# Analysis, Geometry, and Dynamical Systems Seminar

### Some loci of rational cubic fourfolds

We shall report on joint work with Michele Bolognesi and Giovanni Staglianò on the irreducible divisor $\mathcal C_{14}$ inside the moduli space of smooth cubic hypersurfaces in $\mathbb P^5$. A general point of $\mathcal C_{14}$ is, by definition, a smooth cubic fourfold containing a smooth quartic rational normal scroll (or, equivalently, a smooth quintic del Pezzo surfaces) so that it is rational. We shall prove that every cubic fourfold contained in $\mathcal C_{14}$ is rational.

In passing we shall review and put in modern terms some ideas of Fano, yielding a geometric insight to some known results on cubic fourfolds, e.g. the Beauville-Donagi isomorphism, and discuss also the connections of our results with the recent examples about the bad behavior of rationality in smooth families of fourfolds.