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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); $\pi_1(Spec(k), z)=Gal(k)$).

suppose f is a spliting of this short exact sequence , then consider the image of f,denote by Im(f) ,it is a subgroup of $\pi_1(X,x)$.Then is there any understanding of the the subgroup of $\pi_1(X,x)$generated by {Im(f)|all spliting f} ? And when it will it be the full group $\pi_1(X,x)$?

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# Image of spliting of 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); $\pi_1(Spec(k), z)=Gal(k)$).

suppose f is a spliting of this short exact sequence , then consider the image of f,denote by Im(f) ,it is a subgroup of $\pi_1(X,x)$.Then is there any understanding of the the subgroup of $\pi_1(X,x)$generated by {Im(f)|all spliting f} ? And when it will be the full group $\pi_1(X,x)$?