Groups Realizable as Variety Automorphism Groups

Definitions and mathematical statement

For a field $K$ and an algebraic variety $X$ over $K$, let $\operatorname{Aut}_K(X)$ denote the group, under composition, of morphisms $f:X\to X$ over $K$ admitting a morphism inverse over $K$.

The supplied question asks which groups occur as automorphism groups of algebraic varieties. Under an abstract-group interpretation, once a ground field $K$ and a class $\mathcal V_K$ of admissible varieties have been specified, the requested classification is of the groups $G$ satisfying \[ \exists X\in\mathcal V_K\quad G\cong\operatorname{Aut}_K(X). \] Here the isomorphism is an isomorphism of abstract groups. This is a conditional formulation: the record does not specify the ground field, its characteristic, the admissible class of varieties, or whether additional structure on the automorphism group must be preserved. Consequently, it does not determine a unique classification problem.

The original excerpt supplies two contextual examples. Writing $r$ for its number of punctures, it claims that \[ \operatorname{Aut}\!\left(\mathbb A^1\setminus\{a_1,\ldots,a_r\}\right)=\{1\} \qquad(r\geq3) \] for points in general position. The field and the meaning of general position are unspecified. Moreover, automorphisms of the punctured curve need not extend to automorphisms of the affine line, so the intended automorphism group and the stated threshold require mathematical review. This contextual claim is retained as a source claim, not adopted as a verified assertion.

The excerpt also writes $\dot{\mathbb A}=\mathbb A^1\setminus\{0\}$ and records, for the product of $m$ copies, \[ \operatorname{Aut}\left((\dot{\mathbb A})^m\right) \cong \operatorname{GL}_m({\mathbb{Z}})\ltimes(K^\times)^m, \] with the field denoted by $k$ in the source. Under the interpretation as the split torus over $K$ with automorphisms over $K$, the factors represent monomial changes of coordinates with integral invertible exponent matrix and coordinatewise multiplication by nonzero scalars. The source does not explicitly state its field assumptions or the range of $m$. These examples provide context and do not specify the full scope of the classification question.

Short English statement

Classify the groups that can occur as groups of algebraic automorphisms of algebraic varieties. A precise version requires specifying the ground field, the allowed varieties, and the structure to be preserved on the groups. The supplied examples retain unresolved qualifications. The record labels the problem partially solved; this is supplied dated curation, not an independently verified status.

Sources

• Local registry: data/unsolvedmath/problems.json; permanent UnsolvedMath ID $30006253$; exact source problem code catalogue entry. The supplied record is titled "Groups Realizable as Variety Automorphism Groups."
• Supplied report citation: Algebraic Groups (2025), Oberwolfach Reports, pp. 941--1010, DOI 10.4171/owr/2025/20. https://doi.org/10.4171/owr/2025/20. The record attributes its collection to workshop open-problem discussions and names no individual proposer or precise problem page.
• Supplied literature source, labeled "Kraft survey": https://doi.org/10.4171/OWR/2025/20.
• Supplied literature source, labeled "Kraft, automorphism groups of affine varieties": https://arxiv.org/abs/1501.06362.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and states that many classes are realized, including every finite group on a smooth affine curve in characteristic zero, while the general classification remains open. These are supplied dated assessments and claims, not independently verified here. The record itself requests independent verification of time-sensitive status and cited claims.
• The supplied record labels its statement corrected_verified and reports that the official report was checked, the question isolated from surrounding exposition, and its notation repaired. This is a source-verification claim in the supplied metadata; no independent source visit or verification is asserted here.