Timeline for What is the official definition of $\mathcal{M}_g$ as an orbifold, and how much can I ignore it?
Current License: CC BY-SA 4.0
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| Apr 20, 2019 at 14:32 | answer | added | NWMT | timeline score: 3 | |
| Apr 12, 2019 at 15:05 | answer | added | Donu Arapura | timeline score: 13 | |
| Apr 12, 2019 at 14:13 | history | edited | Sean Lawton |
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| Apr 12, 2019 at 13:25 | history | edited | Sean Lawton | CC BY-SA 4.0 |
Highlighted the actual question the OP seemed to want.
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| Apr 12, 2019 at 13:21 | answer | added | Sean Lawton | timeline score: 10 | |
| Apr 12, 2019 at 10:21 | comment | added | user1073 | @Kim - I'm not sure if this is precisely what you're looking for, but there's a nice definition of orbifolds, Riemannian structures on orbifolds, etc based on charts in Section 2 of this paper: arxiv.org/abs/0805.3148 | |
| Apr 12, 2019 at 10:09 | history | edited | Kim | CC BY-SA 4.0 |
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| Apr 12, 2019 at 6:37 | comment | added | Kim | Is there a good definition relying on charts and not functor of points? My understanding is that stacky definitions go the second route, and I was hoping to see something more like the first. | |
| Apr 11, 2019 at 22:06 | comment | added | Qfwfq | Also, you can equivalently work with groupoids (internal to your favorite geometric category). It's a less intrinsic definition then the one with stacks (because it presupposes the choice of an atlas) but it's equivalent (at least if you consider the "right" definition of morphisms for this purpose, i.e. Morita morphisms) and more concrete. Your $\mathcal{M}_g$ would then be (represented by) the complex analytic action groupoid $\Gamma\times\mathcal{T}_g\rightrightarrows \mathcal{T}_g$. | |
| Apr 11, 2019 at 22:04 | comment | added | Alexandre Eremenko | The general definition of an orbifold is in the preprint of Thurston (who introduced the word) Three-dimensional geometry and topology. Part of this preprint is published as a book with the same name. It is a space locally represented as factors of a disk over an action of a finite group. | |
| Apr 11, 2019 at 22:03 | comment | added | Qfwfq | The most useful "official" definition of orbifold is probably a (smooth, or complex analytic, or...) stack with some nice properties (e.g. Deligne-Mumford). There's an older definition of orbifold as "V-manifold", due to Satake if I'm not mistaken, but people -especially those working in the algebro-geometric or complex analytic setting- seem to prefer to use the newer one. | |
| Apr 11, 2019 at 17:19 | history | asked | Kim | CC BY-SA 4.0 |