Some Loci of Rational Cubic Fourfolds

Abstract

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 ${\mathcal{C}}_{14}$.

arXiv comment24 pages. Final, rewritten and expanded version with some new results, to appear on Math. Annalen