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Homogenizing the affine equation gives the projective closureThis is the Klein quartic. It is smooth: if all three projective partial derivatives vanished, thenMultiplying them gives $28X^3{Y}^3{Z}^3=0$. If one coordinate is zero, the displayed equations force all three to vanish, which is impossible in projective space. Thus $C$ is a smooth compactification of $X^{\prime }$. The smooth compactification compatible with the extended coordinate functions is its normalization, and since $C$ is already smooth it is biholomorphic to $C$.

For a direct genus computation, consider the extended coordinate map ${\pi }_x:X\to {\mathbb{C}}_{\infty }$. A generic $x$ leaves the cubic equationso $\mathrm{deg}{\pi }_x=3$. In the affine part, ramification occurs where $F_y=0$. Solving $F=F_y=0$ gives $(0,0)$ and the seven pointsAt the origin, $x=-y^3+O(y^6)$, so the ramification index of a holomorphic map is three and the contribution is two. At each of the other seven points, $F_x=-7/2\ne 0$, $F_y=0$, and $F_{yy}=6y\ne 0$, so the ramification index is two and each contributes one.

There are two points at infinity. Near $P=[1:0:0]$, in coordinates $u=Y/X$, $v=Z/X$, the equation isand $1/x=v$, so $P$ is unramified. Near $Q=[0:1:0]$, in coordinates $r=X/Y$, $s=Z/Y$, the equation isThus $s=-r^3+O(r^{10})$ and $1/x=s/r=-r^2+O(r^9)$, so $Q$ has ramification index two and contributes one. The total ramification is thereforeThe Riemann-Hurwitz formula for the degree-three map to the sphere givesand henceThis also agrees with the genus of a smooth plane curve of degree four.

Let $F:X\to Y$ be a nonconstant holomorphic map of degree $d$ between compact connected Riemann surfaces. At $p\in X$, let $e_p$ be the ramification index of a holomorphic map. Choose a triangulation of $Y$ containing every branch value among its vertices, and lift it to $X$.

If the target triangulation has $V$ vertices, $E$ edges, and $T$ triangles, then the lifted triangulation has $dE$ edges and $dT$ triangles, because no branch value lies in an edge or face interior. For a target vertex $v$, local degree counting givesConsequently the number of vertices above $v$ isWriting $R={\sum }_{p\in X}(e_p-1)$, the Euler characteristic is thereforeSince a compact orientable surface of genus $g$ has Euler characteristic $2-2g$, this Triangulation proof of the Riemann-Hurwitz formula givesor equivalently

Now extend a cubic polynomial $f:\mathbb{C}\to \mathbb{C}$ to a degree-three holomorphic mapof the Riemann sphere. Both genera are zero, so Riemann--Hurwitz givesThe point at infinity is totally ramified with $e_{\infty }=3$, contributing two. Thus the finite points contribute exactly two. There are consequently two cases.

If there is one finite ramification point $\alpha$, it has index three. Choose an affine source map $h$ sending zero to $\alpha$. Thenfor some $c\ne 0$, and integration givesAn affine target map subtracts $d$ and rescales by $3/c$, yielding $z^3$.

Otherwise there are two distinct finite ramification points, each of index two. Choose an affine source map $h$ sending $-1$ and $1$ to them. The derivative of $f\circ h$ is then a nonzero multiple of $z^2-1$:HenceThe affine target map $g(w)=(w-d)/c$ givesTherefore the affine normal forms of a complex cubic polynomial are precisely

In local coordinates centred at $p$ and $f(p)$, a nonconstant rational map has the formThe point $p$ is a ramification point of a holomorphic map when its ramification index of a holomorphic map satisfies $e_p>1$. A branch value of a holomorphic map, also called a branch point in the target, is a value $f(p)$ of some ramification point.

Let $B$ be the finite set of branch values and put $R=f^{-1}(B)$. For every $w\notin B$, all points of $f^{-1}(w)$ have local degree one, so the holomorphic inverse function theorem supplies disjoint neighbourhoods on which $f$ is biholomorphic. Compactness of the fibre lets their target neighbourhoods be intersected to one evenly covered neighbourhood of $w$. Henceis an unramified covering map. Here the displayed restriction requires $R=f^{-1}(B)$; if “ramification point” is reserved only for points with $e_p>1$, then the source deletion must be written $f^{-1}(B)$.

The Monodromy theorem says that analytic continuations along endpoint-fixed homotopic paths have the same terminal germ. Equivalently here, the path lifting theorem lifts a loop $\gamma$ based at $w\notin B$ from each point of $f^{-1}(w)$; taking the endpoint of each lift permutes that fibre. The permutation depends only on $[\gamma ]\in {\pi }_1({\mathbb{C}}_{\infty }\setminus B,w)$, giving the monodromy group of a covering.

For the stated function, make the Möbius change of coordinateThenThe critical points are $t=0,\pm 1/\sqrt{2},\infty$, with branch valuesA loop around $0$ interchanges the two roots born from $t=0$, so its branch cycle is a transposition. A loop around $-1/4$ simultaneously interchanges the two pairs that collide there, so its branch cycle is a product of two disjoint transpositions. With a suitable labelling these areThey generate a transitive group of order eight; their product is a four-cycle. Therefore the full monodromy group isthe dihedral group in its action on the four vertices of a square.