In [this note](http://math.uchicago.edu/~amathew/pi1.pdf) of Akhil MATHEW, when he proves the fundamental group of a smooth projective curve over a algebraic closed field $k$ of characteristic $0$ admits $2g$ topological generators, there are two sentences in the proof of theorem 4.5(page 16): 


*"Indeed, by “noetherian descent,” we note that $k$ is the colimit of the subfields $k'\subset k$ which are
the algebraic closures of finitely generated extensions of $\mathbb{Q}$."*


and 


*"by choosing $k'$ appropriately, noetherian descent allows us to assume that $X\_0'\rightarrow \text{Spec}\,k'$ is itself
a smooth projective curve".*


I want to know how to construct these $k'$ in the first sentence and how to choose the $k'$ such that $X\_0'\rightarrow \text{Spec}\,k'$ is a smooth projective curve?



Fundamental group of a smooth projective curve of char $0$
Regarding the second question, the following is true.



> 
> Let $k$ be a field and let $k'\subset k$ be a subfield. Then a variety over $k'$ is smooth and proper iff its base change to $k$ is.
> 
> 
> 


Proof. In one direction it is enough to note that an arbitrary base change of a smooth (resp. proper) morphism is smooth (resp. proper). In the second direction note that the property of being smooth (resp. proper) is fpqc local on the base and that any field extension is faithfully flat.


This is enough to address the question because a smooth proper curve over a field is projective.


