\documentclass[12pt]{article} \usepackage{amsthm,amsfonts,amsmath,amssymb,latexsym,epsfig} %\usepackage{showkeys} \usepackage{hyperref} \title{Riemann Surfaces and Algebraic Curves} \author{JWR} \date{Tuesday December 11, 2001, 9:03 AM} \newcommand{\ETC}{$\ldots$} % indicates there is more to be written % \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\D}{\mathbb{D}} \newcommand{\E}{\mathbb{E}} \newcommand{\F}{\mathbb{F}} \newcommand{\G}{\mathbb{G}} \renewcommand{\H}{\mathbb{H}} \newcommand{\K}{\mathbb{K}} \renewcommand{\P}{\mathbb{P}} \newcommand{\Q}{\mathbb{Q}} % \newcommand{\fH}{\mathfrak{H}} \newcommand{\fF}{\mathfrak{F}} % \newcommand{\cA}{\mathcal{A}} \newcommand{\cB}{\mathcal{B}} \newcommand{\cBbar}{\overline{\mathcal{B}}} \newcommand{\cC}{\mathcal{C}} \newcommand{\cD}{\mathcal{D}} \newcommand{\cF}{\mathcal{F}} \newcommand{\cE}{\mathcal{E}} \newcommand{\cG}{\mathcal{G}} \newcommand{\cH}{\mathcal{H}} \newcommand{\cI}{\mathcal{I}} \newcommand{\cJ}{\mathcal{J}} \newcommand{\cK}{\mathcal{K}} \newcommand{\cL}{\mathcal{L}} \newcommand{\cM}{\mathcal{M}} \newcommand{\cMbar}{\overline{\mathcal{M}}} \newcommand{\cN}{\mathcal{N}} \newcommand{\cP}{\mathcal{P}} \newcommand{\cO}{\mathcal{O}} \newcommand{\cQ}{\mathcal{Q}} \newcommand{\cR}{\mathcal{R}} \newcommand{\cS}{\mathcal{S}} \newcommand{\cT}{\mathcal{T}} \newcommand{\cU}{\mathcal{U}} \newcommand{\cV}{\mathcal{V}} \newcommand{\cUbar}{\overline{\mathcal{U}}} \newcommand{\cX}{\mathcal{X}} \newcommand{\cZ}{\mathcal{Z}} % \newcommand{\GL}{{\mathrm{GL}}} \newcommand{\U}{{\mathrm{U}}} \newcommand{\SU}{{\mathrm{SU}}} \newcommand{\mcg}{\mathrm{mcg}} \newcommand{\id}{\mathrm{id}} \newcommand{\ad}{\mathrm{ad}} \newcommand{\codim}{\mathrm{codim}} \newcommand{\Perm}{\mathrm{Perm}} \newcommand{\Diff}{\mathrm{Diff}} \newcommand{\Vect}{\mathrm{Vect}} \newcommand{\Stab}{\mathrm{Stab}} \newcommand{\Span}{\mathrm{Span}} \newcommand{\free}{\mathrm{free}} \newcommand{\Aut}{\mathrm{Aut}} \newcommand{\Iso}{\mathrm{Iso}} \newcommand{\Mor}{\mathrm{Mor}} \newcommand{\Fix}{\mathrm{Fix}} \newcommand{\Ord}{\mathrm{Ord}} \newcommand{\Kernel}{\mathrm{Kernel}} \newcommand{\Image}{\mathrm{Image}} \newcommand{\End}{\mathrm{End}} \newcommand{\ind}{\mathrm{ind}} \newcommand{\res}{\mathrm{res}} \newcommand{\Res}{\mathrm{Res}} % \newcommand{\Stretch}{\renewcommand{\arraystretch}{1.5}} % \newcommand{\p}{\partial} \newcommand{\Wedge}{{\textstyle\bigwedge\nolimits}} \newcommand{\jhat}{{\hat{\jmath}}} \newcommand{\pbar}{{\bar\partial}} \newcommand{\eps}{\varepsilon} \newcommand{\Mat}[1]{\left(\begin{array}{#1}} \newcommand{\Rix}{\end{array}\right)} % % \newcommand{\inner}[2]{\left\langle #1,#2\right\rangle} %\newcommand{\proof}[1]{\par\smallskip\par\noindent{\bf Proof#1:\ }} %\newcommand{\QEDSYM}{\mbox{$\square$}} %\newcommand{\QED}{\hfill\QEDSYM} \newcommand{\jdef}[1]{{\bf #1}} \newcommand{\mdef}[1]{#1} % \newcommand{\half}{{\textstyle\frac{1}{2}}} \newcommand{\quarter}{{\textstyle\frac{1}{4}}} \newcommand{\ds}{\displaystyle} % \newtheorem{PARA}{}[section] \newcommand{\para}{\begin{PARA}\rm } \newcommand{\arap}{\end{PARA}} \newtheorem{theorem}[PARA]{Theorem} \newtheorem{corollary}[PARA]{Corollary} \newtheorem{lemma}[PARA]{Lemma} \newtheorem{proposition}[PARA]{Proposition} \newtheorem{definition}[PARA]{Definition} \newcommand{\dfn}{\begin{definition}\rm } \newcommand{\nfd}{\end{definition}} \newtheorem{remark}[PARA]{Remark} \newcommand{\rmk}{\begin{remark}\rm } \newcommand{\kmr}{\end{remark}} \newtheorem{example}[PARA]{Example} \newcommand{\xmpl}{\begin{example}\rm } \newcommand{\lpmx}{\end{example}} \newtheorem{exercise}[PARA]{Exercise} \newcommand{\xrcs}{\begin{exercise}\rm } \newcommand{\scrx}{\end{exercise}} % \newcommand{\note}{[} % These may eventually become end notes. \newcommand{\eton}{] } % \newcommand{\medskipnoindent}{\par\medskip\par\noindent} \newcommand{\Item}[1]{\medskipnoindent\ItemMark{#1}} \newcommand{\ItemMark}[1]{{\bf (#1)}} % \newcommand{\resp}[2]{$\displaystyle \left\{\begin{array}{l}\mbox{#1}\\ \mbox{#2} \end{array}\right\} $} % \newcommand{\mapdown}[1]{\Big\downarrow \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}} \newcommand{\mapup}[1]{\Big\uparrow \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}} \newcommand{\mapright}[1]{\smash{ \mathop{\longrightarrow}\limits^{#1}}} % \begin{document} \maketitle We describe the relation between algebraic curves and Riemann surfaces. An elementary reference for this material is~\cite{GRIFFITHS}. \section{Riemann surfaces} \para A \jdef{Riemann surface} is a smooth complex manifold $X$ (without boundary) of complex dimension one. Let $K\to X$ denote the \jdef{canonical line bundle} so that the fiber $K_p$ over $p\in X$ is the space of complex linear maps from $T_pX$ to $\C$. A section of $K$ is called a \jdef{differential} on $X$. We define $$ \begin{array}{lcl} \cM(X) &=& \mbox{the field of meromorphic functions on } X, \\ \\ \cO(X) &=& \mbox{the ring of holomorphic functions on } X, \\ \\ \cM(X,K) &=& \mbox{the space of meromorphic differentials on } X, \\ \\ \Omega(X) &=& \mbox{the space of holomorphic differentials on } X. \end{array} $$ If $X$ is compact, $\cO(X)=\C$ the constant functions. An element of $\cM(X)$ can be viewed as a holomorphic map to the Riemann sphere (projective line) $$ \P:=\C\cup\{\infty\} $$ and the only holomorphic map which does not arise this way is the constant map which sends all of $X$ to $\infty$. The \jdef{genus} $g$ of a compact Riemann surface $X$ is defined by $$ 2g=\dim_\R\,H^1(X,\R) $$ so the \jdef{Euler characteristic} of $X$ is $\chi(X)=2-2g$. The Riemann Roch Theorem implies that for $X$ compact we have $$ g=\dim_\C(\Omega(X)) $$ the dimension of the space of holomorphic differentials. \arap \para\label{def:order} Let $p\in X$ and $z$ be a local holomorphic coordinate on $X$ with $z(p)=0$. Any $f\in \cM(X)\setminus\{0\}$ has form $$ f(z)=z^kh(z) $$ in the coordinate $z$ where $h$ is holomorphic and $h(0)\ne 0$. The integer $$ \Ord_p(f):= k $$ is independent of the choice of the coordinate $z$; it is called the \jdef{order} of $f$ at $p$. A point $p$ is called a \jdef{zero} of $f$ if $\Ord_p(f)>0$, a \jdef{pole} of $f$ if $\Ord_p(f)<0$, and a \jdef{singularity} of $f$ if it is either a zero or pole, i.e. if $\Ord_p(f)\ne 0$. (One can define analogously the order of a singularity meromorphic section of any holomorphic line bundle but here we only need the notion for differentials.) Thus any $\omega\in \cM(X,K)$ has form $$ \omega = f\,dz $$ where $f\in \cM(X)$. The integer $$ \Ord_p(\omega):=\Ord_p(f) $$ is independent of the choice of the coordinate $z$; it is called the \jdef{order} of $\omega$ at $p$. The complex number $$ \res_p(\omega)=\frac{1}{2\pi i}\oint_{\gamma_p}\omega $$ is independent of the choice of the small circle $\gamma_p$ about $p$ having no pole other than $p$ in its interior; it is called the \jdef{residue} of $\omega$ at $p$. \arap \begin{theorem}[Residue Theorem] Let $X$ be a compact Riemann surface and $\omega\in \cM(X,K)\setminus\{0\}$. Then $$ \sum_{p\in X} \res_p(\omega)=0. $$ \end{theorem} \begin{proof} Away from the singularities we have $\omega=f(z)\,dz$ where $f$ is