Are conical symplectic resolutions Mori dream spaces? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T10:40:55Z http://mathoverflow.net/feeds/question/80313 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/80313/are-conical-symplectic-resolutions-mori-dream-spaces Are conical symplectic resolutions Mori dream spaces? Ben Webster 2011-11-07T17:05:54Z 2011-11-07T21:02:15Z <p>This is one of these questions where it's tempting to just leave it at the title, but let me try to define the objects in question.</p> <p>A <em>conical symplectic resolution</em> is a projective resolution of singularities $X \to Y$ such that </p> <ul> <li>$X$ is algebraically symplectic, </li> <li>$Y$ is affine, and </li> <li>there are compatible $\mathbb{G}_m$-actions on the two varieties which make $Y$ into a cone and act on the symplectic form with positive weight $n$.</li> </ul> <p>Examples include the Springer resolution, a minimal resolution of a rational double point, the Hilbert scheme of points in that space (via the Hilbert-Chow resolution), a hypertoric variety or a Nakajima quiver variety.</p> <p>All of these spaces have something in common: they are (relative) <a href="http://en.wikipedia.org/wiki/Mori_dream_space" rel="nofollow"><em>Mori dream spaces</em></a>. (For a definition of "relative Mori dream space," see <a href="http://alpha.science.unitn.it/~andreatt/symplectic5c.pdf" rel="nofollow">this paper</a>).</p> <p>Thus, I am inclined to wonder:</p> <blockquote> <p>Are all conical symplectic resolutions relative Mori dream spaces? Or am I just not original enough to come up with counter examples?</p> </blockquote>