Irrationality of the first stabilization of a complex smooth cubic threefold

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 $m\ge 2$ and ${\mathrm{Sym}}^{2}X$ is rational.

22 pages