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\title{Riemann Surfaces and Algebraic Curves}
\author{JWR}
\date{Tuesday December 11, 2001, 9:03 AM}

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\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}
