On connected line sets of antiflag class [0,α,q][0,α,q] in AG(n,q)

We investigate the partial linear spaces, fully embedded in an affine space AG(n,q) with the property that for every antiflag {p,L}{p,L}, the number of lines through pp intersecting LL is either 0,α0,α, or qq. Besides some general results we prove a complete classification of those geometries fully embedded in an affine plane of order qq and of the connected geometries with 1<α