We say that two (irreducible) algebraic varieties $X$ and $Y$ are stably birational if $X \times \mathbb{P}^n$ is birational to $Y\times \mathbb{P}^n$ for some $n\ge 0$.

The natural question is then the following: if $X$ and $Y$ are stably birational, then are they birational?

For the case of curves, it is an easy exercise to show that it is true. Does someone has a reference for this? The case of surfaces is also true, I think (again I would be happy to have a reference). For large dimension, I only know the result of Beauville, Colliot-Thélène, Sansuc, Swinnerton-Dyer (Annals of Math 1985), which says that the answer to the above question is no in general.

Are there more recent results/examples in this direction?

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    $\begingroup$ You answer your own "natural question". Is the revised question something like, what is the last dimension in which this holds? (What is the large dimension in which it is known to fail?) $\endgroup$ – Allen Knutson May 7 '16 at 18:21
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    $\begingroup$ I think the question of the post is the question in the last sentence, not the question in the title (=the "natural question"). $\endgroup$ – potentially dense May 7 '16 at 19:21
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    $\begingroup$ The case for curves is V, Exercise 2.1 in Hartshorne; Mohan in a comment on Math.SE provides a sketch of the argument, but I don't know of a reference otherwise of this geometric argument; on the other hand, from the field-theoretic perspective I think this is a result of Nagata and Deveney. There is a more expository paper by Ohm as well. $\endgroup$ – Takumi Murayama May 7 '16 at 19:22
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    $\begingroup$ … for more recent results, this paper by Kang discusses what we know at the moment in §2. In particular, the surface case is open in characteristic $p > 0$; a more recent paper by Belov and Yu showed the case in characteristic $p > 0$ when $Y = \mathbf{A}^2$. $\endgroup$ – Takumi Murayama May 7 '16 at 19:29
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    $\begingroup$ Aside: The recent spectacular advances initiated by Voisin and pursued by Colliot-Thélène/Pirutka, Totaro, Hassett/Pirutka/Tschinkel show that some specific varieties are not stably rational. On the other hand, the results pertaining to birational superrigidity (Pukhlikov, De Fernex,...) seem to establish irrationality only. $\endgroup$ – ACL May 7 '16 at 20:37

A pretty recent result of Kollár, Symmetric powers of Severi-Brauer varieties, classifies products of symmetric powers of a Severi-Brauer variety up-to stable birational equivalence. The description includes Grassmannians and moduli spaces of genus 0 stable maps.

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