  In this paper we investigate the divisor $\mathcal C_{14}$ inside the moduli
space of smooth cubic hypersurfaces in $\mathbb P^5$, whose generic element is
a smooth cubic containing a smooth quartic scroll. Using the fact that all
degenerations of quartic scrolls in $\mathbb P^5$ contained in a smooth cubic
hypersurface are surfaces with one apparent double point, we conclude that
every cubic hypersurface belonging to $\mathcal C_{14}$ 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
$\mathcal C_{14}$.
