Timeline for What is the local structure of a general Artin stack?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Nov 25, 2019 at 7:03 | answer | added | Angelo | timeline score: 4 | |
| Nov 25, 2019 at 2:19 | comment | added | Ben Wieland | A stack which is the quotient by a group $G$ has inertia all subgroups of $G$. A finite dimensional algebraic group cannot contain infinitely many abelian varieties (up to isogeny?). Thus the moduli of (unmarked) genus 1 curves, which has inertia all elliptic curves is not locally a quotient stack. But I think that a finite dimesional family of unipotent groups can be embedded in a single linear group. | |
| Nov 24, 2019 at 19:57 | comment | added | John Pardon | @DanPetersen The assumption in that result that the stabilizer groups are linearly reductive is too restrictive for the setting I'm interested in. For example, it does not hold for the point $\mathbb CP^1\vee\mathbb CP^1$ in the moduli stack of nodal curves. | |
| Nov 24, 2019 at 19:52 | comment | added | Dan Petersen | Sure, I get that you don't want to assume this a priori. I mean rather: why isn't your question answered by the Alper-Hall-Rydh structure theorem discussed in the question you linked to? | |
| Nov 24, 2019 at 15:41 | comment | added | John Pardon | @DanPetersen: By Artin stack I just mean it has a smooth atlas (plus some condition in the diagonal). It's definitely not assumed a priori to be etale locally a quotient stack. | |
| Nov 24, 2019 at 5:40 | comment | added | Dan Petersen | Maybe I don't get the question, but if your stack is etale locally a quotient stack then surely it is also locally a quotient stack for the analytic topology? Since an etale cover can be refined to an analytic open cover. | |
| Nov 23, 2019 at 20:36 | comment | added | Ben Wieland | A DM stack has a coarse space, but it's still an algebraic space, and algebraic spaces can be pathological. | |
| Nov 23, 2019 at 15:50 | comment | added | John Pardon | @Angelo: Thank you, I've rephrased the question to avoid that notion. | |
| Nov 23, 2019 at 15:49 | history | edited | John Pardon | CC BY-SA 4.0 |
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| Nov 23, 2019 at 14:25 | comment | added | Angelo | What do you mean by the "coarse space"? | |
| Nov 23, 2019 at 3:42 | history | asked | John Pardon | CC BY-SA 4.0 |