  Some classes of cubic fourfolds are birational to fibrations over $P^2$,
where the fibers are rational surfaces. This is the case for cubics containing
a plane (resp. an elliptic ruled surface), where the fibers are quadric
surfaces (resp. del Pezzo sextic surfaces). It is known that the rationality of
these cubic hypersurfaces is related to the rationality of these surfaces over
the function field of $P^2$ and to the existence of rational (multi)sections of
the fibrations. We study, in the moduli space of cubic fourfolds, the
intersection of the divisor $C_{8}$ (resp. $C_{18}$) with $C_{14}$, $C_{26}$
and $C_{38}$, whose elements are known to be rational cubic fourfolds. We
provide descriptions of the irreducible components of these intersections and
give new explicit examples of rational cubics fibered in (quartic, quintic) del
Pezzo surfaces or in quadric surfaces over $P^2$. We also investigate the
existence of rational sections for these fibrations. Under some mild
assumptions on the singularities of the fibers, these properties can be
translated in terms of Brauer classes on certain surfaces.
