A couple weeks ago I attended a talk about the Keel-Mori theorem regarding existence of coarse moduli spaces for Deligne-Mumford stacks with finite inertia. Here are some questions that I have been wondering about since then: What are some applications of this theorem? What does it matter if a DM stack has a coarse space? What are examples of things that we can do with the coarse space that we maybe can’t do with the stack? Given (for instance) a moduli problem, what does the existence of a coarse moduli space tell us that the existence of a DM moduli stack doesn’t tell us?
Since the coarse space, if it exists, is probably determined by the stack (is it?), I should probably be asking instead: What can we do more easily or more directly with a coarse space than with a stack?
Here is a bad answer: If we are interested in intersection theory (as in e.g. Gromov-Witten theory), then the existence of the coarse space can help us to circumvent having to develop an intersection theory for stacks. But clearly this is a pretty lame answer.