holomorphic. Hence $\p \omega=0$ (as $dz\wedge dz=0$) and $\pbar\omega=0$ (as $f$ is holomorphic) so $d\omega=0$. Hence for any open subset $\Omega\subset X$ with smooth boundary and such that $\Omega\cup\p\Omega$ contains no pole we have $$ \int_{\p\Omega}\omega=\int_\Omega d\omega=0. $$ Choose a tiny disk $\Delta_p$ about each pole $p$ so that $$ \int_{\p\Delta_p}\omega = 2\pi i \res_p(\omega). $$ For $\Omega=X\setminus\bigcup_p\Delta_p$ we have $$ \int_{\p\Omega}\omega = \sum_p\int_{\p\Omega}\omega. $$ (See Theorem~4.8 on page~18 of~\cite{GRIFFITHS}.) \end{proof} \begin{corollary} Let $X$ be a compact Riemann surface and $f\in \cM(X)\setminus\{0\}$. Then $$ \sum_{p\in X} \Ord_p(f)=0. $$ \end{corollary} \begin{proof} Let $\omega = df/f$. Then $\Ord_p(f)=\res_p(\omega)$. \end{proof} \para \label{deg-1} The \jdef{degree} of a holomorphic map $f:X\to Y$ between compact Riemann surfaces is the sum of the local degrees over the preimage of a given point $y\in Y$. The local degree at$p\in X$ of a holomorphic map is the same as the order of the zero of of the local representative of the map in any holomorphic coordinates $z$ centered at $p$ and $w$ centered at $f(p)$. Thus when $Y=\P$, this local degree at $p\in X$ is $\Ord_p(f)$ if $f(p)=0$ and $-\Ord_p(f)$ if $f(p)=\infty$ so the corollary is also a corollary of the theorem that the degree of a holomorphic map $f:X\to Y$ is well defined, i.e. independent of the choice of $y\in Y$ used to defined it. \arap \begin{theorem}[Poincar\'e-Hopf] Let $X$ be a compact Riemann surface and $\omega\in \cM(X,K)\setminus\{0\}$. Then $$ \sum_{p\in X}\Ord_p(\omega)=-\chi(X) $$ where $\chi(X)$ is the Euler characteristic of $X$. \end{theorem} \begin{proof} In a suitable holomorphic coordinate centered at $p$ we have $$ \omega =z^\nu\,dz $$ where $\nu=\Ord_p(\omega)$ so where $z=x+iy=re^{i\theta}$ we have $$ \Re\omega= r^\nu(\cos(\nu\theta)\,dx-\sin(\nu\theta)\,dy) $$ so the degree of the map $$ \frac{\Re\omega}{|\Re\omega|}:\{|z|=\eps\}\to S^1 $$ is $-\Ord_p(\omega)$. The sum of these degrees is the Euler characteristic by the Poincar\'e Hopf Theorem. (See Theorem~6.5 on page~24 of~\cite{GRIFFITHS}). \end{proof} \begin{theorem}[Weil] Let $X$ be a compact Riemann surface and $f,g\in \cM(X)\setminus\{0\}$. Assume that$(f)$ and $(g)$ are disjoint. Then $$ \prod_{p_\in X} f(p)^{\Ord_p(g)}=\prod_{p_\in X} g(p)^{\Ord_p(f)}. $$ \end{theorem} \begin{proof} See~\cite{GH} page~242. \end{proof} \para The restriction of nonconstant holomorphic map $f:X\to Y$ to the complement of the preimage of the set of critical values is a $d$-sheeted covering space, i.e. if $V\subset Y$ is a sufficiently small open set containing no critical value of $f$, then $f^{-1}(V)$ is a disjoint union of $d$ open sets each mapped diffeomorphicaly to $V$ by $f$. The number $d$ is the degree of $f$ as defined in paragraph~\ref{deg-1}. Near each a critical point $f$ has the form $z\mapsto z^k$ where $k=\deg_p(f)$ is the local degree of the critical point. For this reason a nonconstant holomorphic map is called a \jdef{ramified cover} and the critical points of $f$ are called \jdef{ramification points}. The number $e_p(f)=\deg_p(f)-1$ is called the \jdef{ramification index} so that $e_p(f)>0$ if and only if $p$ is a ramification point of $f$. \arap \begin{theorem}[Riemann Hurwitz]\label{thm:R-H} If $f:X\to Y$ is a holomorphic map between compact Riemann surfaces of degree $d$, then $$ \chi(X)=d\chi(Y)-\sum_{p\in X}e_p(f) $$ where $\chi(X)$ is the Euler characteristic of $X$. \end{theorem} \begin{proof} Triangulate $X$ and $Y$ so