Some Loci of Rational Cubic Fourfolds

TL;DR

This paper studies a special divisor in the moduli space of smooth cubic fourfolds, showing that all such fourfolds containing a quartic scroll are rational and analyzing the structure of the Pfaffian locus within this divisor.

Contribution

It identifies the divisor of cubic fourfolds containing quartic scrolls as rational and explores the non-openness of the Pfaffian locus within this divisor.

Findings

All cubic fourfolds in the divisor are rational.

Degenerations of quartic scrolls lead to surfaces with one apparent double point.

The Pfaffian locus is not open in the divisor er the constructed examples.

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 $\mathcal C_{14}$.