Some Loci of Rational Cubic Fourfolds

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 $\mathcal C_{14}$.Comment: 24 pages. Final, rewritten and expanded version with some new results, to appear on Math. Annale

New examples of rational Gushel-Mukai fourfolds

We construct new examples of rational Gushel-Mukai fourfolds, giving more evidence for the analog of the Kuznetsov Conjecture for cubic fourfolds: a Gushel--Mukai fourfold is rational if and only if it admits an associated K3 surface.Comment: Minor changes. Exposition improved. To appear in Mathematische Zeitschrif