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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
comment 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
comment 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
comment 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
comment 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
comment 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
comment 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
comment Basics on anabelian geometry and Grothendieck's section conjecture
seems very redundant with mathoverflow.net/questions/108860/… to me.
Jan
24
comment 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
comment 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
comment linearly reductive group acting on $X$
What are $\mathcal{F}$,$\mathcal{G}$ ?
Jan
10
answered Spin group as an automorphism group
Dec
21
comment 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
comment homotopy exact sequence for the étale fundamental group
In my opinion this part of SGA1 does need no translation, it is perfectly clear.