Let \({\mathcal C}\) be the moduli space of smooth cubic hypersurfaces of \(\mathbb{P}^5\) (cubic fourfolds); if \({\mathcal U} \subset \mathbb{P}^{5}\) is the open set parameterizing those cubic hypersurfaces which are smooth, we have that \({\mathcal C}\) is given by the orbits space defined on \({\mathcal U}\) by the action of \(SL_6\) on \(\mathbb{P}^5\).
There is a divisor \({\mathcal C}_8 \subset {\mathcal C}\) which parameterizes the cubic fourfolds that contain a plane; it is conjectured that the general element in \({\mathcal C}_8\) is not rational, but no irrational example has been found yet. The main result of this paper is the following, which shows that it is not true that the \(generic\) element of \({\mathcal C}_8\) is irrational:
There is a countably infinite collection of divisors in \({\mathcal C}_8\) that parameterize rational cubic fourfolds. Each of these is a codimension-two subvariety of \({\mathcal C}\).
What is used is that a cubic fourfold containing a plane is birational to a smooth quadric surface over \(k(\mathbb{P}^2)\) (projecting from the plane \(P\) to \(\mathbb{P}^2\) we have quadric surfaces as fibers; this gives to the blow-up of the fourfold along \(P\) a structure of quadric surface bundle). If the quadric over \(k(\mathbb{P}^2)\) is rational, so is the fourfold.
The countable union of divisor is determined using Hodge theory on quadric surfaces.