Bounding the degree of generic sharp transitivity

We show that a generically sharply t-transitive permutation group of finite Morley rank on a set of rank r satisfies t≤ r+2 provided the pointwise stabilizer of a generic $(t-1)$-tuple is an L-group, which holds, for example, when the stabilizer is solvable or when r≤ 5. This makes progress on the Borovik-Cherlin conjecture that every generically $(r+2)$-transitive permutation group of finite Morley rank on a set of rank r is of the form PGLᵣ₊₁(F) acting naturally on Pʳ(F). Our proof is assembled from three key ingredients that are independent of the main theorem -- these address actions of Alt(n) on L-groups of finite Morley rank, generically $2$-transitive actions with abelian point stabilizers, and simple groups of rank $6$.