Rational maps from a smooth curve to a proper scheme are morphisms

Let $K=k(X)=\mathrm{Frac}{\mathcal{O}}_{X,x}$ for every closed point $x$, so that $K$ is simultaneously the function field of $X$ and the fraction field of each of the local rings considered below [F11, F1]. The complement $Z=X\setminus U$ is a proper closed subset of $X$: it is closed, and proper because $U$ is nonempty open and contains the generic point $\eta$ [F1]. By [F3] the set $Z$ is finite and every $x\in Z$ is a closed point of $X$.

Since $Y$ is proper over $k$, the structure morphism $Y\to \mathrm{Spec}k$ is separated, of finite type and universally closed, hence in particular quasi-separated; uniqueness of lifts in the valuative criterion will use the separatedness of $Y$, and existence will use properness in the form of [F8].

Fix $x\in Z$. Choose an affine open $\mathrm{Spec}A\subseteq X$ containing $x$, so that $A$ is a finite-type $k$-domain and ${\mathcal{O}}_{X,x}=A_{{\mathfrak{m}}_x}$ for the maximal ideal ${\mathfrak{m}}_x$. By [F5] applied to $\mathfrak{q}={\mathfrak{m}}_x$, and since $\kappa ({\mathfrak{m}}_x)$ is a finite extension of $k$ (whence ${\mathrm{trdeg}}_k\kappa ({\mathfrak{m}}_x)=0$), the local dimension at $x$ is $\dim {\mathcal{O}}_{X,x}$. That local dimension is $1$: every open neighbourhood of the closed point $x$ contains the generic point $\eta$ [F1], so each such neighbourhood has chain dimension one, and $X$ itself has dimension one [F1]. Hence $\dim {\mathcal{O}}_{X,x}=1$. The ring ${\mathcal{O}}_{X,x}$ is Noetherian because $A$ is a finite-type algebra over the field $k$ and localisations of Noetherian rings are Noetherian [F14]; it is regular because $x$ is a point of the smooth $k$-scheme $X$, hence a point of $X_k$ in the notation of [F4]. Therefore [F6] applies under Choice [F13] and ${\mathcal{O}}_{X,x}$ is a discrete valuation ring.

Consequently ${\mathcal{O}}_{X,x}=V_v$ for a discrete valuation $v$ of $K$, so ${\mathcal{O}}_{X,x}$ is a valuation ring with fraction field $K$ and contains an element of value $1$ [F6, F11]; the valuative diagrams used below take $R={\mathcal{O}}_{X,x}$.

The composite $\mathrm{Spec}K\to U\stackrel{{\phi }_U}{\to }Y$ is defined because $\eta \in U$ [F1], and it is a $k$-morphism; together with the structure morphism $\mathrm{Spec}{\mathcal{O}}_{X,x}\to \mathrm{Spec}k$ it makes the square of a valuative diagram for $Y\to \mathrm{Spec}k$ commute, the two composites $\mathrm{Spec}K\to Y\to \mathrm{Spec}k$ and $\mathrm{Spec}K\to \mathrm{Spec}{\mathcal{O}}_{X,x}\to \mathrm{Spec}k$ both being the structure morphism of $\mathrm{Spec}K$ [F7].

By [F8] and the properness of $Y$, under Choice [F13] this valuative diagram has a unique lift $g_x:\mathrm{Spec}{\mathcal{O}}_{X,x}\to Y$, whose restriction to the generic point $\mathrm{Spec}K$ is the given map $\mathrm{Spec}K\to U\to Y$ [F7].

Spread out the local lift to a neighbourhood. Choose an affine open $\mathrm{Spec}A\subseteq X$ containing $x$, and write $\mathfrak{m}\subset A$ for the maximal ideal corresponding to $x$. Then ${\mathcal{O}}_{X,x}=A_{\mathfrak{m}}$, and the map $g_x$ is equivalently a $k$-algebra homomorphism $\rho :B\to A_{\mathfrak{m}}$ for any affine open $V=\mathrm{Spec}B\subseteq Y$ containing the image of the closed point of $\mathrm{Spec}{A}_{\mathfrak{m}}$. Such a chart exists, and its preimage under $g_x$ is an open subscheme of the local spectrum containing its closed point; the only such open is all of $\mathrm{Spec}{A}_{\mathfrak{m}}$. By [F15], choose $k$-algebra generators $b_1,\dots ,b_n$ of $B$. Write each $\rho (b_i)=a_i\mathrm{/}{t}_i$ with $a_i\in A$ and $t_i\mathrm{\notin }\mathfrak{m}$, and put $s={\prod }_i{t}_i\mathrm{\notin }\mathfrak{m}$. Then every $\rho (b_i)$ lies in $A_s$. Since $A$ is a domain and $s\mathrm{\ne }0$, the localization map $A_s\to A_{\mathfrak{m}}$ is injective. Present $B$ as a quotient of $k[y_1,\dots ,y_n]$ using the chosen generators; every defining relation maps to zero in $A_{\mathfrak{m}}$ under $y_i\mapsto \rho (b_i)$, so injectivity shows it already maps to zero in $A_s$. Thus the generator assignment induces a $k$-algebra map $\tilde{\rho }:B\to A_s$. By [F16] this map gives a $k$-morphism ${\psi }_x:D(s)=\mathrm{Spec}{A}_s\to V\subseteq Y$, on an open neighbourhood of $x$, and its restriction to $\mathrm{Spec}{A}_{\mathfrak{m}}$ is $g_x$.

