  In this paper we consider cubic 4-folds containing a plane whose discriminant
curve is a reduced nodal plane sextic. In particular, we describe the singular
points of such cubic 4-folds and we give an estimate of the rank of the free
abelian group generated by the equivalence classes of the algebraic cycles of
codimension 2. Moreover, we describe some conditions on the geometry of the
plane sextics so that all the associated cubic 4-folds are singular and we
construct a family of smooth rational cubic 4-folds whose discriminant curve is
reduced but reducible.