that the ramification points are vertices and the map $f$ is simplicial and use the fact that the Euler characteristic $\chi$ is the number of vertices minus the number of edges plus the number of faces in any triangulation. See~\cite{GRIFFITHS} page~92. \end{proof} \section{Algebraic curves} \para\label{variety} An \jdef{projective algebraic variety} $X$ is a subset of a complex projective space $\P^N$ of form $$ X = \{x\in \P^N: F_1(x)=\cdots=F_k(x)=0\} \eqno(*) $$ where $F_1,\ldots,F_n$ are homogeneous polynomials. An \jdef{affine algebraic variety} is a subset of a complex affine space $\C^N$ of form $$ Y = \{y\in \C^N: f_1(y)=\cdots=f_k(y)=0\}. $$ For every polynomial $f(y_1,\ldots,y_N)$ there is a unique homogeneous polynomial $F(x_0,x_1,\ldots,x_N)$ of the same degree such that $$ f(y_1,\ldots,y_N)=F(1,y_1,\ldots,y_N), $$ so every affine variety corresponds to a projective variety. We use the term {\em algebraic variety} ambiguously to mean either {\em projective algebraic variety} or {\em affine algebraic variety}. (There is an abstract notion of {\em algebraic variety} which embraces both projective and affine algebraic varieties as special cases.) \arap \para An algebraic variety is \jdef{irreducible} iff it is not the union of two distinct varieties. Every algebraic variety $X$ may be written as $$ X=X_1\cup X_2\cup\cdots \cup X_k $$ where the $X_i$ are irreducible and $X_i\ne X_j$ for $i\ne j$; this decomposition is unique up to a reindexing. The varieties $X_i$ are called the \jdef{irreducible components} of $X$. \arap \para Let $X$ be an algebraic variety. A point $p\in X$ is called a \jdef{smooth point} iff it has a neighborhood $U$ such that $U\cap X$ is a holomorphic submanifold. A point which is not smooth point is called a \jdef{singular point}. For an irreducible variety the dimension of $U\cap X$ is independent of the choice of the smooth point $p$ and is called the \jdef{dimension} of $X$. An \jdef{algebraic curve} is an algebraic variety each of whose irreducible components has dimension one; a \jdef{plane algebraic curve} is an algebraic curve of codimension one, i.e. an algebraic curve which is a subset of $\P^2$. \arap \para Every compact Riemann surface admits a holomorphic embedding into $\P^3$. (See~\cite{GRIFFITHS} page~213.) A closed holomorphic submanifold of $\P^N$ is a smooth algebraic variety (Chow's Theorem, see~\cite{GH} page~187); hence every Riemann surface is isomorphic to a smooth algebraic curve. \arap \para Let $C\subseteq \P^N$ be an algebraic curve and $S\subseteq C$ be the set of singular points of $C$. A \jdef{normalization} of $C$ is a holomorphic map $$ \sigma:X\to\P^N $$ from a compact Riemann surface $X$ such that $\sigma(X)=C$, $\sigma^{-1}(S)$ is finite and the restriction $$ X\setminus\sigma^{-1}(S)\to C\setminus S $$ is bijective. (Since the restriction is a holomorphic map between Riemann surfaces it follows that it is biholomorphic.) \arap \begin{theorem}[Normalization Theorem] Every algebraic curve admits a normalization. The normalization is unique up to isomorphism in the following sense: If $\sigma:X\to\P^N$ and $\sigma':X'\to\P^N$ are normalizations of the same curve $C$, then the unique continuous map $\tau:X\to X'$ satisfying $\sigma'=\tau\circ\sigma$ is (a bijection and) biholomorphic. \end{theorem} \begin{proof} See~\cite{GRIFFITHS} page~5 and page~68. \end{proof} \rmk The number $k$ in equation~$(*)$ of paragraph~\ref{variety} is always greater than or equal to the codimension of $X$; a variety which has form~$(*)$ with $k$ equal to the codimension is