Is there an example of a smooth projective variety $X$ such that $X$ is irrational, but $X\times X$ is rational?
For instance, is $X\times X$ irrational for a smooth cubic threefold $X$?
Is there an example of a smooth projective variety $X$ such that $X$ is irrational, but $X\times X$ is rational?
For instance, is $X\times X$ irrational for a smooth cubic threefold $X$?
For the first question I am not that pessimistic. At least there are candidates as follows: Recall that $Z$ is stably rational if there is $n\ge0$ such that $Z\times\mathbf A^n$ is rational. Now suppose there is such a $Z$ such that the minimal $n$ is $\ge2$. I would be extremely surprised if that didn't exist. Then put $Y=Z\times\mathbf A^{n-2}$ and $X=Y\times\mathbf A^1$. By assumption, $X$ is not rational but $X\times\mathbf A^1$ is. Let $d=\dim X\ge1$. Then $$ X\times X\cong Y\times \mathbf A^1\times X\cong Y\times\mathbf A^{d+1}\cong X\times\mathbf A^1\times\mathbf A^{d-1}\cong\mathbf A^{d+1}\times\mathbf A^{d-1}\cong\mathbf A^{2d} $$ A candidate for $Y$ would be the non-rational $3$-fold constructed by Beauville et al. for which Shepherd-Barron proved that $Y\times\mathbf A^2$ is rational. I don't know whether $Y\times\mathbf A^1$ is rational or not.