I recently worked through most of the proof of the rationality of the moduli of genus 3 curves, which seemed to have the following structure:


1. Every nonhyperelliptic genus 3 curve is a smooth plane quartic.
2. The plane quartics form a projective space.
3. Apply GIT to this projective space and the $PGL(3)$ action.
4. Prove that this quotient is rational.


I've seen somewhat similarly structured arguments before. So my question:



> 
> When is a GIT quotient rational?
> 
> 
> 


In particular, are quotients of $\mathbb{P}^n$ by $PGL\_k$ rational, under some reasonable hypotheses?


Are there any natural invariants that are preserved by quotients (again, with reasonable conditions, or of the above form)?



Rationality of GIT quotients
A useful general result is the 'no-name lemma' stating that when a reductive group $G$ acts linearly on two vector spaces $V$ 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, $V/G \times \mathbb{C}^m$ and $W/G \times \mathbb{C}^n$ are birational for some $m$ and $n$).


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


Clearly, the following implications hold:


rational $\implies$ stably rational $\implies$ 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 vector space having an almost free $PGL\_n$-action is couples of $n\times n$ 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 $420=2^2\times3\times5\times7$ then such quotients are stably rational. For couples of matrices under simultaneous conjugation rationality is known for $n\leq 4$ 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 square-free $n$ by a result of David Saltman.


