Solvable Automorphism Groups of Varieties
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866916009589342208 |
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| author | Cantat, Serge Kraft, Hanspeter Regeta, Andriy van Santen, Immanuel |
| author_facet | Cantat, Serge Kraft, Hanspeter Regeta, Andriy van Santen, Immanuel |
| contents | 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 Jonquières subgroup. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_13515 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Solvable Automorphism Groups of Varieties Cantat, Serge Kraft, Hanspeter Regeta, Andriy van Santen, Immanuel Algebraic Geometry Group Theory Primary 14J50, Secondary 14L30, 14R20 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 Jonquières subgroup. |
| title | Solvable Automorphism Groups of Varieties |
| topic | Algebraic Geometry Group Theory Primary 14J50, Secondary 14L30, 14R20 |
| url | https://arxiv.org/abs/2605.13515 |