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If we have a variety, $X$, over a field, $k$, and $x$ is a geometric point of $X$, and let $\bar x$ be a geometric point of $X_{k^s} := X \times_k k^s$ above $x$ then we have the following short exact sequence:

$1 \rightarrow \pi_1(X_{k^s}, \bar x) \rightarrow \pi_1(X,x) \rightarrow Gal(k) \rightarrow 1$

Implicit in this is a choice of $k^s$ (if you want, this is a choice of geometric point, $z$, on $Spec(k)$; $\pi_1(Spec(k), z)=Gal(k)$).

I'm wondering how to interpret the splitting of this short exact sequence, and more specifically: what is the significance of choosing different splittings? I'm having a hard time picturing intuitively how to think of this splitting.

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You might find Kirsten Wickelgren's thesis of interest: www.math.harvard.edu/~kwickelg/papers/LCS.pdf She studies obstructions to splittings that arise from group cohomology. More specifically, the obstructions come from an analysis of the lower cental series of the etale fundamental group of X. –  Dan Ramras May 4 '10 at 3:07
    
Minhyong Kim's expository articles on this are also very good. –  JSE Jun 29 '10 at 15:44
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2 Answers

A splitting can be obtained by a $k$-rational point of $X$. In some (interesting) cases a section necessarily comes from a point and Grothendieck conjectured it in very general situations (this is part of what is called anabelian geometry).

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Are splittings given by different k-points necessarily different? Given the point, how would you define the splitting? –  Makhalan Duff May 3 '10 at 22:00
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A k-rational point is just a section of the structure map X->k, the splitting would then be defined by the functoriality of the construction of the fundamental group (provided that compatible base points have been chosen). It is known that the map {rational points} -> {sections of pi_1(X,x)->Gal(k)} is injective. Grothendieck's famous Section Conjecture roughly states that it is bijective. –  Lars May 3 '10 at 22:42
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I should add: Because of the base point issue, one rather gets a map {rational points} -> {conjugacy classes of sections}, which Grothendieck conjectures to be bijective in case $X$ is a projective curve. For affine curves there also is a conjecture, but one has to take into account the points at infinity in a suitable manner. –  Lars May 3 '10 at 23:02
    
Just a note: the injectivity Lars mentions is for the case where X is a curve of genus bigger than 1. In general, the map certainly need not be injective; i.e. take X to be some variety with X(C) simply connected, so that the sequence above is just 1->1->G_k->G_k->1; there may be lots of points in X(k), but they all yield the same splitting. –  JSE Jun 29 '10 at 15:43
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For me, philosophically, the splitting of the short exact sequence means that etale coverings of $X$ basically come in two flavors: Geometric coverings (classified by $\pi_1(X_{\bar{k}})$ which is sometimes also called "geometric fundamental group of $X$") and arithmetic coverings (classified by $Gal(k)$). All coverings can be obtained by "combining" geometric and arithmetic coverings.

Another similar interpretation is the following: By passing to the limit over all galois coverings of $X$ (more precisely, over the system of pointed galois coverings of $(X,x)$) one obtains a universal covering scheme $\hat{X}$. As a set, the fiber over the base point is the profinite set $\pi_1(X,x)$! Similarly one can construct the universal covering of $X_{\bar{k}}$ and of $Spec(k)$ (which is just $Spec(k^{sep})$. The fibers over the fixed base point of these covering schemes are $\pi_1(X_{\bar{k}})$ and $Gal(k)$ respectively (as sets). The splitting of the short exact sequence now gives information about the fiber of the universal covering of $X$ in terms of points coming from the fibers of $X_{\bar{k}}$ and $Spec(k)$.

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