Equivariant Embeddings of Commutative Linear Algebraic Groups of Corank One

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

2010 Mathematics Subject Classification: Primary 14M17, 14M25, 14M27; Secondary 13N15, 14J50

Keywords and Phrases: Toric variety, Cox ring, locally nilpotent derivation, Demazure root

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