Solvable Automorphism Groups of Varieties

TL;DR

This paper investigates the structure of solvable automorphism groups of varieties, proving they are algebraic under certain conditions and exploring their properties for quasi-affine varieties.

Contribution

It establishes that solvable subgroups generated by irreducible families are algebraic and characterizes their placement within automorphism groups of varieties.

Findings

Solvable subgroups generated by irreducible families are algebraic.

Connected solvable subgroups are contained in Borel subgroups with derived length ≤ n+1.

Quasi-affine varieties with certain Borel subgroups are isomorphic to affine space.

Abstract

Let $X$ be a variety of dimension $n$, and let $\mathrm{Aut}(X)$ be its automorphism group. When $X$ is quasi-affine, we prove that a solvable subgroup of $\mathrm{Aut}(X)$ that is generated by an irreducible family of automorphisms containing the identity is an algebraic subgroup. Our main applications concern arbitrary varieties. First, every connected solvable subgroup of $\mathrm{Aut}(X)$ is contained in a Borel subgroup and its derived length is $\leq n+1$. Second, the notion of solvable and unipotent radicals are well defined for any subgroup of $\mathrm{Aut}(X)$. Third, if $X$ is quasi-affine and connected and $\mathcal{B} \subset \mathrm{Aut}(X)$ is a Borel subgroup of derived length $n+1$, then $X$ is isomorphic to the affine $n$-space $\mathbb{A}^n$ and $\mathcal{B}$ is conjugate to the Jonqui\`eres subgroup.