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?