It is a well known fact that there exist a so called threefold category equivalence between following tree categories:
1) the category of finitely generated field extensions $K/k$ of transcendence degree 1
2) category of normal integral curves (over $k$) and dominant rational maps
3) compact Riemann surfaces and holomorphic maps
By taking the generic fiber (resp the field of meromorphic functions) it's quite easy to obtain a functor from 2) and 3) to 1).
Could anybody sketch the construction of the functors from 1) to 2) and from 1) to 3)?
My considerations:
for 1) to 2):
here I have two problems namely
i) how to define the functor $K \to X_K$
ii) and if we have a finite morphism (=inclusion) of fields $\phi: K \subset L$ of transcendence degree $1$ and assume I know how to construct $X_K$ and $X_L$ in i) then there exist open sets $U \subset X_K, V \subset X_L$ to which $\phi$ the field map $\phi$ can be extended: $f_{\phi}: V \to U$. We obtain a rational morphism $X_L \dashrightarrow X_K$.
How can this dashed arrow be extended to a well defined map $X_L \to X_K$?
Regarding 1) to 2): no idea
remark: a curve is here defined as a 1-dimensional proper k-scheme