# 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:

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)?

• 1: every 'non-hyperelliptic' genus 3 curve is a smooth quartic. 3: that would be PGL(3), not PGL(4). – VA. Feb 22 '10 at 21:46
• This is not an answer, but maybe helpful: Probably you have come across this yourself, but in case you have not, Chris Boehning's habilitation may be useful: uni-math.gwdg.de/boehning/invariant_theory_book.pdf – olli_jvn Feb 22 '10 at 22:23
• @VA Thanks for the corrections. Is there a 2? – Charles Siegel Feb 23 '10 at 1:23

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.

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.