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Let $C$ be an irreducible algebraic plane curve of degree $n$, given by an equation $f ( X , Y ) = 0$ where $f$ is an irreducible bivariate polynomial of degree $n$ over a ground field $k$ (cf. also [[Algebraic curve|Algebraic curve]]). For simplicity $k$ is assumed to be algebraically closed (cf. also [[Algebraically closed field|Algebraically closed field]]), although most of what is said below can be suitably generalized without that assumption. For the basic field theory involved, see [[#References|[a6]]] (or the modernized version [[#References|[a4]]]) and [[#References|[a3]]]. For much of the geometry to be discussed, see [[#References|[a5]]] and [[#References|[a8]]]; in particular, for the idea of points at infinity of $C$, see [[#References|[a1]]]. For an interplay between the geometry and the algebra, see [[#References|[a2]]].

One starts by analyzing when the curve $C$ can be rationally parametrized. For example, the unit circle $X ^ { 2 } + Y ^ { 2 } = 1$ has the rational parametrization

\begin{equation*} X = \frac { 1 - t ^ { 2 } } { 1 + t ^ { 2 } } , Y = \frac { 2 t } { 1 + t ^ { 2 } }. \end{equation*}

Likewise, the cuspidal cubic $Y ^ { 2 } = X ^ { 3 }$ has the rational parametrization $X = t ^ { 2 }$ and $Y = t ^ { 3 }$. However, the non-singular cubic $Y ^ { 2 } = X ^ { 3 } - 1$ does not have any rational parametrization. To obtain the parametrization of the circle, one cuts it by a line of slope $t$ through the point $( - 1,0 )$ and notes that it meets the circle in the variable point

\begin{equation*} \left( \frac { 1 - t ^ { 2 } } { 1 + t ^ { 2 } } , \frac { 2 t } { 1 + t ^ { 2 } } \right). \end{equation*}

For the cuspidal cubic one takes a line through the cusp $( 0,0 )$ and notes that it meets the cubic in the variable point $( t ^ { 2 } , t ^ { 3 } )$. This works because a line meets a circle in $2$ points, and it meets a cubic in $3$ points. In case of a cuspidal cubic two intersections are absorbed in the cusp. In case of a non-singular cubic there is no such point for the absorption. Generalizing this one can show that the curve $C$ cannot have more than $( n - 1 ) ( n - 2 ) / 2$ double points and if it does have that many, then it can be parametrized rationally. To this end one first notes that a bivariate polynomial of degree $m$ has

\begin{equation*} \left( \begin{array} { c } { m + 2 } \\ { 2 } \end{array} \right) = \frac { ( m + 2 ) ( m + 1 ) } { 2 } \end{equation*}

coefficients and hence the dimension of the system $S$ of curves of degree $n - 2$ passing through $( n - 1 ) ( n - 2 ) / 2$ double points of $C$ is $n ( n - 1 ) / 2 - 1 - ( n - 1 ) ( n - 2 ) / 2 = n - 2$. Next, by the [[Bezout theorem|Bezout theorem]] (which is the oldest theorem in [[Algebraic geometry|algebraic geometry]]), a curve of degree $n$ and a curve of degree $m$, having no common component, meet in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007034.png"/> points, counted properly. In the proper counting a double point of $C$ should be counted twice. Thus, the number of free points in which $C$ meets a curve $D$ of degree $n - 2$ passing through the $( n - 1 ) ( n - 2 ) / 2$ double points of $C$ is $n ( n - 2 ) - ( n - 1 ) ( n - 2 ) = n - 2$. Since $n - 2$ is also the dimension of the system $S$, the members of $S$ which pass through $n - 3$ fixed simple points of $C$ form a pencil, i.e., a one-parameter family $D _ { t }$, a variable member of which meets $C$ in one variable point whose coordinates are single-valued, and hence rational, functions of $t$. If $C$ had an extra double point, then one can take a value $t_0$ of $t$ so that $D _ { t _ { 0 } }$ goes through it and this would make the properly counted intersections of $C$ and $D _ { t _ { 0 } }$ to be $&gt; n ( n - 2 )$, contradicting the Bezout theorem because $C$ is irreducible and $D _ { t _ { 0 } }$ has smaller degree.

