Methods of showing a variety is stably rational

Question

As anyone who follows the algebraic geometry tag on arXiv will probably know, there has been a lot of papers recently showing various varieties are non-stably rational. What I am interested in however is ways of showing a variety is stably rational.

Obviously, showing a variety is rational would do the trick. There are examples of stably rational varieties which are non-rational, the first as far I'm aware was in "Variétés stablement rationnelles non rationnelles" by A. Beauville, J.-L. Colliot-Thélène, J.-J. Sansuc, and P. Swinnerton-Dyer. Another proof came later in "Stably rational irrational varieties" by N. Shepherd-Barron. The variety and it's proof arises out of group actions and geometric invariant theory, an area which I am not very knowledgeable on, so I can't really give a better description.

The second potential method I've found is from the result of Larsen and Lunts on the Grothendieck ring of varieties. I.e. If $X$ is a smooth and complete variety, then it is stably birational if and only if $[X] = 1 \text{ mod } \mathbb{L} $, where $[X]$ is it's class in the Grothendieck ring and $\mathbb{L} = [\mathbb{A}^1] $. This seems to be quite a nice criterion, but calculating classes explicitly in the Grothendieck ring seems to be in general rather hard. I have yet to see this be used to give an example of a non-rational, stably rational variety.

Are there any other potential methods I have not mentioned?

Answer 1

Though it applies to quite particular varieties, one might mention the so-called "no-name method": let $G$ be a reductive group which admits an almost free representation $V$ such that the quotient $V/G$ is rational ("almost free" means that there is a $G$-stable Zariski open subset of $V$ on which $G$ acts freely). Then for all almost free representations $W$ of $G$, the quotient $W/G$ is stably rational. This applies to many groups, e.g. $GL(n)$ (take for $V$ the space of $n\times n$ matrices, with $GL(n)$ acting by multiplication), $SL(n)$, $SO(n)$ etc. A good reference is Dolgachev's survey Rationality of fields of invariants in the Proceedings of the Bowdoin AMS conference.