The restriction of ${\psi }_x$ to the generic point of $D(s)$ is the generic map $\mathrm{Spec}K\to U\to Y$ of step 4.1, because ${\psi }_x$ restricts to $g_x$ on $\mathrm{Spec}{A}_{\mathfrak{m}}$ and $g_x$ restricts to that map.

Agreement near $x$. Put $W=D(s)\cap U$, a nonempty open subscheme of $X$ containing $\eta$, reduced as an open subscheme of the reduced scheme $X$ [F1]. Both ${\psi }_x{\mathrm{\mid }}_W$ and ${\phi }_U{\mathrm{\mid }}_W$ are $k$-morphisms $W\to Y$. To see they are equal, form $h=({\psi }_x{\mathrm{\mid }}_W,{\phi }_U{\mathrm{\mid }}_W):W\to Y{\times }_{k}Y$. Since $Y$ is separated over $k$, the diagonal $\mathrm{\Delta }:Y\to Y{\times }_{k}Y$ is a closed immersion [F12], so $Z:=h^{-1}(\mathrm{\Delta }(Y))$ is a closed subscheme of $W$. By step 6.1 the restriction of $h$ to the generic point $\mathrm{Spec}K\to W$ factors through $\mathrm{\Delta }$, so $Z$ contains the image of $\mathrm{Spec}K$, namely $\eta$; since $\eta$ is dense in $W$ [F1], the underlying space of $Z$ is all of $W$. As $W$ is reduced, a closed subscheme with the same underlying space equals $W$: on each local ring ${\mathcal{O}}_{W,p}$ a proper ideal $I$ with $\mathrm{Spec}({\mathcal{O}}_{W,p}\mathrm{/}I)=\mathrm{Spec}{\mathcal{O}}_{W,p}$ would be contained in the nilradical, which vanishes. Hence $Z=W$, so $h$ factors through the diagonal and ${\psi }_x{\mathrm{\mid }}_W={\phi }_U{\mathrm{\mid }}_W$.

Gluing. The morphisms ${\phi }_U:U\to Y$ and ${\psi }_x:D(s_x)\to Y$, one for each $x\in Z$ (where $D(s_x)$ is the neighbourhood produced in step 5.1), are defined on an open cover $U\cup {\bigcup }_{x\in Z}D(s_x)=X$ of $X$: indeed $X\setminus U=Z$ [step 1.1]. They are compatible: ${\psi }_x{\mathrm{\mid }}_W={\phi }_U{\mathrm{\mid }}_W$ on $W=D(s_x)\cap U$ by step 7.1, and for $x\mathrm{\ne }{x}^{\mathrm{\prime }}$ the two morphisms ${\psi }_x,{\psi }_{x^{\mathrm{\prime }}}$ agree on the nonempty open $(D(s_x)\cap D(s_{x^{\mathrm{\prime }}}))\cap U$, which is dense in the reduced scheme $D(s_x)\cap D(s_{x^{\mathrm{\prime }}})$ because $X$ is irreducible [F1]; hence ${\psi }_x={\psi }_{x^{\mathrm{\prime }}}$ on the overlap by [F9]. By [F10] the compatible morphisms glue to a unique $k$-morphism $F:X\to Y$ with $F{\mathrm{\mid }}_U={\phi }_U$.

Uniqueness and the correspondence. If $F^{\mathrm{\prime }}\text{\hspace{0.17em}⁣}:X\to Y$ is a second $k$-morphism with $F^{\mathrm{\prime }}{\mathrm{\mid }}_U={\phi }_U$, then $F$ and $F^{\mathrm{\prime }}$ agree on the nonempty open, hence dense, subscheme $U$ of the reduced scheme $X$, so $F=F^{\mathrm{\prime }}$ by [F9] applied with $W=X$, $V=U$ and $S=\mathrm{Spec}k$. Thus each representative $(U,{\phi }_U)$ extends to exactly one $k$-morphism $X\to Y$; conversely every $k$-morphism $X\to Y$ is a representative of a rational map with domain $X$, and the equivalence of two extensions is detected on $U$ by [F9], so the resulting map from rational maps to $k$-morphisms is a bijection.

Every assertion of the statement holds: existence of extensions over the finite set $Z=X\setminus U$ is step 8.1, surjectivity onto $k$-morphisms and injectivity (uniqueness) are step 9.1, and the description of the local rings as discrete valuation rings is step 1.3. The Axiom of Choice enters through [F3] at step 1.1, [F4] and [F5] at step 1.3, [F6] at step 1.3, [F8] at step 4.1, and [F9] at steps 7.1–9.1; the remaining cited inputs are choice-free.