Equivariant embeddings of commutative linear algebraic groups of corank one

TL;DR

This paper classifies equivariant embeddings of certain commutative algebraic groups as toric varieties, linking group actions to Demazure roots and describing their orbit structures.

Contribution

It proves that all equivariant G-embeddings of G=G_m^r×G_a are toric varieties and characterizes the group actions via Demazure roots.

Findings

All such G-embeddings are toric varieties.

Group actions correspond to Demazure roots.

Orbit structures are explicitly described.

Abstract

Let K be an algebraically closed field of characteristic zero, G_m=(K\{0},*) be its multiplicative group, and G_a=(K,+) be its additive group. Consider a commutative linear algebraic group G=G_m^r\times G_a. We study equivariant G-embeddings, i.e. normal G-varieties X containing G as an open orbit. We prove that X is a toric variety and all such actions of G on X correspond to Demazure roots of the fan of X. In these terms, the orbit structure of a G-variety X is described.