Two points of the boundary of toric geometry

TL;DR

This paper explores boundary aspects of toric geometry, including deformations of affine toric varieties to address ramification issues and limits of equivariant maps to analyze additive preorders.

Contribution

It introduces two new boundary-related insights in toric geometry: deformation techniques for affine varieties and limits of birational maps for preorder spaces.

Findings

Deformation of affine toric varieties can lead to non-toric germs.

Limits of equivariant birational maps relate to the structure of additive preorders.

Provides new methods to study ramification and preorder spaces in toric geometry.

Abstract

This note presents two observations which have in common that they lie at the boundary of toric geometry. The first one because it concerns the deformation of affine toric varieties into non toric germs in order to understand how to avoid some ramification problems arising in the study of local uniformization in positive characteristic, and the second one because it uses limits of projective systems of equivariant birational maps of toric varieties to study the space of additive preorders on ${\mathbf Z}^r$ for $r\geq 2$.