Timeline for What can we do with a coarse moduli space that we can't do with a DM moduli stack?
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20 events
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| Nov 11, 2010 at 10:57 | answer | added | steve | timeline score: 0 | |
| Apr 1, 2010 at 17:36 | vote | accept | Kevin H. Lin | ||
| Mar 14, 2010 at 4:20 | comment | added | JBorger | But I don't yet see an example of a question phrased naturally in terms of the original moduli problem, rather than its representing object, which is best studied using the coarse moduli space. Admittedly, the moduli problem and the representing object determine each other, so maybe this distinction is not real. But I do think there's something to it. (See my question about Yoneda properties.) Similarly, you might want to study subgroups of the abelianization of a given group $G$, and that could be really interesting. But does it tell you more about maps from $G$ to abelian groups? | |
| Mar 14, 2010 at 4:17 | comment | added | JBorger | I'm still not sure what I think about all this. Without a doubt, it would be useful to have the coarse moduli space when studying any concept expressed in terms of maps from the moduli stack to schemes, and such concepts do come up often in practice (at a minimum because some of them are agreed by everyone to be interesting). For example, probably Weil cohomology theories can be expressed in these terms. | |
| Mar 14, 2010 at 0:01 | comment | added | JBorger | @Kevin: Hmmm. Good point. | |
| Mar 13, 2010 at 21:27 | comment | added | Harry Gindi | @Prof. Conrad: Thanks for the suggestion and correction. | |
| Mar 13, 2010 at 16:13 | history | edited | BCnrd | CC BY-SA 2.5 |
This theorem is due to K-M alone, or else give credit to Rydh too.
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| Mar 13, 2010 at 16:11 | answer | added | BCnrd | timeline score: 13 | |
| Mar 13, 2010 at 15:57 | comment | added | Emerton | 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. | |
| Mar 13, 2010 at 11:22 | comment | added | BCnrd | @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. | |
| Mar 13, 2010 at 10:19 | comment | added | Kevin Buzzard | "you wouldn't think it would come up too often"---but how many papers mention the j-invariant! ;-) | |
| Mar 13, 2010 at 9:36 | comment | added | JBorger | I don't know of any answers to the question, and here's a philosophical remark (similar to others I've made on MO) which I think explains why: The primary purpose in life of a moduli space/stack/whatever is to receive maps. The defining universal property is in these terms, and it's what is used again and again. But the universal property of the coarsification is about maps out of the stack, as mentioned above. So you wouldn't think it would come up too often. | |
| Mar 13, 2010 at 9:02 | comment | added | Kevin Buzzard | @fpqc: I only vaguely remember what Deligne said. I'd only just graduated and barely knew what I was talking about (when it comes to stacks I still only barely know what I'm talking about to be honest). I am happy to insert the word "affine" and perhaps you're saying that this must have been his point---he might well have said "affine open". My perception of his point at the time was that sometimes passing to an etale cover was losing too much information in some way. But I didn't have the sense to ask for a specific example. | |
| Mar 13, 2010 at 8:14 | comment | added | Harry Gindi | Isn't it part of the definition of a DM stack that it's covered by étale monomorphisms (Zariski-open immersions) of schemes? I'm not too familiar with the terminology, so did you mean that the neighborhood also has to be affine? | |
| Mar 13, 2010 at 7:52 | comment | added | Kevin Buzzard | 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. | |
| Mar 13, 2010 at 6:51 | answer | added | Brian Jurgelewicz | timeline score: 1 | |
| Mar 13, 2010 at 4:16 | comment | added | S. Carnahan♦ | 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. | |
| Mar 13, 2010 at 3:46 | history | edited | Kevin H. Lin | CC BY-SA 2.5 |
added 250 characters in body; edited title; added 15 characters in body
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| Mar 13, 2010 at 3:09 | history | edited | Kevin H. Lin | CC BY-SA 2.5 |
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| Mar 13, 2010 at 1:44 | history | asked | Kevin H. Lin | CC BY-SA 2.5 |