A note on higher order Gauss maps

Abstract

We study Gauss maps of order $k$, associated to a projective variety $X$ embedded in projective space via a line bundle $L.$ We show that if $X$ is a smooth, complete complex variety and $L$ is a $k$-jet spanned line bundle on $X$, with $k\geq 1,$ then the Gauss map of order $k$ has finite fibers, unless $X=\mathbb{P}^n$ is embedded by the Veronese embedding of order $k$. In the case where $X$ is a toric variety, we give a combinatorial description of the Gauss maps of order $k$, its image and the generic fibers.