Explicit Generators for Higher-Dimensional Cremona Groups

Definitions and mathematical statement

For a field $k$, let ${\mathbb{P}}^n_k$ denote projective $n$-space over $k$, and define the Cremona group \[ \operatorname{Bir}_k({\mathbb{P}}^n_k)=\{f\colon{\mathbb{P}}^n_k\dashrightarrow{\mathbb{P}}^n_k : f\text{ is defined over }k\text{ and has a rational inverse defined over }k\}, \] with group operation given by composition. The source writes $\operatorname{Bir}({\mathbb{P}}^n)$ without specifying the ground field, its characteristic, or whether it is algebraically closed. Here $k$ is a placeholder for this unresolved datum, not an additional universally quantified hypothesis.

The source asks: for each integer $n\geq4$, find a simple explicit generating set $S_n\subseteq\operatorname{Bir}_k({\mathbb{P}}^n_k)$, giving an analogue of the Noether--Castelnuovo theorem. The generation requirement means \[ \forall f\in\operatorname{Bir}_k({\mathbb{P}}^n_k),\quad \exists r\in{\mathbb{Z}}_{\geq0},\quad \exists s_1,\ldots,s_r\in S_n,\quad \exists \varepsilon_1,\ldots,\varepsilon_r\in\{-1,1\},\qquad f=s_1^{\varepsilon_1}\circ\cdots\circ s_r^{\varepsilon_r}, \] where the empty composition is the identity. No finiteness or countability condition on $S_n$ is supplied.

The words "simple explicit" and the intended scope of the analogy with the Noether--Castelnuovo theorem are not defined in the record. Thus the displayed generation condition specifies only the algebraic requirement; the intended restrictions on the description or form of the generators remain unresolved. Together with the missing ground-field hypotheses, this prevents a fully precise formulation of the requested generating theorem.

Short English statement

For every dimension $n\geq4$, describe simple explicit birational transformations of projective space from which every birational self-map can be obtained by finitely many compositions and inverses. The ground field and the meaning of "simple explicit" are unspecified. The supplied record labels the problem open; this draft does not independently verify that status.

Sources

• Local UnsolvedMath registry: permanent ID $30003860$, source code catalogue entry, title "Explicit Generators for Higher-Dimensional Cremona Groups." The supplied record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year 2018. No individual proposer is identified.
• Supplied primary provenance: Subgroups of Cremona Groups (2019), Oberwolfach Reports, pp. 1685--1743, DOI 10.4171/owr/2018/28. https://doi.org/10.4171/owr/2018/28. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. Its bibliographic year 2019 and its 2018 report identifiers are retained without reconciliation.
• Supplied literature reference: Cantat--Hudson--Déserti, Generators for the Cremona group. https://perso.univ-rennes1.fr/serge.cantat/Articles/hudson-pan-derksen.pdf.
• Supplied literature reference: CNRS, Cremona groups in higher dimension. https://www.insmi.cnrs.fr/fr/cnrsinfo/groupes-de-cremona-en-dimension-superieure.
• Supplied literature reference: Oberwolfach Report 28/2018. https://ems.press/content/serial-article-files/46750.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the dimension-three characteristic-zero case is known, whereas no general satisfactory generating theorem is known for $n\geq4$. This is supplied dated curation derived from the OWR material, not an independently verified assessment. The record explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied registry assigns the statement status corrected_verified and reports a manual semantic, typographic, and LaTeX review. These are source-provided review labels, not mathematical verification performed for this draft.