Sets of class [q+1, 2q+1, 3q+1]3 in PG(4, q)

Abstract The article gives a combinatorial characterisation of a ruled cubic surface in PG ( 4 , q ) . In PG ( 4 , q ) , q odd, q ≥ 9 , a set K of q 2 + 2 q + 1 points is a ruled cubic surface if and only if K has class [ q + 1 , 2 q + 1 , 3 q + 1 ] 3 such that every plane that contains four points of K contains at least q + 1 points of K .