### Abstract

We introduce an intrinsic property for a projective variety as follows: there exists an embedding into some projective space such that the Gauss map is of rank zero, which we call (GMRZ) for short. It turns out that (GMRZ) imposes strong restrictions on rational curves on projective varieties: In fact, using (GMRZ), we show that, contrary to the characteristic zero case, the existence of free rational curves does not imply that of minimal free rational curves in positive characteristic case. We also focus attention on Segre varieties, Grassmann varieties, and hypersurfaces of low degree. In particular, we give a characterisation of Fermat cubic hypersurfaces in terms of (GMRZ), and show that a general hypersurface of low degree does not satisfy (GMRZ).

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

Pages (from-to) | 2645-2661 |

Number of pages | 17 |

Journal | Advances in Mathematics |

Volume | 224 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2010 Aug |

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### Keywords

- Gauss map
- Hypersurface
- Inseparable
- Minimal free rational curve
- Normal bundle

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Advances in Mathematics*,

*224*(6), 2645-2661. https://doi.org/10.1016/j.aim.2010.02.017

**Projective varieties admitting an embedding with Gauss map of rank zero.** / Fukasawa, Satoru; Furukawa, Katsuhisa; Kaji, Hajime.

Research output: Contribution to journal › Article

*Advances in Mathematics*, vol. 224, no. 6, pp. 2645-2661. https://doi.org/10.1016/j.aim.2010.02.017

}

TY - JOUR

T1 - Projective varieties admitting an embedding with Gauss map of rank zero

AU - Fukasawa, Satoru

AU - Furukawa, Katsuhisa

AU - Kaji, Hajime

PY - 2010/8

Y1 - 2010/8

N2 - We introduce an intrinsic property for a projective variety as follows: there exists an embedding into some projective space such that the Gauss map is of rank zero, which we call (GMRZ) for short. It turns out that (GMRZ) imposes strong restrictions on rational curves on projective varieties: In fact, using (GMRZ), we show that, contrary to the characteristic zero case, the existence of free rational curves does not imply that of minimal free rational curves in positive characteristic case. We also focus attention on Segre varieties, Grassmann varieties, and hypersurfaces of low degree. In particular, we give a characterisation of Fermat cubic hypersurfaces in terms of (GMRZ), and show that a general hypersurface of low degree does not satisfy (GMRZ).

AB - We introduce an intrinsic property for a projective variety as follows: there exists an embedding into some projective space such that the Gauss map is of rank zero, which we call (GMRZ) for short. It turns out that (GMRZ) imposes strong restrictions on rational curves on projective varieties: In fact, using (GMRZ), we show that, contrary to the characteristic zero case, the existence of free rational curves does not imply that of minimal free rational curves in positive characteristic case. We also focus attention on Segre varieties, Grassmann varieties, and hypersurfaces of low degree. In particular, we give a characterisation of Fermat cubic hypersurfaces in terms of (GMRZ), and show that a general hypersurface of low degree does not satisfy (GMRZ).

KW - Gauss map

KW - Hypersurface

KW - Inseparable

KW - Minimal free rational curve

KW - Normal bundle

UR - http://www.scopus.com/inward/record.url?scp=77953292111&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77953292111&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2010.02.017

DO - 10.1016/j.aim.2010.02.017

M3 - Article

AN - SCOPUS:77953292111

VL - 224

SP - 2645

EP - 2661

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 6

ER -