Projective injections of geometries and their affine extensions (Q1346999)

In this remarkable paper, the authors describe a geometrical method to extend a given geometry under very mild conditions. The new geometry has as diagram the old diagram extended by one node, which is joined by an edge with label to exactly one node of the old diagram. The label symbolizes the class of rank 2 geometries with point set the set of points of an affine space, line set the set of lines of that affine space belonging to a prescribed set of parallel classes, and natural incidence relation. When the affine space is actually an affine plane, and when one prescribes only two parallel classes, then a (non-thick) generalized quadrangle is obtained. It is also shown how flag-transitive extensions may be constructed.