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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.

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Regarding the parenthetical question in your second paragraph, the coarse space (i.e., the morphism $\mathcal{X} \to X$, where $X$ is an algebraic space) is unique up to unique isomorphism if it exists, by the universal property of coarse spaces. – S. Carnahan Mar 13 '10 at 4:16
Deligne one said to me "sometimes it's necessary to have a Zariski-open neighbourhood of a point". I don't know what situation he was thinking about, but my understanding of his point that a point in an algebraic stack only has an etale neighbourhood, rather than a Zariski open neighbourhood. – Kevin Buzzard Mar 13 '10 at 7:52
"you wouldn't think it would come up too often"---but how many papers mention the j-invariant! ;-) – Kevin Buzzard Mar 13 '10 at 10:19
@fpqc: a DM stack covered by open subschemes is a scheme. This is immediate from basic considerations with definitions. Even for algebraic spaces one does not have open scheme neighborhoods of all "points". That is the whole difficulty of doing geometry with them, and underlies their added flexibility beyond working with schemes. I don't know from where you have been reading about these fancy things, but I strongly recommend that you learn basics about algebraic spaces very well before exploring the theory of stacks. Otherwise it's like running a marathon without tying your shoelaces. – BCnrd Mar 13 '10 at 11:22
I think Kevin's remark about the $j$-invariant is pretty pertinent here. Having a coarse moduli space means that the objects being classified are determined (over algebraically closed fields) by some parameters (the points of a variety). That seems pretty important information, when it is true. After all, as Kevin indicates, the basic tool for talking about elliptic curves is the $j$-invariant. – Emerton Mar 13 '10 at 15:57
up vote 10 down vote accepted

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.

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@Brian: Mazur's paper is a great thing to bring up. Stacky Y_0(p) isn't a scheme, but a lot of time is spent in that paper analysing scheme Y_0(p) (and of course the local equations at the ss points are different: XY=p vs XY=p^e). So in some sense the stack and the scheme really are quite different objects (the scheme isn't always even regular!). I wonder what happens if you "try to prove Mazur's theorem using only stacks", i.e. you're not "allowed to use" the scheme Y_0(p) with local equation XY=p^e at the ss points, but you can have the stack and also the scheme Y_1(N;p) for N>=4 etc. – Kevin Buzzard Mar 13 '10 at 16:46
@Kevin: I think it would be awkward at best, and probably impossible to pull off. For starters, he makes essential use the Neron model of the Jacobian, which is influenced by the structure of the minimal regular resolution, whereas the moduli stack is regular (and it seems unlikely that a non-scheme stack would give rise to a "Jacobian" with useful properties like for the coarse scheme). Also, since $X_0(p)$ is "nowhere" a scheme (even over $\mathbf{Q}$) and there's no link to rational points on $Y_1(N,p)$ with $N \ge 4$, extra level structure doesn't seem to help. – BCnrd Mar 13 '10 at 17:09
Oh my goodness yes of course the special fibre of the Neron model of the Jacobian is controlled by the reduction of the curve and of course it sees all of those e's doesn't it. So in fact this makes James Borger's comment even more bewildering: "you wouldn't think it comes up too often" but in fact those e's explain all sorts of things, like "dihedral" (non-Eisenstein) errors in the theory of mod p modular forms and so on (e.g. the level 13 counterexample to Serre's conjecture if you demand that the char of the char 0 form is the Teichmuller lift of the mod p form). – Kevin Buzzard Mar 13 '10 at 17:57

May I suggest rephrasing the question to something like: Are there any results in the theory of stacks that rely on the existence of an underlying coarse moduli space?

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This isn't really as specific as the question as it currently stands. – Harry Gindi Mar 13 '10 at 7:21

Regarding Borger's philosophical remark. Regular functions and sections of geometric line bundles map from a source space to some target space. This would seem a fair philosophical reason why coarse moduli spaces should be important, and moreover they require less theory to study. Also if one thinks of problems involving semi-stable reduction or potentially good reduction, monodromy actions on sections of an etale cover of the base (e.g. the n torsion points of an Abelian variety) are often the central objects study. So here some sort of dual principle is going on: the group scheme of n torsion points maps to the base; but their sections map from the base and form part of the functor of points. And the functor of points determines the original scheme. I guess the slogan is ``sections matter''.

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