Algebraic (2,2)-transformation groups

TL;DR

This paper classifies algebraic transformation groups over algebraically closed fields that act transitively but not primitively on varieties, under specific sharp transitivity conditions on blocks and points.

Contribution

It determines all such algebraic groups with specified transitivity properties, extending the understanding of their structure and actions.

Findings

Classified all algebraic (2,2)-transformation groups under given conditions.

Identified the structure of groups acting sharply 2-transitively on block varieties.

Described the action of these groups on pairs of independent points.

Abstract

This paper contains the more significant part of the article with the same title that will appear in the Volume 12 of Journal of Group Theory (2009). In this paper we determine all algebraic transformation groups $G$, defined over an algebraically closed field $\sf k$, which operate transitively, but not primitively, on a variety $\Omega$, provided the following conditions are fulfilled. We ask that the (non-effective) action of $G$ on the variety of blocks is sharply 2-transitive, as well as the action on a block $\Delta$ of the normalizer $G_\Delta$. Also we require sharp transitivity on pairs $(X,Y)$ of independent points of $\Omega$, i.e. points contained in different blocks.