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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.\n  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 $\\mathrm{Aut}(X)$. 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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.\n  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 quas"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"When X is quasi-affine, a solvable subgroup of Aut(X) that is generated by an irreducible family of automorphisms containing the identity is an algebraic subgroup. Every connected solvable subgroup of Aut(X) is contained in a Borel subgroup and its derived length is ≤ n+1. If X is quasi-affine and connected and B is a Borel of derived length n+1, then X ≅ A^n and B is conjugate to the Jonquières subgroup.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The family generating the solvable subgroup is irreducible (in the sense of algebraic geometry, presumably Zariski-irreducible) and contains the identity; the base field is algebraically closed of characteristic zero (standard but unstated in the abstract).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Solvable subgroups generated by irreducible families in Aut of quasi-affine varieties are algebraic, implying connected solvable subgroups have derived length at most n+1 and that maximal Borels on A^n are Jonquières groups.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Solvable subgroups of automorphism groups on quasi-affine varieties are algebraic when generated by irreducible families containing the identity.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"b5f83fe2f1cb568e4f2e7d9b9b9bc961d7b0cc6bd959ca3531ad426fffd44f20"},"source":{"id":"2605.13515","kind":"arxiv","version":1},"verdict":{"id":"a22ae8a1-46e5-447b-b87a-a533ab078e00","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T17:53:41.532292Z","strongest_claim":"When X is quasi-affine, a solvable subgroup of Aut(X) that is generated by an irreducible family of automorphisms containing the identity is an algebraic subgroup. Every connected solvable subgroup of Aut(X) is contained in a Borel subgroup and its derived length is ≤ n+1. If X is quasi-affine and connected and B is a Borel of derived length n+1, then X ≅ A^n and B is conjugate to the Jonquières subgroup.","one_line_summary":"Solvable subgroups generated by irreducible families in Aut of quasi-affine varieties are algebraic, implying connected solvable subgroups have derived length at most n+1 and that maximal Borels on A^n are Jonquières groups.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The family generating the solvable subgroup is irreducible (in the sense of algebraic geometry, presumably Zariski-irreducible) and contains the identity; the base field is algebraically closed of characteristic zero (standard but unstated in the abstract).","pith_extraction_headline":"Solvable subgroups of automorphism groups on quasi-affine varieties are algebraic when generated by irreducible families containing the identity."},"references":{"count":87,"sample":[{"doi":"","year":2008,"title":"Milne, James S. , title=. 2008 , note=","work_id":"1eaf8ac8-e6e3-43d2-b020-d54db6071584","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2003,"title":"Grothendieck, Alexander and Raynaud, Mich. 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