Rational fibered cubic fourfolds

Some classes of cubic fourfolds are birational to fibrations over \({\mathbb {P}}^2\), 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 \({\mathbb {P}}^2\) 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 \({\mathcal {C}}_{8}\) (resp. \({\mathcal {C}}_{18}\)) with \({\mathcal {C}}_{14}\), \({\mathcal {C}}_{26}\) and \({\mathcal {C}}_{38}\), 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 \({\mathbb {P}}^2\). 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.