MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

If $X$ is a proper curve of genues $g$ over an algebraically closed field $K$ of characteristic $0$, and $U$ an open subset, say obtained by removing $n$ closed points from $X$, then by comparison with the complex topology (more precisely by the Riemann Existence Theorem) one can derive that $\pi_1^{et}(U)$ is isomorphic to the profinite completion of the group $$\lt a_1,\ldots,a_g, b_1,\ldots,b_g,c_1,\ldots,c_n|[a_1,b_1]\cdot\ldots\cdot[a_g,b_g]c_1\ldots c_n\gt$$ As far as I know, there is no algebraic proof for this fact.