Zac Maeder-Wolland: Some Daily AG Problems

Hartshorne Exercise IV.1.3.

First, we show that it is always possible to embed such a scheme $X$ into a proper regular curve, $\bar X$. Let $U\subset X$ be an affine open subscheme which has a closed immersion $U\ra \A{k}{N}$ by NNL. This yields an embedding $U\ra \PP{k}{N}$. Let $\bar X$ be the normalization of the closure $\bar U\subset \PP{k}{N}$ of $U$. The normalization of a variety is a finite map between varieties that is an isomorphism over the normal locus. Hence, we see that $\bar X$ is a variety that is proper (a composition of proper maps) and normal. It is smooth since a 1 dimensional normal scheme is regular (under noetherian hypothesis, {normal, local rings of dimension 1} = {DVRs} = {regular, local, dim 1 rings}), and the morphism $\bar X\ra\bar U $ is an isomorphism over $U$ because $U$ is regular, thus normal. We therefore have an embedding $U\ra \bar X$, which we claim extends to an embedding $\iota: X\ra \bar X$. Indeed, we use the valuation criterion for properness as follows. For any point $p\in X-U$ Consider the commutative diagram,

$$\spec{\Bbb K(X)}\ra \bar X$$

$$\downarrow\quad\quad\quad\quad\quad\quad\quad\quad\:\:\downarrow$$

$$\spec{\OO_{X,p}}\longrightarrow \spec k.$$

It yields a unique map $\spec{\OO_{X,p}}\xrightarrow{h} \bar X$. $h$ corresponds to a $k$-algebra homomorphism $k[x_1,…,x_r]/I=:R\xrightarrow{\varphi}\OO_{X,p}$ with $\op{im}h\subsetV:=\spec R\subset\bar X$ an open neighborhood (because the image of $h$ is just 2 points). Let $p\inW=\spec A\subset X$ be an affine open neighborhood. $\varphi$ corresponds to a choice of $\varphi(x_i)=\frac{a_i}{g_i}\in\OO_{X,p}$ such that $p\in\{g_i\ne0\}$ and such that, for every $t\in I$, $\exists s\in A-p$ with $st(\varphi(x_1),…,\varphi(x_r))$. Taking products of all such $s,g_i\in A$ yields an element $g\in A$ for which $p\in\{g\ne 0\}\subset A$ and gives a morphism $\tilde h: \{g\ne 0\}\raV\ra\bar X$.

The top row in the commutative diagram is actually the composition $\spec{\Bbb K(X)}\raU\ra\bar X$ with image in $V$. In particular the homomorphism that is $\varphi$ followed by the fraction field, $R\ra\OO_{X,p}\ra\Bbb K(X)$ is the same as $R\ra\OO_X(U)\ra\Bbb K(X)$. Above we showed that $\varphi$ is in fact a composition $R\raA[\frac 1g]\ra\OO_{X,p}$, which tells us that localization at the generic point of $X$ equalizes the homomorphisms $R\raA[\frac 1g]$ and $R\ra\OO_X(U)$. That is morphisms $U\ra\bar X$ and $\{g\ne 0\}$ are generically equal on $U\cap\{g\ne 0\}$. We conclude that $U\ra\bar X$ and $V\ra\bar X$ are the same where they meet, and therefore glue to give a morphism $U\cupV\ra\bar X$ that restricts to the original embedding $U\ra X$. Since $X-U$ is closed, there are only finitely many points, so repeating this for every one of them does the job.
Now, consider the set $\bar X- X$: every point in it is sent somewhere in $\bar U-U$. Because $\bar U-U$ is closed in $\bar U$ and thus finite, and $\bar X\ra \bar U$ is finite and thus has finite fibers, we conclude $\bar X -X$ is a finite set of closed points, $\{P_1,…,P_r\}$. Let $f\in \Bbb K(X)$ be the rational function found in the previous problem, with poles on these points and regular everywhere else. Then $f$ induces a morphism $f:\bar X\ra \PP{[X:Y]}{1}$ (as every rational map of smooth proper curves extends to one) such that $f^{-1}(\{Y= 0\})={P_1,…,P_r}$, so that $f^{-1}(\{Y\ne 0\})= X$. Indeed, at any point $p\in \bar X$ we write $f$ locally as a ratio $\frac gh\in \Bbb K(U)$, where $p\in U$ an open affine neighborhood. Then the morphism $f$ sends $p$ to the ratio $[g(p):h(p)]$, and $h(p)=0\iff h\notin\OO_{\bar X,p}\iff$ $f$ has a pole at $p$.
Because $\bar X$ is a smooth and proper curve and $\PP{k}{1}$ is a curve, the image of $f$ is either a point which is clearly impossible, or $f$ is finite and surjective. Then it follows that $X$ is affine.