Equivariant embeddings of commutative linear algebraic groups of corank one

Abstract

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