Rationality of GIT quotients 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:


*

*Every nonhyperelliptic genus 3 curve is a smooth plane quartic.

*The plane quartics form a projective space.

*Apply GIT to this projective space and the $PGL(3)$ action.

*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)?
 A: There is a very nice (if somewhat dated - it predates Katsylo's work of M3) survey  of the problem by Dolgachev in the AG Bowdoin volume. Here is the google books link.
A: 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.
