Riemann showed (not proved rigorously) that there is a correspondence between compact riemann surfaces and algebraic function fields in one variable (does anyone know the year?).
To construct the algebraic function field of the compact riemann surface take the field of meromorphic functions M(M). Main question: Seen as an antiequivalence between categories, what is the inverse functor? Or explicitly, how to construct a riemann surface X from its function field M(X), preferably as a polynomial in one variable with coefficients depending on another variable. If there are singularities on the recovered surface, is there a systematic way to remove them? A simpler but useful question is: how many generators do M(X) have, and how does the genus g(X) depend on M(X)? (notice that i am not restricting X to sit in any particular space).
Here is the part i am not so good at but it shows a possible solution: I think the function field K(X) can always be written as K[x](y)/< P(x,y) > where P is an irreducible polynomial. This would be fine if any function field of X has uniquely this form. Then the equation i am looking for should be the extracted P(x,y)=0. Did this make the problem any simpler? And still, what is the functor inverse to M()?
If this is really impossible maby a parametric representation of X from M(X) is possible.
Reading another post on mathoverflow i think i am asking for higher genus Weierstrass Pe-functions related to the generators, that should be constructed in terms of Riemann theta-functions. As i understand it, for X in the g(X)=1 case K=C(Pe,Pe') (for a given lattice in C), and X is parametrized by Weierstrass Pe and Pe'-functions. But that didn't help me since there were no formulas or references for higher genera. Here is one reference for something that looks like higher genus Pe-functions.
Aside: since Riemann showed there is this correspondence, birational equivalence should be the same as biholomorphic equivalence for compact riemann surfaces. I have never even seen a theorem about it. How is the situation for general riemann surfaces?
Note: i will from now on write trdeg(F) for the transcenden degree of a field extension F over K.
A generalization to higher dimensions (background: infered from wikipedia). Is this true (even for singular algebraic varieties or only smooth algebraic varieties)?
Every algebraic variety X over a field K has a function field K(X) that is the field of rational functions on X, is a field extension of K that is finitely generated, and has trdeg(K(X))=dim(X) (both over K). And, all such field extensions of K with finite trdeg(K(X)) are the function field of some algebraic variety X over K.
And, question: is there a categorical antiequivalence between
the the category of algebraic function fields of trdeg(K(X)) variables that are extensions of K, and ring homomorphism as morphisms, and the category of algebraic varieties over K of dimension trdeg(K(X)) and rational functions between them as morphisms?
Finally, does this correspondence hold locally for schemes over a field?