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.



are moduli stacks deligne-mumford stacks in general
If the objects under consideration all have finite automorphism groups, you should expect your stack to be Deligne-Mumford. Otherwise, it isn't Deligne-Mumford; but that is no cause for alarm.


