Solvable Automorphism Groups of Varieties
Serge Cantat, Hanspeter Kraft, Andriy Regeta, Immanuel van Santen

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 be a variety of dimension , and let be its automorphism group. When is quasi-affine, we prove that a solvable subgroup of 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 is contained in a Borel subgroup and its derived length is . Second, the notion of solvable and unipotent radicals are well defined for any subgroup of . Third, if is quasi-affine and connected and is a Borel subgroup of derived length , then is isomorphic to the affine -space and is conjugate to the Jonqui\`eres subgroup.
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