On the automorphism group of a toral variety

Abstract

Let $\BK$ be an algebraically closed field of characteristic zero. An affine algebraic variety $X$ over $\BK$ is toral if it is isomorphic to a closed subvariety of a torus $(\BK^*)^d$. We study the group $\Aut(X)$ of regular automorpshims of a toral variety $X$. We prove that if $T$ is a maximal torus in $\Aut(X)$, then $X$ is a direct product $Y\times T$, where $Y$ is a toral variety with a trivial maximal torus in the automorphism group. We show that knowing $\Aut (Y)$, one can compute $\Aut(X)$. In the case when the rank of the group $\BK[Y]^*/\BK^*$ is $\dim Y + 1$, the group $\Aut(Y)$ can be described explicitly.