4
$\begingroup$

Let $X$ be a proper Deligne-Mumford stack, whose normalization, $X'$, is a global quotient stack (that is, a stack of the form [W/GL_n],where W is an algebraic space) with a projective scheme as a coarse moduli space.

Question:Is it true that $X$ is also a global quotient stack?

$\endgroup$
6
  • 1
    $\begingroup$ Two questions on your question: 1) the normalization of a DM-stack is well-defined because normalization is an étale-local property, right ? is it well-defined for Artin stacks ? 2) What are the reasons for your proper/projective assumptions ? Do you have easy counterexamples without them ? $\endgroup$
    – ACL
    Mar 28, 2014 at 8:25
  • $\begingroup$ @ACL: Regarding the need for "proper", one silly observation is that one does, at least, need "quasi-compact". Otherwise the disjoint union of $B\mu_n$ over all integers $n$ is a counterexample. But this counterexample doesn't address, say, "proper" versus "quasi-compact and locally finitely presented over the base". $\endgroup$ Mar 28, 2014 at 11:59
  • $\begingroup$ @ACL: Also, I think the counterexamples of Edidin, Hassett, Kresch and Vistoli show why one should assume that $X$ is separated. Still, I don't see why one should prefer "proper" to "separated, quasi-compact and locally finitely presented" for this question. $\endgroup$ Mar 28, 2014 at 13:37
  • $\begingroup$ @Jason Starr: You are right. People believe that a stack with the conditions you mentioned above should be a quotient stack. In my situation I can also assume these extra conditions. $\endgroup$
    – matthew
    Mar 28, 2014 at 14:03
  • 2
    $\begingroup$ @ACL: in fact normalization commutes with smooth localization, hence is well-defined for Artin stacks. This is proved in Laumon and Moret-Bailly's book. $\endgroup$ Mar 28, 2014 at 21:06

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.