A split short exact sequence of algebraic fundamental groups 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.
 A: 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).
A: 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 $Spec(k)$.
