ag.algebraic geometry - Rationality of GIT quotients

A useful general result is the 'no-name lemma' stating that when a reductive group G acts linearly on two vectorspaces V and W 'almost freely' (that is, the stabilizer subgroup of a general point is trivial), then the GIT-quotients V/G and W/G are stably rational (that is, V/G x C^m and W/G x C^m are birational for some m and n).

Btw. Katsylo used it in the rationality of genus 3 curves you mentioned.

C;early, the following implications hold

rational ==> stably rational ==> unirational

and counterexamples to the other implications exist (Artin-Mumford for a unirational non stably rational variety and Colliot-Thelene, Sansuc and Swinnerton-Dyer for a non-rational stably rational one).

As to PGL_n : here the 'canonical' example of a vectorspace having an almost free PGL_n-action is couples of nxn matrices under simultaneous conjugation. Hence, by the NNL any other almost free GIT-quotient is stably rational to it.

Here the best result known is that when n divides 420=2^2x3x5x7 then such quotients are stably rational. For couples of matrices under simultaneous conjugation rationality is known for n<= 4 but even for the cases n=5 and n=7 only stably rationality is known. 'Retract rationality' (a lot weaker than stable rationality) is known for all squarefree n by a result of David Saltman.