Smooth Proper Curves Divisors Genus and Ramification

Delete empty members of the finite cover. Each remaining $A_i$ is a domain, since $U_i$ is a nonempty affine open of the integral scheme $C$ [F15], and $A_i$ is a finitely generated $k$-algebra because $C$ is of finite type over $k$. By [F4] the fraction field $\mathrm{Frac}(A_i)$ is canonically identified with $K={\mathcal{O}}_{C,\eta }$, and these identifications are compatible on overlaps, so all of them may be regarded as subfields of one copy of $K$. Each $U_i\cap U_j$ is affine by [F5], since $C$ is separated over $\mathrm{Spec}k$ and all affine opens of $C$ lie over the single affine open $\mathrm{Spec}k$.

Let $B_i\subseteq K$ be the integral closure of $A_i$ in $K$ [F9]. By [F1], $B_i$ is a finite $A_i$-module and a domain with fraction field $K$. The integral-closure theorem [F17] makes $B_i$ integrally closed. Since $k$ is a principal ideal domain, [F8] makes each finite-type $A_i$ Noetherian; [F19] then makes the module-finite $A_i$-algebra $B_i$ Noetherian. Applying both directions of [F18] to $B_i$, all of its prime localizations are integrally closed, so $B_i$ is normal in the Noetherian-ring sense [F10].

For $0\mathrm{\ne }f\in A_i$ the integral closure of the principal localisation $(A_i{)}_f$ in $K$ is exactly $(B_i{)}_f$ and $(B_i{)}_f$ is a finite $(A_i{)}_f$-module, by [F2]. Consequently the ring $B_i$ attached to the chart is determined on each principal open $D(f)\subseteq U_i$ by that open alone, namely as $(B_i{)}_f$ inside $K$.

Gluing data. Fix $i,j$ and put $W=U_i\cap U_j$, which is affine by [F5]. Let $V_{ij}$ be the inverse image of $W$ in $\mathrm{Spec}(B_i)$. We construct compatible identifications locally on $W$. For each point $w\in W$, choose principal opens $D(f)\subseteq U_i$ and $D(g)\subseteq U_j$ containing $w$ and contained in $W$; such choices exist by [F13]. On $D(f)$ the restriction of $g$ is a regular function, hence is represented by an element of $(A_i{)}_f$; write it as $c\mathrm{/}{f}^r$. Then $D(f)\cap D(g)=D(fc)$ as an open of $U_i$. Similarly, on $D(g)$ the restriction of $f$ is represented by $d\mathrm{/}{g}^s$ in $(A_j{)}_g$, so the same intersection is $D(gd)$ as an open of $U_j$. Thus $D(fc)=D(gd)$ is a common principal-open neighborhood of $w$ in $W$. Its coordinate rings, computed in either chart, are the same subring of $K$, since both are $\mathrm{\Gamma }(D(fc),{\mathcal{O}}_C)$. By [F2], the integral closures of this ring in $K$ are respectively $(B_i{)}_{fc}$ and $(B_j{)}_{gd}$, so these localizations are equal inside $K$. The identity of that ring induces an isomorphism between the corresponding opens in the two normalization charts. These common opens cover $W$; the isomorphisms agree on further intersections because every ring map is the identity inside $K$. They therefore glue to an isomorphism $V_{ij}\to V_{ji}$ over $W$. The same identity-in-$K$ argument gives inverse maps and the cocycle condition on triple overlaps. No single distinguished open of $U_i$ is assumed to be represented by one element of $A_j$.

Gluing. By [F3] the affine schemes $\mathrm{Spec}(B_i)$, equipped with the open subschemes $V_{ij}$ and the compatible isomorphisms ${\phi }_{ij}$, glue to a scheme $C^{\mathrm{n}\mathrm{u}}$ on which the charts $\mathrm{Spec}(B_i)$ form an open affine cover.

The morphism $\nu$. Each inclusion $A_i\subseteq B_i$ induces a $k$-morphism $\mathrm{Spec}(B_i)\to U_i$ by [F12]. On the common principal-open refinements from step 1.4, the chart isomorphism is induced by the identity of the localized integral-closure ring inside $K$; both composites to $C$ are therefore the same map to the overlap $W$. The chart morphisms agree on the open overlaps and glue by [F6] to $\nu :C^{\mathrm{n}\mathrm{u}}\to C$. The transition maps are these normalization-chart isomorphisms over $W$, not inclusions of one chart into the other.

$C^{\mathrm{n}\mathrm{u}}$ is integral, normal and has function field $K$. Every chart $\mathrm{Spec}(B_i)$ is integral and has generic point with local ring $K$. Any two remaining $U_i,U_j$ meet in a nonempty open because $C$ is irreducible. The inverse image of that overlap contains the generic point of each normalization chart, and the transition maps identify those generic points by the identity of $K$. They therefore give one point $\eta$ lying in every chart. It is dense in each chart because each $B_i$ is a domain, hence dense in $C^{\mathrm{n}\mathrm{u}}$; this proves global irreducibility. The charts are reduced, so the glued scheme is reduced and therefore integral. Their finite affine cover has Noetherian coordinate rings by step 1.2, so $C^{\mathrm{n}\mathrm{u}}$ is Noetherian; [F18] makes every local ring integrally closed, hence the scheme is normal by [F20]. The common generic local ring is $K$, so $k(C^{\mathrm{n}\mathrm{u}})=K$.

