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

From an irreducible complete immersed curveX in a projective space ? other than a line, one obtains a curveX ^′ in a Graasmann manifoldG of lines in ? that is the image ofX under the Gauss map, which is defined by the embedded tangents ofX. The main result of this article clarifies in case of positive characteristic what curvesX have the sameX′: It is shown thatX is uniquely determined byX′ ifX, or equivalentlyX′, has geometric genus at least two, and that for curvesX ₁ andX ₂ withX ₁ ≠X ₂ in ?, ifX′₁ =X′₂ inG and eitherX ₁ orX ₂ is reflexive, then bothX ₁ andX ₂ are rational or supersingular elliptic; moreover, examples of smoothX ₁ andX ₂ in that case are given.