curves and rational maps

curves.tex

@@ -22,11 +22,164 @@ \section{Introduction}
 A reference covering algebraic curves over the complex numbers is
 the book \cite{ACGH}.
 
+\medskip\noindent
+What we already know. Besides general algebraic geometry, we
+have already proved some specific results on algebraic curves.
+Here is a list.
+\begin{enumerate}
+\item We have discussed affine opens of and ample invertible sheaves on
+$1$ dimensional Noetherian schemes in
+Varieties, Section \ref{varieties-section-dimension-one}.
+\item We have seen a curve is either affine or projective
+in Varieties, Section \ref{varieties-section-curves}.
+\item We have discussed degrees of locally free modules on
+proper curves in Varieties, Section \ref{varieties-section-divisors-curves}.
+\item We have discussed the Picard scheme of a nonsingular projective
+curve over an algebraically closed field in
+Picard Schemes of Curves, Section \ref{pic-section-introduction}.
+\end{enumerate}
+
+
+
+
+
+\section{Curves and function fields}
+\label{section-curves-function-fields}
+
+\noindent
+In this section we elaborate on the results of
+Varieties, Section \ref{varieties-section-varieties-rational-maps}
+in the case of curves.
 
+\begin{lemma}
+\label{lemma-extend-over-dvr}
+Let $k$ be a field. Let $X$ be a curve and $Y$ a proper variety.
+Let $U \subset X$ be a nonempty open and let $f : U \to Y$ be a morphism.
+If $x \in X$ is a closed point such that $\mathcal{O}_{X, x}$
+is a discrete valuation ring, then there exists an open
+$U \subset U' \subset X$ containing $x$ and a morphism of
+varieties $f' : U' \to Y$ extending $f$.
+\end{lemma}
 
+\begin{proof}
+This is a specical case of
+Morphisms, Lemma \ref{morphisms-lemma-extend-across}.
+\end{proof}
 
+\begin{lemma}
+\label{lemma-extend-over-normal-curve}
+Let $k$ be a field. Let $X$ be a normal curve and $Y$ a proper variety.
+The set of rational maps from $X$ to $Y$ is the same as the set
+of morphisms $X \to Y$.
+\end{lemma}
 
+\begin{proof}
+This is clear from Lemma \ref{lemma-extend-over-dvr}
+as every local ring is a discrete valuation ring
+(for example by Varieties, Lemma \ref{varieties-lemma-regular-point-on-curve}).
+\end{proof}
 
+\begin{lemma}
+\label{lemma-extend-to-completion}
+Let $k$ be a field. Let $X \to Y$ be a morphism of varieties
+with $Y$ proper. There exists a factorization $X \to \overline{X} \to Y$
+where $X \to \overline{X}$ is an open immersion
+and $\overline{X}$ is a projective curve.
+\end{lemma}
+
+\begin{proof}
+This is clear from Lemma \ref{lemma-extend-over-dvr}
+and Varieties, Lemma \ref{varieties-lemma-reduced-dim-1-projective-completion}.
+\end{proof}
+
+\noindent
+Here is the main theorem of this section. We will say a morphism
+$f : X \to Y$ of varieties is {\it constant} if the image $f(X)$
+consists of a single point $y$ of $Y$. If this happens then
+$y$ is a closed point of $Y$ (sicne the image is a constructible set
+and varieties are Jacobson schemes).
+
+\begin{theorem}
+\label{theorem-curves-rational-maps}
+Let $k$ be a field. The following categories are canonically equivalent
+\begin{enumerate}
+\item The category of finitely generated field extensions $K/k$ of
+transcendence degree $1$.
+\item The category of curves and dominant rational maps.
+\item The category of normal projective curves and nonconstant morphisms.
+\item The category of nonsingular projective curves and nonconstant morphisms.
+\item The category of regular projective curves and nonconstant morphisms.
+\end{enumerate}
+\end{theorem}
+
+\begin{proof}
+The equivalence between categories (1) and (2) is the restriction of the
+equivalence of
+Varieties, Theorem \ref{varieties-theorem-varieties-rational-maps}.
+Namely, a variety is a curve if and only if its function field has
+transcendence degree $1$, see for example
+Varieties, Lemma \ref{varieties-lemma-dimension-locally-algebraic}.
+
+\medskip\noindent
+The categories in (3), (4), (5) are the same. First of all, the
+terms ``regular'' and ``nonsingular'' are synonyms, see
+Properties, Definition \ref{properties-definition-regular}.
+Being normal and regular are the same thing for Noetherian
+$1$-dimensional schemes
+(Properties, Lemmas \ref{properties-lemma-regular-normal} and
+\ref{properties-lemma-normal-dimension-1-regular}). See
+Varieties, Lemma \ref{varieties-lemma-regular-point-on-curve}
+for the case of curves. Thus (3) is the same as (5)
+
+\medskip\noindent
+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). 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 \ref{lemma-extend-over-dvr}
+this extends to all of $X$ and we obtain a morphism in the category (5).
+Thus we see that there is a fully faithfull functor (5)$\to$(1).
+
+\medskip\noindent
+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 \ref{varieties-lemma-normalization-locally-algebraic}).
+Then we choose $X \to \overline{X}$ with
+$\overline{X} \setminus X = \{x_1, \ldots, x_n\}$ as in
+Varieties, Lemma \ref{varieties-lemma-reduced-dim-1-projective-completion}.
+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 \ref{varieties-lemma-dim-1-projective-completion}
+and then normalize using
+Varieties, Lemma \ref{varieties-lemma-normalization-locally-algebraic}.
+Doing it this way we avoid using the somewhat tricky
+Varieties, Lemma \ref{varieties-lemma-relative-normalization-normal-codim-1}.)
+\end{proof}
+
+\begin{definition}
+\label{definition-normal-projective-model}
+Let $k$ be a field. Let $X$ be a curve.
+A {\it nonsingular projective model of $X$}
+is a pair $(Y, \varphi)$ where $Y$ is a nonsingular projective
+curve and $\varphi : k(X) \to k(Y)$ is an isomorphism
+of function fields.
+\end{definition}
+
+\noindent
+A nonsingular projective model is determined up to unique
+isomorphism by Theorem \ref{theorem-curves-rational-maps}.
+Thus we often say ``the nonsingular projective model''.
+We usually drop $\varphi$ from the notation.
+Warning: it needn't be the case that $Y$ is smooth over $k$
+(but this can only happen in positive characteristic).