Theorem 53.2.6. Let $k$ be a field. The following categories are canonically equivalent

1. The category of finitely generated field extensions $K/k$ of transcendence degree $1$.

2. The category of curves and dominant rational maps.

3. The category of normal projective curves and nonconstant morphisms.

4. The category of nonsingular projective curves and nonconstant morphisms.

5. The category of regular projective curves and nonconstant morphisms.

6. The category of normal proper curves and nonconstant morphisms.

Proof. The equivalence between categories (1) and (2) is the restriction of the equivalence of Varieties, Theorem 33.4.1. Namely, a variety is a curve if and only if its function field has transcendence degree $1$, see for example Varieties, Lemma 33.20.3.

The categories in (3), (4), (5), and (6) are the same. First of all, the terms “regular” and “nonsingular” are synonyms, see Properties, Definition 28.9.1. Being normal and regular are the same thing for Noetherian $1$-dimensional schemes (Properties, Lemmas 28.9.4 and 28.12.6). See Varieties, Lemma 33.43.8 for the case of curves. Thus (3) is the same as (5). Finally, (6) is the same as (3) by Varieties, Lemma 33.43.4.

If $f : X \to Y$ is a nonconstant morphism of nonsingular projective curves, then $f$ sends the generic point $\eta$ of $X$ to the generic point $\xi$ of $Y$. Hence we obtain a morphism $k(Y) = \mathcal{O}_{Y, \xi } \to \mathcal{O}_{X, \eta } = k(X)$ in the category (1). If two morphisms $f,g: X \to Y$ gives the same morphism $k(Y) \to k(X)$, then by the equivalence between (1) and (2), $f$ and $g$ are equivalent as rational maps, so $f=g$ by Lemma 53.2.2. Conversely, suppose that we have a map $k(Y) \to k(X)$ in the category (1). Then we obtain a morphism $U \to Y$ for some nonempty open $U \subset X$. By Lemma 53.2.1 this extends to all of $X$ and we obtain a morphism in the category (5). Thus we see that there is a fully faithful functor (5)$\to$(1).

To finish the proof we have to show that every $K/k$ in (1) is the function field of a normal projective curve. We already know that $K = k(X)$ for some curve $X$. After replacing $X$ by its normalization (which is a variety birational to $X$) we may assume $X$ is normal (Varieties, Lemma 33.27.1). Then we choose $X \to \overline{X}$ with $\overline{X} \setminus X = \{ x_1, \ldots , x_ n\}$ as in Varieties, Lemma 33.43.6. Since $X$ is normal and since each of the local rings $\mathcal{O}_{\overline{X}, x_ i}$ is normal we conclude that $\overline{X}$ is a normal projective curve as desired. (Remark: We can also first compactify using Varieties, Lemma 33.43.5 and then normalize using Varieties, Lemma 33.27.1. Doing it this way we avoid using the somewhat tricky Morphisms, Lemma 29.53.16.) $\square$

Comment #1856 by Keenan Kidwell on

In the last sentence of the third text block of the proof, "faithfull" should be "faithful."

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