We know that if $X$ is a smooth complex projective variety and we assume that there is a dominant morphism $f : X \to Y$ with $Y$ and the general fibers of $f$ rationally connected. Then $X$ itself is rationally connected. (This is
established as a corollary of Graber, Harris and Starr’s theorem).


My question is why this fails for rational varieties of higher dimension. I saw in some works about rationally connected varieties that "there exists" some examples of $\mathbb{P}^1$-fibrations over $\mathbb{P}^2$ that are not rational, but I couldn't find these examples explicitely.


Can you give me an idea or references, please?
Thank you very much in advance!



$\mathbb{P}^1$-fibrations over $\mathbb{P}^2$ that are not rational
First, a $\mathbb{P}^1$-bundle over $\mathbb{P}^2$ is by definition rational, because it is locally trivial. What you want is probably a fibration over $\mathbb{P}^2$ with general fibre rational (but not the generic fibre), and which is not rational. For this, consider conic bundles over $\mathbb{P}^2$. The discriminant is the curve of $\mathbb{P}^2$ where the conic becomes singular. If the discriminant has a large degree, then the 3fold tha you get is birationnaly rigid and the conic bundle structure is in fact unique. This is a theorem of Sarkisov, which gave rise to a so called Sarkisov program, very useful for these questions. You have the same kind of result for del Pezzo fibration over the line, the general fibre is rational and the target too but the 3fold us not.


Have a look at page 283 of the book "birationnally rigid varieties" by A. Pukhlikov. On google books, the page is available . It gives good idea of the story.


