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

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.