### Abstract

From an irreducible complete immersed curve X in a projective space ℙ other than a line, one obtains a curve X^{ ′} in a Graasmann manifold G of lines in ℙ that is the image of X under the Gauss map, which is defined by the embedded tangents of X. The main result of this article clarifies in case of positive characteristic what curves X have the same X′: It is shown that X is uniquely determined by X′ if X, or equivalently X′, has geometric genus at least two, and that for curves X_{ 1} and X_{ 2} with X_{ 1} ≠X_{ 2} in ℙ, if X′_{1} =X′_{2} in G and either X_{ 1} or X_{ 2} is reflexive, then both X_{ 1} and X_{ 2} are rational or supersingular elliptic; moreover, examples of smooth X_{ 1} and X_{ 2} in that case are given.

Original language | English |
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Pages (from-to) | 249-258 |

Number of pages | 10 |

Journal | Manuscripta Mathematica |

Volume | 80 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1993 Dec |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**On the space curves with the same image under the gauss maps.** / Kaji, Hajime.

Research output: Contribution to journal › Article

*Manuscripta Mathematica*, vol. 80, no. 1, pp. 249-258. https://doi.org/10.1007/BF03026550

}

TY - JOUR

T1 - On the space curves with the same image under the gauss maps

AU - Kaji, Hajime

PY - 1993/12

Y1 - 1993/12

N2 - From an irreducible complete immersed curve X in a projective space ℙ other than a line, one obtains a curve X ′ in a Graasmann manifold G of lines in ℙ that is the image of X under the Gauss map, which is defined by the embedded tangents of X. The main result of this article clarifies in case of positive characteristic what curves X have the same X′: It is shown that X is uniquely determined by X′ if X, or equivalently X′, has geometric genus at least two, and that for curves X 1 and X 2 with X 1 ≠X 2 in ℙ, if X′1 =X′2 in G and either X 1 or X 2 is reflexive, then both X 1 and X 2 are rational or supersingular elliptic; moreover, examples of smooth X 1 and X 2 in that case are given.

AB - From an irreducible complete immersed curve X in a projective space ℙ other than a line, one obtains a curve X ′ in a Graasmann manifold G of lines in ℙ that is the image of X under the Gauss map, which is defined by the embedded tangents of X. The main result of this article clarifies in case of positive characteristic what curves X have the same X′: It is shown that X is uniquely determined by X′ if X, or equivalently X′, has geometric genus at least two, and that for curves X 1 and X 2 with X 1 ≠X 2 in ℙ, if X′1 =X′2 in G and either X 1 or X 2 is reflexive, then both X 1 and X 2 are rational or supersingular elliptic; moreover, examples of smooth X 1 and X 2 in that case are given.

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UR - http://www.scopus.com/inward/citedby.url?scp=65749309694&partnerID=8YFLogxK

U2 - 10.1007/BF03026550

DO - 10.1007/BF03026550

M3 - Article

VL - 80

SP - 249

EP - 258

JO - Manuscripta Mathematica

JF - Manuscripta Mathematica

SN - 0025-2611

IS - 1

ER -