Question

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)?

Answer 1

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 2

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.