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line bundle descends?
The statement "By a result of Kempf a line bundle $L$ on $\mathbb{P}^3$ descends to the quotient if and only if the stabilizer $S_x$ of each point $x\in \mathbb{P}^3$ acts trivially on the fiber $L_x$." is not very precise. You probably mean : an equivariant line bundle, else the action on the fiber is not even defined. Could you give a reference ? 
Jan 19 
comment 
Stack of curves and universal deformations
Deformation theory from the point of view of fibered categories Mattia Talpo, Angelo Vistoli arxiv.org/abs/1006.0497 
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 FriedJarden 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 
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Finite etale atlas for DeligneMumford stacks
About your first comment: it seems better to me to give the simplest possible example than a class including pointless complications such as nongeneric 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 DeligneMumford 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
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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 
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 ManinDrinfeld 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 
comment 
What is your picture of the flat topology?
"don't forget to buy my products" means "vote for my answer" ? 