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. 



What can we do with a coarse moduli space that we can't do with a DM moduli stack?
An example is Deligne's theorem on the existence of good notion of quotient $X/G$ of a separated algebraic space $X$ under the action of a finite group $G$, or relativizations or generalizations (with non-constant $G$) due to D. Rydh. See Theorem 3.1.13 of my paper with Lieblich and Olsson on Nagata compactification for algebraic spaces for the statement and proof of Deligne's result in a relative situation, and Theorem 5.4 of Rydh's paper "Existence of quotients..." on arxiv or his webpage for his generalization. 


Note that in the above, there is no mention of DM stacks, but they come up in the proof! The mechanism to construct $X/G$ (say in the Deligne situation or its relative form) is to prove existence of a coarse space for the DM stack $[X/G]$ via Keel-Mori and show it has many good properties to make it a reasonable notion of quotient. Such quotients $X/G$ are very useful when $X$ is a scheme (but $X/G$ is "only" an algebraic space), such as for reducing some problems for normal noetherian algebraic spaces to the scheme case; cf. section 2.3 of the C-L-O paper. I'm sure there are numerous places where coarse spaces are convenient to do some other kinds of reduction steps in proofs of general theorems, such as reducing a problem for certain DM stacks to the case of algebraic spaces. 


Also, Mazur used a deep study of the coarse moduli scheme associated to the DM stack $X\_0(p)$ in his pioneering study of torsion in and rational isogenies between elliptic curves over $\mathbf{Q}$ (and these modular curves show up in numerous other places). But those specific coarse spaces are schemes and can be constructed and studied in more concrete terms without needing the fact that they are coarse spaces in the strong sense of the Keel-Mori theorem, so I think the example of Deligne's theorem above is a "better" example. 


