Thanks a lot Qing for the very slick argument! It is completely self contained and the key result that you use is that there is no non-constant rational map going form $\mathbb{P}^1\rightarrow E$ which as you pointed is a consequence of Luroth's theorem. Cool!

Of course, one could verify by a local computation that a blow up at one point does not change $h^1(\mathcal{O})$ but then one would have to show that a birational map between $X_1$ and $X_2$ could be obtained by a sequence of blow ups which a priori looks as a more difficult question than the original.

I'm quite happy with your proof but I think there should be a more elementary proof which is self contained. You see the whole point of this question is to come up with a birational invariant and if we assume from the outset that $h^1(\mathcal{O})$ is a birational invariant then it (almost) kills the problem.

So I guess the first $X_2$ should read as $X_1$. Is it completely obvious that $h^1(\mathcal{O}_X)$ is a birational invariant? After all $\mathcal{O}_X$ is the sheaf of regular functions on $X$ which a priori could change under birational maps.

@Angelo, thanks for your very simple and instructive example!

I needed my map $f$ to be a closed embedding and not just proper.

Hi @Angelo, thanks for pointing my mistake, I know what is wrong!

Thanks Anon. Can you give me a precise state? What is the analogue $\mathbb{P}^1-\\{0,1,\infty\}}$ in higher dimension?