# $\mathbb{P}^1$-fibrations over $\mathbb{P}^2$ that are not rational

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!

• The title suggests what you are really asking, but the body text of the question is still rather unclear. What fails? – user5117 Mar 7 '14 at 12:19
• I presume that he means that the result is false if you replace "rationally connected" with "rational". – Jérémy Blanc Mar 7 '14 at 13:07

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.

• Thank you very much for your answer and the corrections. I will edit the question in order to be well posed. – Pedro Montero Mar 7 '14 at 11:22
• You are welcome. Yes, the title looks better now. – Jérémy Blanc Mar 7 '14 at 11:26

Note that a conic bundle is rationally connected. It is still an open problem to establish if rational connectedness and unirationality are equivalent notions or not. An example of a non unirational conic bundle would be a counterexample. On the other hand the unirationality of conic bundle is an important open problem as well. There some partial results, under some hypothesis on the discriminant here http://arxiv.org/pdf/1403.7055.pdf.

Here is an example of a morphism with rational target and rational fiber.

Let $π : X\rightarrow \mathbb{F}_e$ be a standard $3$-fold conic bundle. Let $\Delta$ be the discriminant curve. Assume that $\Delta \sim aC_0 + bF$, with $3 < a \leq 7$, then $X$ is unirational but not rational.

This is Corollary $1.3$ in http://arxiv.org/pdf/1403.7055.pdf.