Finite Flag - Transitive An - 2 . 2 . L * . L Geometries

If F is a geometry for this rank 3 diagram and F' is its shadow geometry with respect to the central node of the diagram, then f satisfies the Intersection Property (IP), it belongs to the following diagram where L and L* denote the classes of linear spaces and dual linear spaces, respectively. According to a well established habit, we only consider residually connected and firm geometries. That is, we include residual connectedness and firmness in the definition of geometry, as in [25]. The classification theorem will be stated in Section 3 (Theorem 1). The examples to be mentioned in that classification will be described in Section 2. We leave the rank 3 case out of the scope of this paper. When n = 3, the diagram An-2.L*.L looks as follows: