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 general element is a smooth cubic containing a smooth quartic rational normal scroll. By showing that all degenerations of quartic scrolls in $${\mathbb {P}}^5$$ contained in a smooth cubic hypersurface are surfaces with one apparent double point, we prove that every cubic hypersurface contained in $${\mathcal {C}}_{14}$$ is rational. Combining our proof with the Hodge theoretic definition of $${\mathcal {C}}_{14}$$ , we deduce that on a smooth cubic fourfold every class $$T\in {\text {H}}^{2,2}(X,\mathbb {Z})$$ with $$T^2=10$$ and $$T\cdot h^2=4$$ is represented by a (possibly reducible) surface of degree four which has one apparent double point. As an application of our results and of the construction of some explicit examples, we also prove that the Pfaffian locus is not open in $${\mathcal {C}}_{14}$$ .