Clearly the etale fundamental group of $\mathbb{P}^1\_{\mathbb{C}} \setminus \{a\_1,...,a\_r\}$ doesn't depend on the $a\_i$'s, because it is the profinite completion of the topological fundamental group. Does the same hold for when I replace $\mathbb{C}$ by a finite field? How about an algebraically closed field of positive characteristic?


(note that I'm talking about the full $\pi\_1$ and not the prime-to-$p$ part)



Does the etale fundamental group of the projective line minus a finite number of points over a finite field depend on the points?
It is a result of Tamagawa that for two affine curves $C\_1, C\_2$ over finite fields $k\_1,k\_2$ any continuous isomorphism $\pi\_1(C\_1)\rightarrow \pi\_1(C\_2)$ arises from an isomorphism of schemes $C\_1\rightarrow C\_2$. Hence, if $\pi\_1( \mathbb{P}^1\setminus\{a\_1,\ldots, a\_r\})$ were independent of the choice of the $a\_i$, then the isomorphism class of the schemes $\mathbb{P}^1\setminus\{a\_1,\ldots, a\_r\}$ would be independent of the choice of $a\_1,\ldots,a\_r$.


Tamagawa's result is Theorem 0.6 in this paper:


The Grothendieck conjecture for affine curves, A Tamagawa - Compositio Mathematica, 1997
<http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=298922>


In the case of an algebraically closed field, the answer is also that the fundamental group depends on the choice of the points that are being removed. Again by a theorem by Tamagawa:
If $k$ is the algebraic closure of $\mathbb{F}\_p$, and $G$ a profinite group not isomorphic to $(\hat{\mathbb{Z}}^{(p')})^2\times \mathbb{Z}\_p$, then there are only finitely many $k$-isomorphism classes of smooth curves $C$ with fundamental group $G$ (the restriction on $G$ excludes ordinary elliptic curves).


This can be found in 


Finiteness of isomorphism classes of curves in positive characteristic with prescribed fundamental groups,
A Tamagawa - Journal of Algebraic Geometry, 2004


