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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: 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

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
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
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

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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