In the first part, based on the preprint arXiv:2305.03481 (to appear in Comptes Rendus Math.), we show that even within a class of varieties where the Brauer–Manin obstruction is the only obstruction to the local-to-global principle for the existence of rational points (Hasse principle), this obstruction, even in a stronger, base change invariant form, may be insufficient for explaining counter-examples to the local-to-global principle for rationality. We exhibit examples of toric varieties and rational surfaces over an arbitrary global field $k$ each of those, in the absence of the Brauer obstruction, is rational over all completions of $k$ but is not $k$-rational.

In the second part, based on a work in progress (in collaboration with Jean-Louis Colliot-Thélène), for every global field $k$ and every $n \ge 3$ we give an example of a birational involution of $\mathbb P^n_k$ (that is an element $g$ of order $2$ in the Cremona group $\mathrm{Cr}(n, k))$ such that $\bullet$ $g$ is not linearizable; $\bullet$ $g$ is linearizable in all $\mathrm{Cr}(n, k_v)$.

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