Rationality of GIT quotients - MathOverflow

Rationality of GIT quotients

2010-02-22T20:31:32Z

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)?

Answer by lieven lebruyn for Rationality of GIT quotients

2010-02-23T06:40:09Z

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.

Answer by David Lehavi for Rationality of GIT quotients

2010-02-23T07:49:43Z

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.