The Borovik-Cherlin conjecture holds in ACF

We show that every faithful, transitive, and generically $(n+2)$-transitive action of a connected group $G$ on an irreducible variety $X$ of dimension $n>0$, all defined over an algebraically closed field $F$, is isomorphic to the natural action of the projective linear group $PG{L}_{n+1}(F)$ on the projective space ${\mathbb{P}}^n(F)$. More precisely, we establish the Borovik-Cherlin conjecture for permutation groups $(G,X)$ definable in models of $ACF$.