Incidence Structures whose Planes are Nets

A d-net is a connected semilinear incidence structure π such that (D1) every plane is a net, (D2) the intersection of two subspaces is connected, (D3) if two planes in a 3-space have a point in common then they have a second point in common, and (D4) the minimum number of points which generate π is d. Let V be a vector space over a skew field F, and W a subspace of finite codimension d. Let P, L be the set of d-, (d - 1)-dimensional subspaces respectively of V whose intersection with W is the zero vector. The incidence structure (P,L⊇) is called an attenuated space. We show every d-net for finite d ⩾ 3 is an attenuated space. We also characterize d-nets (together with AG(d, 2)) as those incidence structures belonging to the diagram where signifies a projective plane and signifies a net