Abstract Nonsingular Curves

Sorry for resurrecting such an old post, but Section §1.6 of Hartshorne annoys me in the sense that the concept of "abstract nonsingular curve" is not used very much (if at all) in algebraic geometry. We provide an alternative version of that section (specifically, Lemma 1.6.5 through Theorem 1.6.9) that avoids use of said object.

We start with a replacement of Proposition 1.6.8.

Proposition. Let $\phi: X \to Y$ be a rational map from a normal curve $X$ to a projective variety $Y$. Then $\phi$ extends to a (unique) morphism $X \to Y$.

Proof. We may assume that $\phi$ is dominant (replace $Y$ with $\overline{\phi(X)}$), that $Y \subseteq \mathbb{P}^n$, and that $Y$ is not contained in any of the coordinate hyperplanes. Let $U$ be an open subset of $X$ on which $\phi$ is represented by a morphism, and let $P \in X \setminus U$. Since $X$ is normal, $\mathcal{O}_{P,X}$ is a DVR; let $v$ be the associated discrete valuation on $K(X)$. We have that $1 = \phi^*(x_0/x_0), \phi^*(x_1/x_0), \dots, \phi^*(x_n/x_0)$ are nonzero elements of $K(X)$; pick $i$ such that $v(\phi^*(x_i/x_0))$ is minimal. Then, for all $j$,$$v\left(\phi^*\left({{x_j}\over{x_i}}\right)\right) = v\left(\phi^*\left({{x_j}\over{x_0}}\right)\right) - v\left(\phi^*\left({{x_i}\over{x_0}}\right)\right) \ge 0,$$so $\phi^*(x_j/x_i)$ lies in $\mathcal{O}_{P, X}$ for all $j$. Let $V$ be an open neighborhood of $P$ in $X$ such that $\phi^*(x_j/x_i)$ is regular on $V$ for all $j$. Letting $U_i \subseteq \mathbb{P}^n$ denote the open affine $\{x_i \neq 0\}$, we have that the image of the composition$$A(Y \cap U_i) \hookrightarrow K(Y) \overset{\phi^*}{\to} K(X)$$is contained in $\mathcal{O}(V)$, so $\phi^*$ determines a morphism $\phi': V \to Y$ that induces the same map $K(Y) \to K(X)$ as $\phi$. Therefore, $\phi'$ and $\phi$ are the same rational map, and therefore $V$ is contained in the maximal domain of $\phi$. In this way, $\phi$ can be extended to a morphism $X \to Y$. $$\tag*{$\square$}$$

Lemma. Let $W$ be a variety. Then its normalization map $\pi: \widetilde{W} \to W$ is surjective.

Proof. Let $P \in W$. By the lying-over theorem (Atiyah-MacDonald Theorem 5.10 or Eisenbud Proposition 4.15), since $A(\widetilde{W})$ is integral over $A(W)$, it contains a prime ideal whose restriction to $A(W)$ is the maximal (hence prime) ideal $I(P)$. Replacing this prime with a maximal ideal containing it, if necessary we have a maximal ideal $\mathfrak{a}$ for which $\mathfrak{a} \cap A(W) = I(P)$. This maximal ideal corresponds to a point lying over $P$. $$\tag*{$\square$}$$

Theorem. Every curve is birational to a nonsingular projective curve. Moreover, if $C$ is a curve and $\phi: C \to \widetilde{C}$ is a birational map to a nonsingular projective curve $\widetilde{C}$, then $\phi$ is represented by a morphism on the (open) set of nonsingular points of $C$.

Proof. Let $C$ be a curve. We may assume that it is quasi-projective. It is birational to its projective closure, so we may assume that $C$ is projective. Let $\{U_1, \dots, U_m\}$ be a covering of $C$ by open affines. For each $i$, let $\pi_i: V_i \to U_i$ be the normalization (Exercise 1.3.17). This map is birational (since $A(U_i) \subseteq A(V_i) \subseteq K(U_i)$), so there are nonempty open subsets $U_i' \subseteq U_i$ and $V_i' \subseteq V_i$ such that $\pi_i$ induces an isomorphism $V_i' \overset{\cong}{\to} U_i'$.

Each $V_i$ is affine, say $V$ is closed in $\mathbb{A}^{n_i}$ for all $i$. Let $Y_i$ be its projective closure in $P^{n_i}$. We have a rational map$$\theta: C\,\, -\!\! \rightarrow Y_1 \times \dots \times Y_m \subseteq \mathbb{P}^{n_1} \times \dots \times \mathbb{P}^{n_m},$$represented by a morphism on the set $U := \bigcap U_i'$. Let $Y$ be the closure of the graph. Then $Y$ is a projective curve, and we have morphisms $\phi: Y \to C$ and $\psi_i : Y \to Y_i$ for all $i$. As noted earlier, $\phi$ is birational, and is an isomorphism over $U$. In particular, $Y$ is birational to $C$.

In addition, $Y \cap (U \times \prod \pi_i^{-1} (U))$ is the (closed) graph of the morphism$$(\pi_1^{-1}, \dots, \pi_m^{-1}) : U \to \prod Y_i,$$so this nonempty open subset of $Y$ is isomorphic to $U$ and to $\pi_i^{-1}(U)$ for all $i$; hence all $\psi_i$ are also birational. Moreover, we have commutative diagrams

in which the maps on the left induce the maps on the right.

It remains only to show that $Y$ is nonsingular.

Pick $P \in Y$, and choose $i$ such that $\phi(P) \in U_i$. Let $W$ be an open affine neighborhood of $P$ in $\phi^{-1}(U_i)$, and let $\alpha: \widetilde{W} \to W$ be its normalization. By the lemma, there is a point $\tilde{P} \in \widetilde{W}$ such that $\alpha(\widetilde{P}) = P$. We have a commutative diagram

in which $\beta$ exists because $\widetilde{W}$ is normal, $\phi \circ \alpha$ is dominant, and $\pi_i$ is the normalization. However, $\beta$ coincides with $\psi_i \circ \alpha$ on some nonempty open set, since they are both associated to the same map of function fields. Therefore, $\beta = \psi_i \circ \alpha$, so in particular, $\psi_i(P) = \psi_i(\alpha(\tilde{P})) = \beta(\tilde{P})$ lies in $V_i$

Therefore, the local ring $\mathcal{O}_{P, Y}$ dominates $\mathcal{O}_{\psi_i(P), V_i}$. But the latter is normal, hence a DVR (by Theorem 6.2A), hence a valuation ring. By maximality of valuation rings with respect to dominance (Theorem 6.1A), $\mathcal{O}_{P, Y}$ equals $\mathcal{O}_{\psi_i(P), V_i}$; in particular it is regular (by 6.2A again).

This implies the first assertion of the theorem. The second assertion follows immediately from the proposition.$$\tag*{$\square$}$$

If $k$ is an algebraically closed field, then the procedure described in Hartshorne's text establishes an (anti)equivalence of categories between:

1. The category whose objects are finitely generated field extensions of $k$ of transcendence degree 1, and whose morphisms are $k$-algebra homomorphisms.
2. The category whose objects are nonsingular projective $k$-curves, and whose morphisms are morphisms of $k$-schemes.

In particular, if you have a nonsingular projective $k$-curve, the corresponding function field is unique up to isomorphism, and two such curves are isomorphic if and only if they are birational.

If you start with a curve that is not projective, then its normalization admits an open immersion to a projective curve, and that curve is unique up to isomorphism.