# 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. – dhagbert Apr 29 '13 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. – Jeremy Pecharich Apr 29 '13 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". – André Henriques Apr 29 '13 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. – David Roberts Apr 30 '13 at 3:44
• Why voting to close? The question has a simple answer: 'no'. – Matthieu Romagny Apr 30 '13 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 $p^2|N$. – user28172 May 1 '13 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. – Jason Starr May 1 '13 at 10:20