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10h

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Lefschetz on étale fundamental group for quasiprojective varieties
@Giulia: "I do expect the kernel to be a pro$p$ group: as said there is a Lefschetz theorem for the tame part, in particular for the primeto$p$ part, hence all that can go wrong lives in a subgroup whose primeto$p$ quotient is zero." This is not quite correct. The tame fundamental group is the quotient of the fundamental group by the normal subgroup generated by all $p$subgroups. However, it can happen that this normal subgroup is not itself a $p$group, e.g., if $n> p>2$, then the alternating group $\mathfrak{A}_n$ is generated by $p$cycles. 
11h

answered  Lefschetz on étale fundamental group for quasiprojective varieties 
12h

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Onedimension Algebraic groups
No, that does not answer the question. 
12h

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Onedimension Algebraic groups
For you, is an algebraic group always reduced? 
12h

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Lefschetz on étale fundamental group for quasiprojective varieties
@Giulia: I address the discrepancy in a comment to Will's answer. This very much depends on the meaning of "general". If you mean "the geometric generic point of the parameter space of hyperplane sections", then I am correct. On the other hand, if you mean "there exists a dense, Zariski open subset of the paramter space of hyperplane sections", then Will is correct. Personally, I think Will is correct and I am correct, just for different questions. 
12h

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Lefschetz on étale fundamental group for quasiprojective varieties
@Giulia. Will is correct, but also I am (sort of) correct. Let $X\subset \mathbb{P}^n$ be a quasiprojective variety over an algebraically closed field $k$. By Bertini's theorem, for every connected, finite, etale cover $Y\to X$, there exists a dense, Zariski open subset $U\subset (\mathbb{P}^n_k)^\vee$ such that for every $[H] \in U(k)$, also $Y\times_{\mathbb{P}^n_k} H \to X\times_{\mathbb{P}^n_k} H$ is a connected, finite, etale cover. Thus, for the geometric generic point $[H]$ of $(\mathbb{P}^n_k)^\vee$, the induced map $\pi^1(X\times_{\mathbb{P}^n_k} H) \to \pi^1(X)$ is surjective. 
1d

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Lefschetz on étale fundamental group for quasiprojective varieties
Surjectivity for the algebraic fundamental group (the only one defined in positive characteristic) follows from Bertini's theorem as proved, for instance, in Jouanolou's book. 
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splitting property of etale covering
@DanielLoughran. I had the same thought, but I suspect the OP meant to add a hypothesis that the cover is finite and etale. 
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splitting property of etale covering
For your first question, you need to add some kind of connectedness hypothesis for $Y$; otherwise there are trivial counterexamples. For your second question, what do you mean by "general $X$"? Is $X$ henselian? 
Feb
4 
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Nonuniversally trivial Chow group of zerocycles on Fano hypersurfaces
The paper by Totaro is beautiful, and the papers by Koll'ar and Voisin that it builds upon are also beautiful. You should just read those papers. For your first question: inside $X\times_{\text{Spec}\ (\mathbb{C})} X$ consider the cycles of the diagonal $\Delta$ and $X\times\{ x_0 \}$ for some $x_0\in X(\mathbb{C})$. Now restrict on the first factor of $X\times_{\text{Spec}\ \mathbb{C}} X$ to $\text{Spec}\ \mathbb{C}(X)$. If the pullback $0$cycles were rationally equivalent, then that would give an integral decomposition of the diagonal in $X\times_{\text{Spec}\ \mathbb{C}} X$. 
Feb
4 
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Nonuniversally trivial Chow group of zerocycles on Fano hypersurfaces
One choice that works is $F = \mathbb{C}(X)$, the function field of $X$. 
Feb
4 
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GIT quotient of variety with finite quotient singularities
You are correct, I was wrong. If you remove the origin, then the semistable locus of the new (quasiaffine) scheme equals the properly stable locus of the original scheme. The quotient stack of the entire quasiaffine scheme is a DeligneMumford stack whose coarse moduli space is a nonseparated scheme: the quadric cone with doubled vertex. So my example above is wrong. 
Feb
4 
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Fiber of the specialization map of Picard groups
There is a map from the group of $R$valued points to the group of $k$valued points. 
Feb
4 
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GIT quotient of variety with finite quotient singularities
I will look at your answer below. But in my example, I can always replace $\mathbb{A}^4$ by $\mathbb{A}^4\setminus \{(0,0,0,0)\}$. That does not change the GIT quotient. 
Feb
4 
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Does anyone know if there is a generalization of symplectic Kodaira dimension beyond 4manifolds?
I do not know about the Kodaira dimension, but I believe that some other birational aspects have been extended to dimension 6 by TianJun Li, Yongbin Ruan and Weiyi Zhang. 
Feb
3 
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GIT quotient of variety with finite quotient singularities
You certainly need to restrict to the properly stable locus $X^s$. Already the cone over a smooth quadric surface, considered as a quotient of $\mathbb{A}^4$ by a $\mathbb{G}_m$action, shows that the result cannot hold on the semistable locus. 
Feb
3 
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Conjugacy scheme, fppf versus GIT
Xuhua He and I did a GIT analysis of the stable and semistable locus for the conjugation action on the wonderful compactification of $G$. I realize that $\mathfrak{g}$ is different, but probably the same techniques apply. In particular, I believe that the regular locus is contained in the GIT stable locus, which should imply that the GIT quotient is a uniform geometric quotient on the regular locus. 
Feb
3 
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Fiber of the specialization map of Picard groups
What? There is a morphism from the relative Picard stack of $X_k/k$ to the relative Picard stack of $X_R/R$. Are you asking about the group homomorphism from the group of $R$valued points of the Picard stack (modulo isomorphism) to the group of $k$valued points? How are you thinking of those groups as schemes? 
Feb
2 
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Galois cohomology of a nonabelian group over a function field
@Sasha. Just to make that a little less cryptic: Steinberg's theorem proves that every $G$torsor over $F$ reduces to a maximal subtorus of the linear algebraic group over $F$, $G\times_{\text{Spec}(k)}\text{Spec}(F)$. Some such tori are of the form $T\times_{\text{Spec}(k)}\text{Spec}(F)$; these are split tori. However, there are also nonsplit maximal tori in $G\times_{\text{Spec}(k)}\text{Spec}(F)$. 
Feb
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Galois cohomology of a nonabelian group over a function field
@Sasha. "Do I understand correctly, that if $G$ is a connected split(!) reductive group over $F$, then its first Galois cohomology will be zero, where $F$ is completely arbitrary?" That is not correct. Already the Hamilton Quaternions give a nontrivial element of $H^1_{\text{Gal}}(\mathbb{R},\textbf{PGL}_2)$. Steinberg proves that torsors reduce to tori, but he does not prove that they reduce to split tori (so you do not get to use Hilbert's Theorem 90). 