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Dec 19 |
awarded | Yearling |
Nov 15 |
comment |
Etale fundamental group of a curve in characteristic $p$
@jacob: for an affine curve, we just know if a given finite group $G$ arises as a quotient of $\pi_1$ or not and nothing more. In particular the cardinality of the set of $G$-covers is not known for general $G$. And this data would be probably insufficient to recover the $\pi_1$ anyway. To sum up: no the $\pi_1$ is not known for any curve (complete or not) over any field of positive characteristic except for the trivial cases of $\mathbb P^1$ and elliptic curves. |
Nov 13 |
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Etale fundamental group of a curve in characteristic $p$
@jacob I am not sure I understand your third sentence. About the last sentence : the $\pi_1$ of a proper curve is invariant by algebraically closed field extension. So fixing a specific base field does not change the question for proper curves. And the answer is no, the $\pi_1$ is not known. By specialization theory we just know it is topologically of finite type. For affine curves the situation is even worse since the $\pi_1$ is not topologically of finite type. |
Nov 13 |
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Etale fundamental group of a curve in characteristic $p$
@Joël I agree "completely wrong" was a bit strong :) but I disagree with your interpretation of the question. Clearly the question is : "is the $\pi_1$ known up to isomorphism ?". An answer to your weak version wouldn't answer the following question : "given a fixed finite group $G$, what is the cardinality of the set of $G$-covers" ? |
Nov 13 |
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Etale fundamental group of a curve in characteristic $p$
@jacob I think I remember you can find an example in Fried-Jarden Field Arithmetic. |
Nov 13 |
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Etale fundamental group of a curve in characteristic $p$
@Joël : what you wrote is completely wrong. Abyankhar's conjecture describes the set of finite quotients of the $\pi_1$, but since in the affine case the $\pi_1$ is not topologically of finite type, this is not enough information to get back the $\pi_1$. For instance, the $\pi_1$ of $\mathbb A^1$ is not known in positive characteristic, even if one knows that its finite quotients are the quasi-$p$ groups. In fact to my knowledge there is no example of an hyperbolic curve whose $\pi_1$ is "known" in positive characteristic. |
Nov 2 |
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Finite etale atlas for Deligne-Mumford stacks
About your first comment: it seems better to me to give the simplest possible example than a class including pointless complications such as non-generic stabilizers. About your second comment : I was careful about what you mention. Your quotient stack is interesting but is definitely not a stack of roots. |
Oct 31 |
answered | Finite etale atlas for Deligne-Mumford stacks |
Jul 5 |
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Topological fundamental group of a variety
you could have a look at arxiv.org/abs/1311.6086 On the fundamental group of a variety with quotient singularities Indranil Biswas, Amit Hogadi |
Jun 22 |
revised |
Injectivity of Section conjecture
added 104 characters in body |
Jun 22 |
revised |
Injectivity of Section conjecture
added 104 characters in body |
Jun 22 |
answered | Injectivity of Section conjecture |
Jan 31 |
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Basics on anabelian geometry and Grothendieck's section conjecture
seems very redundant with mathoverflow.net/questions/108860/… to me. |
Jan 24 |
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linearly reductive group acting on $X$
Then, this holds. You just derive the formula for $i=0$ using (for instance) Grothendieck's spectral sequence for the derived functors of a composite functor, and the fact that $(\cdot)^G$ is exact. There is an elementary direct proof of course. |
Jan 24 |
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another question on the Manin-Drinfeld theorem
You could have a look at the book Modular Units Daniel S. Kubert, Serge Lang specifically the chapter The Cuspidal Divisor Class Group on X(N) Good luck ! |
Jan 24 |
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linearly reductive group acting on $X$
What are $\mathcal{F}$,$\mathcal{G}$ ? |
Jan 10 |
answered | Spin group as an automorphism group |
Dec 21 |
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What is your picture of the flat topology?
"don't forget to buy my products" means "vote for my answer" ? |
Dec 19 |
awarded | Yearling |
Dec 2 |
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homotopy exact sequence for the étale fundamental group
In my opinion this part of SGA1 does need no translation, it is perfectly clear. |