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Sometime ago I overheard a conversation in which someone said "studying functions fields is the same thing as studying algebraic curves".

After looking it up, I've found these two results (I'll always assume $k$ algebrically closed):

i) Any two projective curves $X,Y$ are birrationaly equivalent $\Leftrightarrow k(X)\simeq k(Y)$

ii) Two non-singular projective curves $X,Y$ are isomorphic $\Leftrightarrow X,Y$ are birrationally equivalent.

From these two, the phrase above doesn't seem accurate, because given a function field, it's not obvious that it correspond to $k(X)$ for some $X$. Besides, there is the issue of non-singularity of $X$, which I don't know how to deal with.

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    $\begingroup$ If $L/k(t)$ is a finite extension then let $R$ be the integral closure of $k[t]$ in $L$, it is a finitely generated $k$-algebra $R = k[f_1,\ldots,f_n] \cong k[T_1,\ldots,T_n]/I$ where $V(I)$ is a smooth affine curve. Its projective closure is possibly not smooth at the points above $t=\infty$ and there are some algorithms of desingularization to obtain a smooth projective curve with function field $L$. $\endgroup$
    – reuns
    Commented Jul 3, 2019 at 23:37
  • $\begingroup$ @reuns Is the blow-up one of the algorithms you're talking about? $\endgroup$
    – rmdmc89
    Commented Jul 4, 2019 at 0:16

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They were probably referring to a certain equivalence of categories. Let me (try to) formulate it for $k = \mathbb{C}$ if I manage to recall correctly:

The following categories are equivalent:

i) The category of algebraic curves with dominant rational maps

ii) The category of smooth projective curves with dominant morphisms

iii) The category of riemann surfaces with non-constant holomorphic maps

vi) The category of finitely generated (as $\mathbb{C}$-algebra) field extensions $\mathbb{C} \subset L$ of $\text{trdeg}_{\mathbb{C}}(L) = 1$ with homomorphisms of $\mathbb{C}$-algebras

We can also state vi) as

vi') The category of function fields in one variable with homomorphisms of $\mathbb{C}$-algebras

I think you can find that statement (or a similar one) in Introduction to Compact Riemann Surfaces and Dessins d'Enfants by Girondo and González-Diez.

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