On the automorphism group of a toral variety

TL;DR

This paper investigates the automorphism groups of toral varieties, revealing their structure as products with tori and providing methods to compute these groups explicitly in certain cases.

Contribution

It establishes the structure of automorphism groups of toral varieties as products with tori and offers explicit descriptions when the rank condition is met.

Findings

Automorphism group decomposes as a product with a torus.

Knowing automorphisms of a factor helps compute the whole group.

Explicit description of automorphisms when rank condition holds.

Abstract

Let $\mathbb{K}$ be an algebraically closed field of characteristic zero. An affine algebraic variety $X$ over $\mathbb{K}$ is toral if it is isomorphic to a closed subvariety of a torus $(\mathbb{K}^*)^d$. We study the group $\mathrm{Aut}(X)$ of regular automorpshims of a toral variety $X$. We prove that if $T$ is a maximal torus in $\mathrm{Aut}(X)$, 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 $\mathrm{Aut}(Y)$, one can compute $\mathrm{Aut}(X)$. In the case when the rank of the group $\mathbb{K}[Y]^*/\mathbb{K}^*$ is $\dim Y + 1$, the group $\mathrm{Aut}(Y)$ can be described explicitly.