Timeline for Finite etale atlas for Deligne-Mumford stacks
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 3, 2014 at 16:21 | vote | accept | Ariyan Javanpeykar | ||
| Oct 31, 2014 at 13:56 | answer | added | Niels | timeline score: 4 | |
| Oct 30, 2014 at 20:17 | answer | added | Daniel Litt | timeline score: 9 | |
| Oct 30, 2014 at 18:54 | comment | added | Ariyan Javanpeykar | @JasonStarr I changed the question hoping that the new question can be answered. Hope you don't mind your comments look "irrelevant" now (but aren't of course!). | |
| Oct 30, 2014 at 18:52 | history | edited | Ariyan Javanpeykar | CC BY-SA 3.0 |
Completely changed the question
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| Oct 23, 2014 at 13:57 | comment | added | Ariyan Javanpeykar | Ok, sorry about that. I was just confused and worried for a second. :) | |
| Oct 23, 2014 at 13:56 | comment | added | Jason Starr | My second comment shows that the example in my first comment is not a counterexample. I do not give any counterexample (I proposed an example, but it turns out not to be a counterexample). | |
| Oct 23, 2014 at 13:52 | comment | added | Ariyan Javanpeykar | @JasonStarr Are you giving a counterexample in your second comment? I don't quite follow. The assumption in the question is that any atlas (finite etale or just etale surjective) of $X$ is hyperbolic. In the case of orbifold curves this certainly implies that X is hyperbolic, as an orbifold curve with a hyperbolic finite etale atlas has universal covering $\mathbb H$. | |
| Oct 23, 2014 at 13:31 | comment | added | Jason Starr | My example doesn't work: consider the homomorphism $\alpha:\pi_{1}(\mathbb{C}P^1\setminus\{0,1,\infty\}) \to \mathfrak{S}_8$ that sends the loop around $\infty$ to $(1234)(5678)$, sends the loop around $1$ to $(15)(28)(37)(46)$, and sends the loop around $0$ to $(18)(27)(36)(45)$, so that the product of the loops goes to the identity. The image is a transitive subgroup. By Riemann-Hurwitz, the finite etale cover of $X$ is $\mathbb{P}^1$. | |
| Oct 23, 2014 at 13:20 | comment | added | Jason Starr | Take $X$ a stacky $\mathbb{P}^1$ with $1/2$-structure at $0$ and $1$, and $1/4$-structure at $\infty$. This is a specialization of non-hyperbolic stacks that have $1/2$-structure at each of $4$ points; these have finite etale covers by elliptic curves. But the specializations of those etale covers are not smooth curves. So I suspect that $X$ is hyperbolic. However, there is a finite (non-etale) cover that is an elliptic curve: namely, first double-cover by $\mathbb{P}^1$ branched over $0$, $\infty$, then cover by an elliptic curve branched over the preimages of $0$, $1$, $\infty$. | |
| Oct 23, 2014 at 12:56 | history | asked | Ariyan Javanpeykar | CC BY-SA 3.0 |