called a \jdef{complete intersection}. The \jdef{twisted cubic} $$ x_0x_3=x_1x_2, \qquad x_0x_2=x_1^2, \qquad x_1x_3=x_2^2 $$ (so called because its affine part may be parameterized by the equations $x_i=t^i$) is a smooth algebraic curve in $\P^3$ which is not a complete intersection. \kmr \para Every plane algebraic curve $C$ is a complete intersection (see~\cite{GH} page~13) and thus has form $$ C = \{[x_0,x_1,x_2]\in \P^2:F(x_0,x_1,x_2)=0\} $$ where $F$ is a complex homogeneous polynomial; the polynomial $F$ is called a \jdef{defining polynomial} for $C$. Every curve has a defining polynomial of minimal degree, i.e. one with no repeated factors; this polynomial is unique up to multiplication by a nonzero constant. It is easy to see that a point of $C$ is a smooth point if and only if it is regular point of the minimal degree defining polynomial, and that an algebraic plane curve is irreducible if and only if it has a defining polynomial which is irreducible. \arap \para By \jdef{affine coordinates} at a point $p\in\P^2$ we mean coordinates $(x,y)$ of form $$ x=\frac{a_{10}x_0+a_{11}x_2+a_{12}x_2}{a_{00}x_0+a_{01}x_2+a_{02}x_2}, \qquad y=\frac{a_{20}x_0+a_{21}x_2+a_{22}x_2}{a_{00}x_0+a_{01}x_2+a_{02}x_2}, $$ where the matrix $(a_{ij})$ is invertible, the numerators vanish at $p$, and the denominators do not. (Every choice of affine coordinates establishes a correspondence between projective plane curves and affine plane curves as in paragraph~\ref{variety}. \arap \para Let $C\subseteq \P^2$ be an algebraic curve, $p\in C$, $(x,y)$ be affine coordinates at $p$, and $f(x,y)$ the defining polynomial of $C$ in these coordinates. Since $p\in C$ we have $f(0,0)=0$. We call $p$ a \jdef{$k$tuple point} of $C$ iff $d^jf(0,0)=0$ for $j=1,2,\ldots, k-1$ and $d^kf(0,0)\ne 0$. A $k$tuple point is also called a \jdef{simple point} if $k=1$, a \jdef{double point} if $k=2$, a \jdef{triple point} if $k=3$, etc. A point is a smooth point if and only if it is a simple point. Let $p$ be a $k$tuple point. The homogeneous polynomial $$ f_k(x,y):= \left.\frac{d^k}{dt^k} f(tx,ty)\right|_{t=0} $$ factors into linear factors. The point $p$ is called an \jdef{ordinary point} iff these factors are distinct. \arap \begin{theorem}\label{nodalNormalization} Let $X$ be a compact Riemann surface. Then there is an algebraic curve $C\subseteq\P^2$ and a normalization $\sigma:X\to C$ such that (1)~the map $\sigma$ is an immersion, and (2)~the only singularities of $C$ are ordinary double points. \end{theorem} \begin{proof} See~\cite{GRIFFITHS} page~213. \end{proof} \begin{theorem}[The Genus Formula] Let $C\subset\P^2$ be an irreducible plane curve whose only singularities are double points. Then $$ g=\frac{(d-1)(d-2)}{2}-\delta $$ where $g$ is the genus of its normalization, $d$ is the degree of its irreducible defining polynomial, and $\delta$ is the number of double points. \end{theorem} \begin{proof} Project $C$ onto a projective line $\P^1$ from a point not on $C$. Using suitable affine coordinates we see that the number of critical points of this projection is $d(d-1)$. Apply the Riemann Hurwitz formula (Theorem~\ref{thm:R-H}) to the composition of this projection with the normalization map. For more details see~\cite{GRIFFITHS} page~213. \end{proof} \begin{thebibliography}{99} \bibitem{GRIFFITHS} P. A. Griffiths: {\em Introduction to Algebraic Curves}, AMS Translations of Math. Monographs {\bf 76} 1989. \bibitem{GH} P. A. Griffiths \& J. Harris: {\em Principles of Algebraic Geometry}, Wiley Interscience, 1978. \end{thebibliography} \end{document}