A note on higher order Gauss maps

TL;DR

This paper investigates higher order Gauss maps of projective varieties, establishing conditions for finiteness of fibers and providing a combinatorial description for toric varieties, advancing understanding of their geometric and combinatorial properties.

Contribution

It proves finiteness of higher order Gauss map fibers for smooth varieties with k-jet spanned line bundles, except for Veronese embeddings, and describes these maps combinatorially for toric varieties.

Findings

Gauss map of order k has finite fibers unless the variety is a Veronese embedding.

For toric varieties, the Gauss map and its image are described combinatorially.

Identifies conditions under which higher order Gauss maps are finite or have special structure.

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.