Some loci of rational cubic fourfolds

In this paper we investigate the divisor C_14 inside the moduli space of smooth cubic hypersurfaces in P^5, whose generic element is a smooth cubic containing a smooth quartic scroll. Using the fact that all degenerations of quartic scrolls in P^5 contained in a smooth cubic hypersurface are surfaces with one apparent double point, we conclude that every cubic hypersurface belonging to C_14 is rational. As an application of our results and of the construction of some explicit examples contained in the Appendix, we also prove that the Pfaffian locus is not open in C_14.