Algebraic (2,2)-transformation groups

In this paper we determine all algebraic trans-formation groups G, defined over an algebraical-ly closed field k, which operate transitively, but not primitively, on a variety ­M, provided the following conditions are fulfilled. We ask that the (non-effective) action of G on the va-riety of blocks is sharply 2-transitive, as well as the action on a block X of the normalizer Gx. Also we require sharp transitivity on pairs (X,Y)of independent points of M­, i.e. points con-tained in different blocks.