraically closed field. The problem we discuss here is whether it is possible for a projective variety X in ℙN to re-embed into some projective space ℙM so as to be non-reflexive with generically smooth Gauss map. Our result is that the answer is affirmative under the assumption that X has dimension at least 3 and the differential of the Gauss map of X in ℙN is identically zero; hence the projective variety X re-embedded in ℙM yields a negative answer to Kleiman-Piene's question: Does the generic smoothness of the Gauss map imply reflexivity for a projective variety? A Fermat hypersurface in ℙN with suitable degree in positive characteristic is known to satisfy the assumption above. We give some new, other examples of X in ℙN satisfying the assumption.

AB - We study the relationship between the generic smoothness of the Gauss map and the reflexivity (with respect to the projective dual) for a projective variety defined over an algebraically closed field. The problem we discuss here is whether it is possible for a projective variety X in ℙN to re-embed into some projective space ℙM so as to be non-reflexive with generically smooth Gauss map. Our result is that the answer is affirmative under the assumption that X has dimension at least 3 and the differential of the Gauss map of X in ℙN is identically zero; hence the projective variety X re-embedded in ℙM yields a negative answer to Kleiman-Piene's question: Does the generic smoothness of the Gauss map imply reflexivity for a projective variety? A Fermat hypersurface in ℙN with suitable degree in positive characteristic is known to satisfy the assumption above. We give some new, other examples of X in ℙN satisfying the assumption.

KW - Gauss map

KW - Generic smoothness

KW - Reflexivity

UR - http://www.scopus.com/inward/record.url?scp=55449123325&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=55449123325&partnerID=8YFLogxK

U2 - 10.1002/mana.200610688

DO - 10.1002/mana.200610688

M3 - Article

AN - SCOPUS:55449123325

VL - 281

SP - 1412

EP - 1417

JO - Mathematische Nachrichten

JF - Mathematische Nachrichten

SN - 0025-584X

IS - 10

ER -