The injectivity of the extended Gauss map of general projections of smooth projective varieties

Abstract

Let X be a smooth n-dimensional projective variety embedded in some projective space ℙ^N over the field ℂ of the complex numbers. Associated with the general projection of X to a space ℙ^N-m (N-m> n+1) one defines an extended Gauss map $\overline{\gamma}\colon\overline{X}\rightarrow\text{Gr}(n;N-m)$ (in case N-m>2n-1 this is the Gauss map of the image of X under the projection). We prove that $\overline{X}$ is smooth. In case any two different points of X do have disjoint tangent spaces then we prove that $\overline{\gamma}$ is injective.