Local structure of Deligne-Mumford stacks - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T02:33:23Z http://mathoverflow.net/feeds/question/57007 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/57007/local-structure-of-deligne-mumford-stacks Local structure of Deligne-Mumford stacks Alex 2011-03-01T14:23:16Z 2011-03-01T14:40:18Z <p>Let $\mathcal{X}$ be a separated Deligne-Mumford stack over an algebraically closed field $k$ and let $X$ be the corresponding coarse moduli space, which we assume to exist. There is a map $p:\mathcal{X}\to X$ of stacks. Is it true that every point of $X$ has an etale neighborhood $U\to X$ such that its pullback under $p$ is the map $[V/G]\to V/G=U$ where $V$ is an affine variety over $k$ on which a finite group $G$ acts. While reading some papers I've got the impression that the authors are implicitly using this statement or a similar one, but I wasn't able to locate a precise statement or reference in the literature. So I would be grateful if someone points me to one.</p> <p>In the example I'm interested in $\mathcal{X}$ is in fact a quotient stack, but I do not want to assume that $char(k)=0$ or that the orders of the stabilizers are coprime with $char(k)$ (unless this follows from the previous conditions?).</p> http://mathoverflow.net/questions/57007/local-structure-of-deligne-mumford-stacks/57010#57010 Answer by David Zureick-Brown for Local structure of Deligne-Mumford stacks David Zureick-Brown 2011-03-01T14:40:18Z 2011-03-01T14:40:18Z <p>This is Lemma 2.2.3 of the paper </p> <p>Abramovich-Vistoli: Compactifying the space of stable maps;</p> <p>see also section 5.4 of the <a href="http://www.math.columbia.edu/~jarod/stacks_guide.pdf" rel="nofollow">"Guide to the stacks literature"</a> by Jarod Alper. </p>