# are moduli stacks deligne-mumford stacks in general

Let M be your favorite moduli stack over the field of complex numbers.

Is it reasonable to expect M to be a Deligne-Mumford stack?

I know this is true for the moduli space of curves of genus g, ppav's and K3 surfaces. I'm just wondering what I should expect when considering other moduli stacks.

• Stable coherent sheaves are simple and so have a $\mathbb{G}_{m}$ of automorphisms. Apr 29, 2013 at 22:00
• My favorite is $Vect_{n,d}(C)$ of vector bundles on a curve of genus $g\geq 2$ of fixed rank and degree up to isomorphism. The automorphism group has positive dimension, hence not a Deligne-Mumford stack. Maybe what you should expect is to replace Deligne-Mumford by Artin. Apr 29, 2013 at 22:09
• I'm voting to close this question as it has no real answer. Or maybe the answer is "it depends on your taste", and on the meaning of "favorite". Apr 29, 2013 at 22:50
• The problem is, one can define a moduli stack (which is Artin) for any groupoid in the category of complex manifolds. Generically these won't be étale, hence the stack won't be Deligne-Mumford. Apr 30, 2013 at 3:44
• Why voting to close? The question has a simple answer: 'no'. Apr 30, 2013 at 8:14

• @Jason: For another example, $X_0(N)$ (appropriately defined as a proper flat Artin stack over $\mathbf{Z}$) is not Deligne-Mumford in characteristic $p$ when $p^2|N$. May 1, 2013 at 2:33
• Correction: As userN points out, I missed the OP's hypothesis that the stack is defined over $\text{Spec}(\mathbb{C})$. For stacks over $\text{Spec}(\mathbb{C})$, I agree with Ravi: every algebraic stack with finite diagonal is Deligne-Mumford. This follows from Artin's theorems. May 1, 2013 at 10:20