Borovik–Cherlin conjecture

Posed by Borovik and Cherlin in 2008, the conjecture predicts that sufficiently generically highly transitive actions of finite Morley rank are the natural projective actions of $\operatorname{PGL}_{n+1}(F)$ on $\mathbb{P}^n(F)$.

Known results

• Freitag and Moosa (2021) proved the classification in $\operatorname{ACF}_0$.
• Freitag, Jimenez, and Moosa (2023) proved the corresponding theorem in $\operatorname{DCF}_0$.
• A 2017 result established a conditional projective classification under additional stabilizer hypotheses.
• A 2024 result proved the bound $t\le r+2$ under an $L$-group hypothesis on a generic stabilizer.

September 2, 2026 ACF announcement

A new arXiv report states that connected, faithful, transitive, generically $(n+2)$-transitive actions on positive-dimensional irreducible varieties over algebraically closed fields are the natural $\operatorname{PGL}_{n+1}$ actions on projective space. This is a claimed resolution only in the ACF-definable setting, not of the full conjecture; independent verification was not found.

Current status (as of September 2026): The ACF-definable case is claimed proved, while the general finite-Morley-rank conjecture outside that setting remains open.