On connected line sets of antiflag class in

We investigate the partial linear spaces, fully embedded in an affine space AG ( n , q ) 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 AG ( 3 , q ) .