Simultaneous Regularization of Cremona Subgroups

Definitions and mathematical statement

Write $\operatorname{Cr}_2(k)=\operatorname{Bir}_k({\mathbb{P}}^2_k)$ for the plane Cremona group over a field $k$, namely the group of birational self-maps of the projective plane defined over $k$. The supplied record does not specify the allowed base fields or the category of surfaces on which regularization is required. To expose these unresolved parameters, let $\mathcal S$ denote the intended class of surfaces over $k$ birational to ${\mathbb{P}}^2_k$; this is a placeholder for missing source qualifications, not an additional hypothesis attributed to the source.

For a subgroup $H\leq\operatorname{Cr}_2(k)$, define regularizability relative to this class by \[ \operatorname{Reg}_{\mathcal S}(H) \quad\Longleftrightarrow\quad \exists S\in\mathcal S\;\exists\text{ a }k\text{-birational map }\phi:{\mathbb{P}}^2_k\dashrightarrow S \quad \phi H\phi^{-1}\subseteq\operatorname{Aut}_k(S). \] An element $g$ is regularizable when $\operatorname{Reg}_{\mathcal S}(\langle g\rangle)$ holds. The surface and conjugating map in this definition may depend on $g$.

Question, with unresolved source parameters displayed. For the intended choices of $k$ and $\mathcal S$, does every finitely generated subgroup $G\leq\operatorname{Cr}_2(k)$ satisfy \[ \left[\forall g\in G,\ \operatorname{Reg}_{\mathcal S}(\langle g\rangle)\right] \quad\Longrightarrow\quad \operatorname{Reg}_{\mathcal S}(G)? \] Thus the conclusion requires one surface and one birational conjugacy that work simultaneously for every element of $G$. The record does not state whether the surfaces must be smooth, projective, or complete, or whether any extension of the base field is permitted; these choices must be checked before this schema can be regarded as a fully specified formulation.

The record also describes the question as asking whether the locally elliptic action of $G$ on the blow-up complex must be globally elliptic. If these terms mean, respectively, that each element has a fixed point and that the whole group has a common fixed point, the corresponding schema for the stated complex $X$ is \[ \left[\forall g\in G,\ \operatorname{Fix}_X(g)\ne\varnothing\right] \quad\Longrightarrow\quad \operatorname{Fix}_X(G)\ne\varnothing, \qquad \operatorname{Fix}_X(G)=\bigcap_{g\in G}\operatorname{Fix}_X(g). \] The construction of the blow-up complex, the precise convention for local ellipticity, and whether fixed vertices or arbitrary fixed points are intended are not supplied. Consequently, this equivalence is retained as a source assertion requiring verification, rather than asserted here as an independently established fact.

Short English statement

The question asks whether a finitely generated group of plane birational transformations can be conjugated simultaneously into the automorphism group of one surface whenever each individual transformation can be regularized on a possibly different surface. The base-field assumptions and the permitted surfaces remain unspecified. The supplied curation labels the problem open and reports a positive result over finite fields; these are dated source assessments requiring independent verification.

Sources

• Local registry: data/unsolvedmath/problems.json, with catalog/problems.json as the supplementary registry; supplied record with permanent UnsolvedMath ID $30004629$, title Simultaneous Regularization of Cremona Subgroups, and exact source problem code catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2021.
• Supplied report citation: Nonpositively Curved Complexes (2022), Oberwolfach Reports, pp. 383--417, DOI 10.4171/owr/2021/8. https://doi.org/10.4171/owr/2021/8. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The citation's 2022 title date and the record's 2021 proposed year are retained without reconciliation.
• Supplied literature reference: Lonjou--Przytycki--Urech, Cremona subgroups. https://doi.org/10.5802/jep.271.
• Supplied literature reference: Genevois--Lonjou--Urech, finite-field regularisation. https://arxiv.org/abs/2110.14605.
• Supplied literature reference: Oberwolfach report. https://doi.org/10.4171/owr/2021/8.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that the regularizability implication is proved over finite fields, while published recent work states that the general-field question remains unknown. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. Time-sensitive status and cited claims require independent verification.
• The supplied registry assigns the statement status corrected_verified and says that the clean statement was checked against the official Oberwolfach report and preserves its hypotheses, notation, and target. This verification claim is part of the supplied provenance; no source visit or independent verification is claimed here.