Irrationality of the first stabilization of a complex smooth cubic threefold

Source abstract

We show that for every smooth complex cubic threefold $Y$, the fourfold $Y\times\mathbb{P}^1$ is irrational. As an application, we construct a smooth stably rational complex Fano variety $X$ which is irrational, while $X^m$ is rational for every $m\geq2$ and $\mathrm{Sym}^2 X$ is rational.