Generic Transitivity and Projective Linear Actions

Definitions and mathematical statement

Write $\operatorname{rk}$ for Morley rank. In the standard definable interpretation, a permutation group $(X,G)$ of finite Morley rank consists of a definable group $G$ acting definably and faithfully on a nonempty definable set $X$, with both having finite Morley rank. The group $G$ is connected if it has no proper definable subgroup of finite index. Its action is transitive if for every $x,y\in X$ there exists $g\in G$ with $g\cdot x=y$. For a positive integer $k$, the action is generically $k$-transitive if its diagonal action on $X^k$, given by \[ g\cdot(x_1,\ldots,x_k)=(g\cdot x_1,\ldots,g\cdot x_k), \] has an orbit of Morley rank $\operatorname{rk}(X^k)=k\operatorname{rk}(X)$.

For an algebraically closed field $K$, let $\mathbb P^n(K)$ be the set of one-dimensional linear subspaces of $K^{n+1}$, and let \[ \operatorname{PGL}_{n+1}(K)=\operatorname{GL}_{n+1}(K)/\{\lambda I_{n+1}:\lambda\in K^\times\} \] act on $\mathbb P^n(K)$ by its natural projective action.

Question. Suppose $(X,G)$ is a transitive permutation group of finite Morley rank, $G$ is connected, $n=\operatorname{rk}(X)$, and the action is generically $(n+2)$-transitive. Must there exist an algebraically closed field $K$, a group isomorphism \[ \varphi:G\longrightarrow\operatorname{PGL}_{n+1}(K), \] and a bijection \[ f:X\longrightarrow\mathbb P^n(K) \] such that \[ f(g\cdot x)=\varphi(g)\cdot f(x) \qquad(g\in G,\ x\in X)? \]

Unresolved qualifications. The displayed conclusion spells out equivalence as equivalence of permutation actions. The supplied record does not specify whether the equivalence must be definable, whether $K$ must be interpretable in the ambient structure, or which parameters are allowed. These potentially stronger requirements remain unresolved. The record supplies no explicit restriction $n\geq1$; as the finite Morley rank of a nonempty set, $n$ is a nonnegative integer, and no positive-rank hypothesis has been added. The ambient structure and its parameter conventions are also unspecified.

Short English statement

Must every transitive action of a connected group of finite Morley rank on a set of rank $n$, with a full-rank orbit on $(n+2)$-tuples, be equivalent to the natural projective linear action over an algebraically closed field? The required definability of this equivalence is unspecified. This is a statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003024$, title: Generic Transitivity and Projective Linear Actions; source problem code: catalogue entry. Collection: Oberwolfach Reports Open Problems. The record gives proposed year 2016 and names no individual proposer.
• Supplied report citation: Model Theory: groups, geometry, and combinatorics (2016), Oberwolfach Reports, pp. 5--50, DOI 10.4171/owr/2016/1. Supplied source and literature link, labelled Oberwolfach Report 1/2016: https://doi.org/10.4171/owr/2016/1. The background describes these reports as records of mathematical workshops and their open-problem discussions; no precise location within the report is supplied for this question.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified. The record labels the problem open, but its assessment says that the exact record contains no mathematical proposition beyond a remark that a conjecture is clear, and therefore retains the open label under a triage rule. This description conflicts with the explicit mathematical question supplied here. The background requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified; the accompanying metadata reports manual semantic, typographic, and LaTeX review for self-containment and fidelity. These are supplied review claims, not independent verification of the source or mathematical correctness.