Provisionally defining the genus $g$ of $C$ (cf. also [[Genus of a curve|Genus of a curve]]) by

\begin{equation*} g = \frac { ( n - 1 ) ( n - 2 ) } { 2 } -\#\text{double points}, \end{equation*}

one always has $g \geq 0$, and $g = 0 \Rightarrow C$ is rational, i.e., has a rational parametrization. To make the reverse implication $\Leftarrow $ also true, one must learn to count the double points properly. To begin with, one must include singularities of $C$ at infinity. Next, by looking at the curve $Y ^ { e } = X ^ { d }$, where $e &gt; d$ with $\operatorname { gcd } ( e , d ) = 1$, which is obviously rational and has a $d$-fold point at the origin and an $( e - d )$-fold point at infinity as its only singularities, one decides to count a $d$-fold point as $d ( d - 1 ) / 2$ double points. Before discussing infinitely near singularities, one notes that the degree $n$ of $C$ can be geometrically characterized as the number of points in which a general line meets it.

Likewise, the multiplicity $d$ of a point $P$ of $C$ can be characterized geometrically by saying that $n - d$ is equal to the number of points in which a generic line through $P$ meets $C$ outside $P$; $P$ is a simple or singular point of $C$ according as $d = 1$ or $d &gt; 1$. Algebraically, by translating the coordinates one may assume $P$ to be the origin $( 0,0 )$, and then $d$ is the order of $f$, i.e., $f$ has terms of degree $d$ but none of degree $&lt; d$. By making the quadratic transformation $X = X ^ { \prime }$ and $Y = X ^ { \prime } Y ^ { \prime }$ one gets $f ( X ^ { \prime } , X ^ { \prime } Y ^ { \prime } ) = X ^ { \prime d } f ^ { \prime } ( X ^ { \prime } , Y ^ { \prime } )$, where $C ^ { \prime }$: $f ^ { \prime } ( X ^ { \prime } , Y ^ { \prime } ) = 0$ is the proper transform of $C$. The exceptional line $X ^ { \prime } = 0$ meets $C ^ { \prime }$ in points $P _ { 1 } , \ldots , P _ { h }$ whose multiplicities $d _ { 1 } , \ldots , d _ { h }$ add up to $\leq d$. These are the points of $C$ in the first neighbourhood of $P$. Points in the first neighbourhoods of these points are the points of $C$ in the second neighbourhood of $P$, and so on. It is easily seen that all points in a high enough neighbourhood of $P$ are simple. Now $P$ is counted as $\delta ( P )$ double points, where

\begin{equation*} \delta ( P ) = \sum \frac { d ( Q ) ( d ( Q ) - 1 ) } { 2 } \end{equation*}

with the summation extended over all points $Q$ in the various neighbourhoods of $P$, including $P$; here $d ( Q )$ is the multiplicity of $Q$; clearly: $\delta ( P ) = 0$ $\Leftrightarrow$ $P$ is a simple point of $C$. One arrives at the exact genus formula

\begin{equation*} g = \frac { ( n - 1 ) ( n - 2 ) } { 2 } - \sum \delta ( P ), \end{equation*}

with summation over all points $P$ of $C$. One always has $g \geq 0$; and $g = 0 \Leftrightarrow C$ is rational.

