Threefold category equivalence: algebraic curves, Riemann surfaces and fields of transcendence degree 1

Question

According to this article,

The category of curves over the complex numbers is equivalent to two other categories: Riemann surfaces and fields of transcendence degree 1 over $\mathbb{C}$.

But I can not find details of this equivalence:

1. Is it correct only for projective algebraic curves and compact Riemann surfaces?
2. Is the category of curves considered up to birational equivalence?
3. Is this field equal both to field of meromorpic functions of surface and coordinate field of curve?

And the most general question, is there an online material on this topic which intended for beginners? It was mentioned as something widely-known in all articles I found.

Answer 1

1. Yes, this is for smooth projective algebraic curves and compact Riemann surfaces. Smoothness is required so that the curve is a complex manifold, and since it is projective then it is a compact complex manifold of dimension 1. Vice versa, a Riemann surface can be embedded into projective space with a very ample divisor. A priori, this embedding is as a complex submanifold. However, a theorem of Chow assures us that a closed submanifold of projective space is actually algebraic.

2. You may forget smooth projective curves and just consider curves, since in every birational class of curves there is only one non-singular curve. This is because if $f:C\to C'$ is a rational morphism between non-singular curves, then it is a morphism. This comes from a general result saying that if $f:X\to\mathbb{P}^n$ is rational and $X$ is a smooth projective variety, then the place where $f$ is not defined is of codimension at least $2$ (see Shafarevich for example).

3. Yes, the field of meromorphic functions is equal to the coordinate field. Miranda proves in his book, for example, that if there is a subfield of meromorphic functions that separates points and tangents, then it is actually equal to the whole meromorphic function field.

Miranda is a good book to start with that talks about this kind of equivalence. Also look for a book that proves Chow's Theorem on submanifolds of projective space.