TY - JOUR

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

AU - Kaji, Hajime

PY - 1993/12/1

Y1 - 1993/12/1

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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U2 - 10.1007/BF03026550

DO - 10.1007/BF03026550

M3 - Article

AN - SCOPUS:65749309694

SN - 0025-2611

VL - 80

SP - 249

EP - 258

JO - Manuscripta Mathematica

JF - Manuscripta Mathematica

IS - 1

ER -