Alternating forms and transitive locally grid geometries

Let V be a vector space over K. For each nonnegative integer k let Pk(V ) be the set of k-subspaces of V . For positive d (≤ dimV ), let Ld(V ) be the set of pairs (U,W ) with U ∈ Pd−1(V ), W ∈ Pd+1(V ), and U ≤ W . We also set B− d (V ) = Pd−1(V ), B + d (V ) = Pd+1(V ), and Bd(V ) = B − d (V ) ∪ B + d (V ). In the case V = K and d = 2, we can view P2(V ) and L2(V ) as the points and lines of the Klein quadric. Then (B− 2 (V ),P2(V ),B + 2 (V )) is the associated rank 3 geometry of type D3 (= A3) and (P2(V ),L2(V ),B2(V )) the associated rank 3 polar space of type C3. We look at several geometries related to these for the Klein quadric. Let Ad(V ) = (Pd(V ),Ld(V )). (Here and throughout, d is some positive integer but dimKV may be infinite unless stated otherwise.) Ad(V ) is a partial linear space with point set Pd(V ) and line set Ld(V ) whose members (U,W ) we often identify with the 1 + |K| distinct d-spaces (that is, incident points) in between the (d − 1)-space U and the (d + 1)-space W . Ad(V ) is called a Grassmann space and, sometimes, a d-Grassmann space. We associate two rank 3 geometries with Ad(V ). The first is the Grassmannian geometry (or d-Grassmannian geometry)