Gauss maps of toric varieties

TL;DR

This paper explores the structure and properties of Gauss maps of toric varieties, providing combinatorial descriptions, developability criteria, and constructions for varieties with specified Gauss map features across different characteristics.

Contribution

It offers a combinatorial framework for understanding Gauss maps of toric varieties and introduces new criteria and constructions applicable in arbitrary characteristic.

Findings

Gauss map structure described via combinatorics in any characteristic

Developability criterion for toric varieties' Gauss maps

Construction methods for toric varieties with prescribed Gauss map data

Abstract

We investigate Gauss maps of (not necessarily normal) projective toric varieties over an algebraically closed field of arbitrary characteristic. The main results are as follows: (1) The structure of the Gauss map of a toric variety is described in terms of combinatorics in any characteristic. (2) We give a developability criterion in the toric case. In particular, we show that any toric variety whose Gauss map is degenerate must be the join of some toric varieties in characteristic zero. (3) As applications, we provide two constructions of toric varieties whose Gauss maps have some given data (e.g., fibers, images) in positive characteristic.