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Suppose a nonconstant holomorphic map $f:X\to {\mathbb{P}}^1$ from a connected compact Riemann surface has degree of a holomorphic map two and exactly four distinct branch values of a holomorphic map. The valency theorem forces each of their fibres to contain one ramification point with $e_p=2$, a simple ramification point of a holomorphic map. HenceThis determines the genus without recognizing a torus in a drawing.

A concrete realization is the compact Riemann surface obtained by adding the two points at infinity to the even-degree hyperelliptic model given by the algebraic curveProjection to $z$ has degree of a holomorphic map two. At a simple zero $a$ of $P$, $w$ is a local coordinate and $z-a=w^2/P^{\prime }(a)+O(w^4)$, so $e_p=2$. At infinity set $u=1/z$ and $s=w/z^2$: then $s^2=(1-4u^2)(1-u^2)$. There are two points with $u=0$, $s=\pm 1$, each unramified since $u$ is a local coordinate. The four finite roots are therefore all the branch values of a holomorphic map.^[ref]

Choose a triangulation of a surface on the Riemann sphere with these four branch values of a holomorphic map as vertices, combinatorially the boundary of a tetrahedron. Its counts are $V=4,E=6,F=4$. Upstairs, each vertex has one preimage and each open edge and face has two lifts: $V_X=4,E_X=12,F_X=8$, giving Euler characteristic $4-12+8=0$. This is a cell schematic, not a geometric immersion of $X$. The Triangulation proof of the Riemann-Hurwitz formula applies the same counting idea after refining a general triangulation of a surface around the branch values of a holomorphic map.^[ref]

A germ of a holomorphic function at $z\in D$ is an equivalence class $[f{]}_z$ of pairs consisting of a neighbourhood $U$ of $z$ and a holomorphic function $f:U\to \mathbb{C}$. Two pairs represent the same germ if their functions agree on some neighbourhood of $z$.

The space of germs of holomorphic functions isFor every holomorphic $f$ on an open $U\subseteq D$, the setis declared open; these sets form a basis. The forgetful mapmaps each ${\tilde{U}}_f$ homeomorphically onto $U$. The inverses of these restrictions give the charts defining the complex structure, so $\pi$ is a local biholomorphism.

The evaluation map isIt is well-defined by the germ equivalence relation. On the chart ${\tilde{U}}_f$, its coordinate expression iswhich is holomorphic. Hence $\mathcal{E}$ is analytic, as stated in evaluation map on a space of germs.

Now putThis gives the germ surface of the square root of z to the eighth minus one. An explicit gluing description is obtained by pairing the eight roots into four adjacent pairs and cutting the plane along four disjoint arcs joining the members of each pair. On the complement of the cuts choose one branch $q(z)$ of $\sqrt{z^8-1}$. Take two copies, labelled by $q$ and $-q$, and glue the upper bank of each cut in one copy to the lower bank in the other, and conversely. The cut interiors are restored by the gluing, while the eight endpoints remain absent because they are not in $D$.

On this surface defineFor each $(z,w)\in R$, the holomorphic function $z^8-1$ is nonzero near $z$, so it has a unique local square root $q$ with $q(z)=w$. DefineThis is well-defined, injective, and analytic in the displayed local sheets; its inverse on its image is $[q{]}_z\mapsto (z,q(z))$. Thus $\Phi$ is the required analytic embedding, and it intertwines both the forgetful and evaluation maps.

To compactify, first add one point above each of the eight roots of unity. These are simple branch points of the double cover. Since the polynomial has even degree eight, the even-degree hyperelliptic model has two distinct, unbranched points above infinity. Consequently

Finally apply the Riemann-Hurwitz formula to the degree-two meromorphic map $\bar{\pi }:\overline{R}\to {\mathbb{C}}_{\infty }$. Its only ramification consists of the eight simple finite branch points, soThereforeThis is the compactification of y squared equals x to the eighth minus one.

For a divisor $D$ on a smooth projective curve $X$, defineThe Riemann-Roch theorem stateswhere $K_X$ is the divisor of a nonzero rational differential, $g$ is the genus of $X$, and $\mathrm{deg}D$ is the sum of the divisor coefficients because the ground field is algebraically closed.

Under the coordinate change $u=1/x$ and $v=y/x^n$, we haveHenceThis is a polynomial, and its value at zero is the nonzero leading coefficient of $f$. Thus the even-degree hyperelliptic model has two points $P_{+},P_{-}$ above $u=0$, distinguished by the two values of $v$.

Consider the rational differentialAt a finite branch point $x=a$, square-freeness gives a local parameter $y$, with $x-a$ a nonzero constant times $y^2$ to first order. Hence $dx/y$ is regular and nonzero there. It is also regular and nonzero at finite unramified points.

At infinity,soBecause $v(P_{\pm })\ne 0$ and $u$ is a local parameter at each point, $\omega$ has a zero of order $n-2$ at both. Therefore the canonical divisor of an even-degree hyperelliptic curve isand

Since $\mathrm{deg}{K}_X=2g-2$, it follows that

The function $x$ has a simple pole at each of $P_{+}$ and $P_{-}$. ConsequentlyThese $n-1$ functions are linearly independent. Riemann-Roch with $D=K_X$ gives $\ell (K_X)=g=n-1$, so

With respect to this basis, the canonical map of a hyperelliptic curve isIt identifies $(x,y)$ with $(x,-y)$ for general $x$, and hence is not an embedding.

If $X$ embedded in ${\mathbb{P}}^2$, its smooth plane image would have degree $d\ge 4$: degrees one and two have genus zero, and degree three has genus one, whereas $g=n-1\ge 2$. But the canonical map of a smooth plane curve of degree $d\ge 4$ is an embedding by adjunction, contradicting the preceding calculation. Therefore