Finiteness. For each $i$ we have ${\nu }^{-1}(U_i)=\mathrm{Spec}(B_i)$ by construction, and $B_i$ is a finite $A_i$-module by [F1]. The cover $C={\bigcup }_i{U}_i$ is a finite affine open cover of the target, so the local criterion [F7] shows that $\nu$ is finite; in particular $\nu$ is affine by the choice-free first half of [F7].

Birationality. The map $\nu$ sends the common generic point $\eta$ of step 2.3 to the generic point of $C$ and induces the identity map $K\to K$ on function fields. It is therefore dominant and birational by [F11].

Initiality. Let $h:Z\to C$ be finite and birational, with $Z$ normal and integral. For each $i$, $h^{-1}(U_i)=\mathrm{Spec}(D_i)$ and $D_i$ is a finite $A_i$-algebra by [F7, F14]. The preimage contains the generic point, so $D_i$ is a domain. Birationality and [F4] identify $\mathrm{Frac}(D_i)$ with $K$; under these identifications the map $A_i\to D_i$ is injective, so regard it as an inclusion. The finite affine covers and [F8], [F19] make $C$ and $Z$ Noetherian; normality of $Z$ says every localization $(D_i{)}_{\mathfrak{q}}$ is an integrally closed domain [F20]. By the converse direction of [F18], $D_i$ itself is integrally closed in $K$. The finite-morphism theorem [F21] makes every element of $D_i$ integral over $A_i$, so $D_i\subseteq B_i$. Conversely, each $b\in B_i$ is integral over $A_i\subseteq D_i$ and lies in $K$, so integral closedness of $D_i$ gives $b\in D_i$. Thus $D_i=B_i$ as subrings of $K$ for every $i$. The identity ring maps on these equal chart algebras induce chart isomorphisms in both directions, and they glue by [F6] to morphisms $\varphi :C^{\mathrm{n}\mathrm{u}}\to Z$ and $\psi :Z\to C^{\mathrm{n}\mathrm{u}}$ over $C$. They are inverse because their restrictions on each affine chart are identities. Any morphism $C^{\mathrm{n}\mathrm{u}}\to Z$ over $C$ induces the identity on the generic function field $K$; its chart ring maps are therefore the identity on $D_i=B_i\subseteq K$, so it equals $\varphi$. The same argument for a morphism $Z\to C^{\mathrm{n}\mathrm{u}}$ makes it equal to $\psi$. Thus both maps are unique, and in particular $\varphi$ proves initiality in the stated direction.

Uniqueness of the normalization. Let $(C^{\mathrm{\prime }},{\nu }^{\mathrm{\prime }})$ be another normal integral scheme finite and birational over $C$. Step 4.1 gives unique maps $u:C^{\mathrm{n}\mathrm{u}}\to C^{\mathrm{\prime }}$ and $v:C^{\mathrm{\prime }}\to C^{\mathrm{n}\mathrm{u}}$ over $C$. Uniqueness forces $v\circ u$ and $u\circ v$ to be the identity maps. Hence $u$ and $v$ are inverse isomorphisms, unique over $C$.

Conclusion and Choice accounting. Steps 1.2 and 1.3 prove finiteness of the affine integral closures and compatibility with principal localization; steps 1.4, 2.1 and 2.2 construct the glued scheme and morphism; steps 3.1 and 3.2 prove that the morphism is finite, affine and birational; step 2.3 proves integrality, normality and the function-field identity; and steps 4.1 and 5.1 prove initiality and uniqueness. The Axiom of Choice is inherited through the normality-locality criterion [F18], the local criterion for finiteness [F7], and the finite-morphism integrality theorem [F21]; these are used in steps 1.2, 3.1, and 4.1.

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.

Part (1), injectivity of the pullback. Let $f:C\to D$ be a dominant $k$-morphism of smooth proper geometrically integral $k$-curves. Then $f({\eta }_C)={\eta }_D$ [F2], and the pullback $f^{\ast }:k(D)={\mathcal{O}}_{D,{\eta }_D}\to {\mathcal{O}}_{C,{\eta }_C}=k(C)$ is a homomorphism of fields [F1, F2]; being a map of fields into a nonzero field it is injective.

Part (1), an injective homomorphism gives a dominant rational map. Let $\phi :k(D)\hookrightarrow k(C)$ be an injective $k$-algebra homomorphism. Choose a finite affine cover $D={\bigcup }_i\mathrm{Spec}{A}_i$ (possible because $D$ is proper, hence quasi-compact [F1]). For each $i$, choose algebra generators $a_{i1},\dots ,a_{i{r}_i}$ of $A_i$. Choose a nonempty affine open $U_i=\mathrm{Spec}{R}_i\subset C$; each rational function $\phi (a_{ij})$ is a fraction in $\mathrm{Frac}(R_i)$, so after choosing a common nonzero denominator $g_i\in R_i$, all these images lie in the actual localization $(R_i{)}_{g_i}$. The induced $k$-algebra map $A_i\to (R_i{)}_{g_i}$ therefore defines a morphism $D(g_i)=\mathrm{Spec}((R_i{)}_{g_i})\to \mathrm{Spec}(A_i)\subseteq D$. These nonempty source opens are dense in the integral curve $C$. On a pairwise overlap $V$, the two maps agree at its generic point because they induce the same function-field map $\phi$. The equalizer is the pullback of the closed diagonal of the separated scheme $D$, so it is a closed subscheme of $V$. Its underlying closed subset contains the generic point, hence is all of the irreducible space $V$; since $V$ is reduced, [F4] makes the equalizer all of $V$. Thus the maps agree on overlaps and [F5] glues them on their union, a dense open subscheme of $C$, to a rational map $C⇢D$. Its induced map on function fields is $\phi$, which is injective, so the rational map is dominant by [F2].