Rationality of GIT quotients - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T02:50:04Z http://mathoverflow.net/feeds/question/16085 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/16085/rationality-of-git-quotients Rationality of GIT quotients Charles Siegel 2010-02-22T20:31:32Z 2010-02-23T07:49:43Z <p>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:</p> <ol> <li>Every nonhyperelliptic genus 3 curve is a smooth plane quartic.</li> <li>The plane quartics form a projective space.</li> <li>Apply GIT to this projective space and the \$PGL(3)\$ action.</li> <li>Prove that this quotient is rational.</li> </ol> <p>I've seen somewhat similarly structured arguments before. So my question:</p> <blockquote> <p>When is a GIT quotient rational?</p> </blockquote> <p>In particular, are quotients of \$\mathbb{P}^n\$ by \$PGL_k\$ rational, under some reasonable hypotheses?</p> <p>Are there any natural invariants that are preserved by quotients (again, with reasonable conditions, or of the above form)?</p> http://mathoverflow.net/questions/16085/rationality-of-git-quotients/16133#16133 Answer by lieven lebruyn for Rationality of GIT quotients lieven lebruyn 2010-02-23T06:40:09Z 2010-02-23T07:18:31Z <p>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).</p> <p>Btw. Katsylo used it in the rationality of genus 3 curves you mentioned.</p> <p>C;early, the following implications hold</p> <p>rational ==> stably rational ==> unirational</p> <p>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).</p> <p>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.</p> <p>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&lt;= 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.</p> http://mathoverflow.net/questions/16085/rationality-of-git-quotients/16138#16138 Answer by David Lehavi for Rationality of GIT quotients David Lehavi 2010-02-23T07:49:43Z 2010-02-23T07:49:43Z <p>There is a very nice (if somewhat dated - it predates Katsylo's work of M<sub>3</sub>) survey of the problem by Dolgachev in the AG Bowdoin volume. Here is the <a href="http://books.google.com/books?id=pLViHKKXu8oC&amp;pg=PA3&amp;lpg=PA3&amp;dq=dolgachev+bowdin&amp;source=bl&amp;ots=OFzPVauTk4&amp;sig=7DSgFKYnNNlPmEpCfFto1j6wxrU&amp;hl=en&amp;ei=7YeDS4HTOoLy0gT60r3OAg&amp;sa=X&amp;oi=book%5Fresult&amp;ct=result&amp;resnum=1&amp;ved=0CD4Q6AEwAA#v=onepage&amp;q=&amp;f=false" rel="nofollow">google books link</a>.</p>