Equivariant embeddings of commutative linear algebraic groups of corank one

Abstract

Let K be an algebraically closed field of characteristic zero, Gm = (K\{0},×) be its multiplicative group, and Ga = (K,+) be its additive group. Consider a commutative linear algebraic group G = (Gm) r × Ga. 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.