### 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 |
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Pages (from-to) | 1412-1417 |

Number of pages | 6 |

Journal | Mathematische Nachrichten |

Volume | 281 |

Issue number | 10 |

DOIs | |

Publication status | Published - 2008 Oct 1 |

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### 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