A singular cubic fourfold containing a plane

Abstract

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 conditions on the geometry of the plane sextics so that all the associated cubic 4-folds are singular. Moreover, we construct a family of smooth rational cubic 4-folds whose discriminant curve is reduced but reducible.