It turns out that $g$ is a birational invariant of $C$, i.e., it remains unchanged when $C$ undergoes a birational transformation (cf. also [[Birational morphism|Birational morphism]]). The residue class ring of the polynomial ring $k [ X , Y ]$ modulo the ideal generated by $f ( X , Y )$ is the affine coordinate ring of $C$ and is denoted by $k [ C ]$. Note that $k [ C ] = k [ x , y ]$ where $x$, $y$ are the images of $X$, $Y$ in $k [ C ]$. The quotient field $k ( C ) = k ( x , y )$ of $k [ C ]$ is the function field of $C$. A birational correspondence between curves $C$ and $C ^ { * }$ is an almost one-to-one correspondence; it is given by a $k$-isomorphism between $k ( C )$ and $k ( C ^ { * } )$. So one should be able to define $g$ directly in terms of $k ( C )$. Following C.G.J. Jacobi one takes any differential of $k ( C )$ (cf. also [[Differential field|Differential field]]), i.e., an expression of type $ud v$ with $u , v \in k ( C )$, and shows that if the differential is not zero, then the number of its zeros minus the number of its poles equals $2 g - 2$. Having brought the point $P$ of $C$ to the origin, its local ring $R ( P )$ is defined to be the subring of $k ( C )$ consisting of all quotients $r ( x , y ) / s ( x , y )$ where $r ( X , Y )$, $s ( X , Y )$ are polynomials with $s ( 0,0 ) \neq 0$ (cf. also [[Local ring|Local ring]]); its unique [[Maximal ideal|maximal ideal]] $M ( R ( P ) )$ consists of the above quotients with $r ( 0,0 ) = 0$. Let ${\frak C} ( P )$ be the conductor of $R ( P )$, i.e., the largest ideal in $R ( P )$ which remains an ideal in the integral closure $R ^ { \prime } ( P )$ of $R ( P )$ in $k ( C )$. It can be shown that $\delta ( P )$ is the length of ${\frak C} ( P )$ in $R ( P )$, i.e., the maximal length of strictly increasing chains of ideals $\mathfrak { C } ( P ) = I _ { 0 } \subset \ldots \subset I _ { \delta } = R ( P )$ in $R ( P )$; moreover, $2 \delta ( P )$ is the length of ${\frak C} ( P )$ in $R ^ { \prime } ( P )$, which is a ubiquitous result having two dozen proofs in the literature. The ring $R ^ { \prime } ( P )$ has a finite number of maximal ideals and localizing $R ^ { \prime } ( P )$ at them gives discrete valuation rings; as $P$ varies over all points of $C$, including those at infinity, these discrete valuation rings vary over the [[Riemann surface|Riemann surface]] $\Re ( C )$ of $C$, i.e., the set of all discrete valuation rings whose quotient field is $k ( C )$ and which contain $k$. Let $\mathfrak { R } ( C , P )$ denote the localizations of $R ^ { \prime } ( P )$ at the various maximal ideals in $R ^ { \prime } ( P )$ (cf. also [[Localization in a commutative algebra|Localization in a commutative algebra]]); one calls $P$ the centre on $C$ of the members of $\mathfrak { R } ( C , P )$; note that $R ^ { \prime } ( P ) = R ( P )$ $\Leftrightarrow$ $P$ is a simple point of $C$, and hence for all except a finite number of points of $C$, the set $\mathfrak { R } ( C , P )$ has exactly one member. For any $T \in \Re ( C )$ and non-zero $r , s \in k ( C )$ one puts

\begin{equation*} \operatorname { ord } _ { T } ( r / s ) = \lambda - \mu, \end{equation*}

with $r T = M ( T ) ^ { \lambda }$ and $s T = M ( T ) ^ { \mu }$; take $\tau \in T$ with $\tau T = M ( T )$ and define

\begin{equation*} \operatorname { ord } _ { T } ( u d v ) = \operatorname { ord } _ { T } ( u d v / d \tau ); \end{equation*}

one calls $\tau$ a uniformizing parameter of $T$. Now the number of zeros minus number of poles of $ud v$ equals $\sum  \text { ord }_{ T } ( u d v )$ taken over all $T$ in $\Re ( C )$. For any point $P$ of $C$, not at infinity, one has Dedekind's formula

\begin{equation*} f _ { Y } ( x , y ) R ^ { \prime } ( P ) = \mathfrak { C } ( P ) \mathfrak { D } ( P , x ), \end{equation*}

where $\mathfrak { D } ( P , x )$ is the different ideal in $R ^ { \prime } ( P )$ defined by saying that $\mathfrak { D } ( P , x ) T = M ( T ) ^ { \epsilon }$ with $\epsilon = \operatorname { ord } _ { T } ( d x / d \tau )$ for every $T \in \Re ( C , P )$.

For $i = 1,2$, let $C_i$ be an irreducible algebraic plane curve such that $k ( C )$ is a finite separable algebraic field extension of $k ( C _ { i } )$ of field degree $\nu _ { i }$ (cf. also [[Extension of a field|Extension of a field]]; [[Separable extension|Separable extension]]). This defines a $( \nu _ { 1 } , \nu _ { 2 } )$ correspondence between $\mathfrak { R } ( C _ { 1 } )$ and $\mathfrak { R } ( C _ { 2 } )$, and hence between $C _ { 1 }$ and $C _ { 2 }$; namely, $T _ { 1 } \in \Re ( C _ { 1 } )$ and $T _ { 2 } \in \Re ( C _ { 2 } )$ correspond if and only if for some $T \in \Re ( C )$ one has $T \cap k ( C _ { 1 } ) = T _ { 1 }$ and $T \cap k ( C _ { 2 } ) = T _ { 2 }$. Let $g_i$ be the genus of $C_i$, let the different $\mathfrak { D } ( C , C _ { i } )$ be the integer-valued function on $\Re ( C )$ whose value at $T$ in $\Re ( C )$ is given by $\operatorname { ord } _ { T } ( d \tau _ { i } / d \tau )$, where $\tau_i$ is a uniformizing parameter of $T \cap k ( C _ { i } )$, and let $\mathfrak { D } _ { i } = \sum \mathfrak { D } ( C , C _ { i } ) ( T )$ with summation over all $T \in \Re ( C )$. Then the Riemann–Hurwitz formula says that

