On connected line sets of antiflag class in

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