Can a singular Deligne-Mumford stack have a smooth coarse space? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T02:54:06Z http://mathoverflow.net/feeds/question/1565 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/1565/can-a-singular-deligne-mumford-stack-have-a-smooth-coarse-space Can a singular Deligne-Mumford stack have a smooth coarse space? David Zureick-Brown 2009-10-21T03:05:17Z 2010-01-03T21:22:37Z <p>Let XX be a Deligne-Mumford stack and let XX \to X be a coarse moduli space. Suppose that X is smooth. Is XX smooth? If not, what is an example? What if XX is of finite type over C (the complex numbers)? What are conditions we can put on XX to make this true?</p> http://mathoverflow.net/questions/1565/can-a-singular-deligne-mumford-stack-have-a-smooth-coarse-space/1584#1584 Answer by Anton Geraschenko for Can a singular Deligne-Mumford stack have a smooth coarse space? Anton Geraschenko 2009-10-21T04:43:56Z 2009-10-21T04:43:56Z <p>The answer is yes, a singular DM stack can have a smooth coarse space. Let U=Spec(k[x,y]/(xy)) be the union of the axes in <b>A</b><sup>2</sup>, and consider the action of G=<b>Z</b>/2 given by switching the axes: x&rarr;y and y&rarr;x. Then take XX to be the stack quotient [U/G]. This is a singular Deligne-Mumford stack (since it has an etale cover by something singular), but its coarse space is <b>A</b><sup>1</sup>, which is smooth.</p> http://mathoverflow.net/questions/1565/can-a-singular-deligne-mumford-stack-have-a-smooth-coarse-space/10622#10622 Answer by shenghao for Can a singular Deligne-Mumford stack have a smooth coarse space? shenghao 2010-01-03T21:22:37Z 2010-01-03T21:22:37Z <p>I think if the coarse moduli space is smooth, so is the DM stack, because XX --> X is a gerbe, which is always smooth (since smoothness can be checked fppf locally on X, and B(G/X) is smooth over X). A stack (or a morphism of stacks, not necessarily representable) is defined to be smooth if one can find a presentation which is smooth over the base. And if it is smooth, then any presentation is smooth. That's why I got confused on Anton's example. Maybe someone can explain this to me. Thanks in advance.</p>