\begin{equation*} 2 g - 2 = \nu _ { i } ( 2 g _ { i } - 2 ) + \mathfrak { D } _ { i }, \end{equation*}

and this gives rise to the Zeuthen formula

\begin{equation*} \nu _ { 1 } ( 2 g _ { 1 } - 2 ) + \mathfrak { D } _ { 1 } = \nu _ { 2 } ( 2 g _ { 2 } - 2 ) + \mathfrak { D } _ { 2 }. \end{equation*}

Now suppose there is a $k$-isomorphism $\phi : k ( C _ { 1 } ) \rightarrow k ( C _ { 2 } )$. Then $T \in \Re ( C )$ is called a fixed place of the correspondence if $T \cap k ( C _ { 2 } ) = \phi ( T \cap k ( C _ { 1 } ) )$. The Chasles–Cayley–Brill formula says that under suitable conditions, the number of these, counted properly, equals $\nu _ { 1 } + \nu _ { 2 } + 2 \gamma g$, where the integer $\gamma$ is called the valence of the correspondence. For details see [[#References|[a7]]], pp. 189–194.

In case $k$ is the field of complex numbers, to describe Riemann's approach one topologizes $\Re ( C )$ to make it into a compact orientable two-dimensional real manifold, and hence into a sphere with $g$ handles (cf. also [[Riemann surface|Riemann surface]]). Likewise, $\mathfrak { R } ( C _ { 1 } )$ is made into a sphere with $g_1$ handles. Triangulate $\mathfrak { R } ( C _ { 1 } )$ by including all the branch points as vertices, and lift this triangulation to a triangulation of $\Re ( C )$. Let $( V _ { 1 } , E _ { 1 } , F _ { 1 } )$ and $( V , E , F )$ be the vertices, edges, faces of the bottom and top triangulations respectively. Then $V = \nu _ { 1 } V _ { 1 } - \mathfrak { D } _ { 1 }$, $E = \nu _ { 1 } E _ { 1 }$, $F = \nu _ { 1 } F _ { 1 }$, and hence by the Euler–Poincaré theorem one obtains

\begin{equation*} 2 g - 2 = \nu _ { 1 } ( 2 g _ { 1 } - 2 ) + \mathfrak { D } _ { 1 }. \end{equation*}

This proves the birational invariance of $g$ and the Riemann–Hurwitz formula. For details, see [[#References|[a2]]] and [[#References|[a4]]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top"> S.S. Abhyankar, "What is the difference between a parabola and a hyperbola" ''Math. Intelligencer'' , '''10''' (1988) pp. 36–43</td></tr><tr><td valign="top">[a2]</td> <td valign="top"> S.S. Abhyankar, "Algebraic geometry for scientists and engineers" , Amer. Math. Soc. (1990) {{MR|1075991}} {{ZBL|0709.14001}} {{ZBL|0721.14001}} </td></tr><tr><td valign="top">[a3]</td> <td valign="top"> S.S. Abhyankar, "Field extensions" G.A. Pilz (ed.) A.V. Mikhalev (ed.) , ''Handbook of the Heart of Algebra'' , Kluwer Acad. Publ. (to appear)</td></tr><tr><td valign="top">[a4]</td> <td valign="top"> C. Chevalley, "Introduction to the theory of algebraic functions of one variable" , ''Math. Surveys'' , '''6''' , Amer. Math. Soc. (1951) {{MR|0042164}} {{ZBL|0045.32301}} </td></tr><tr><td valign="top">[a5]</td> <td valign="top"> J.L. Coolidge, "A treatise on algebraic plane curves" , Clarendon Press (1931) {{MR|0120551}} {{MR|1522528}} {{ZBL|57.0820.06}} </td></tr><tr><td valign="top">[a6]</td> <td valign="top"> R. Dedekind, H. Weber, "Theorie der algebraischen Functionen einer Veränderlichen" ''Crelle J.'' , '''92''' (1882) pp. 181–290</td></tr><tr><td valign="top">[a7]</td> <td valign="top"> S. Lefschetz, "Algebraic geometry" , Princeton Univ. Press (1953)</td></tr><tr><td valign="top">[a8]</td> <td valign="top"> F. Severi, "Vorlesungen über algebraische Geometrie" , Teubner (1921) {{MR|0245574}} {{ZBL|48.0687.01}} </td></tr></table>