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

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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 languageEnglish
Pages (from-to)249-258
Number of pages10
JournalManuscripta Mathematica
Volume80
Issue number1
DOIs
Publication statusPublished - 1993 Dec
Externally publishedYes

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ASJC Scopus subject areas

  • Mathematics(all)

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