Branched Coverings and the Riemann-Hurwitz Formula

Branched coverings are holomorphic maps between Riemann surfaces that are covering maps away from finitely many branch points. The Riemann-Hurwitz formula relates the genera of the surfaces to the branching data.

Branched Coverings

Definition10.7Branched covering

A non-constant holomorphic map $f: S \to T$ between compact Riemann surfaces is a branched covering of degree $n$ if $|f^{-1}(q)| = n$ for all but finitely many $q \in T$ . The exceptional points are the branch values and their preimages are the branch points (or ramification points).

Definition10.8Ramification index

At a point $p \in S$ , the ramification index $e_p$ is the local degree of $f$ at $p$ : in local coordinates, $f$ looks like $z \mapsto z^{e_p}$ . The point $p$ is a ramification point if $e_p > 1$ . The branch divisor is $R = \sum_p (e_p - 1)\cdot p$ .

ExampleExamples of branched coverings

$f(z) = z^n: \hat{\mathbb{C}} \to \hat{\mathbb{C}}$ has degree $n$ , branched at $0$ and $\infty$ (each with $e = n$ ).
The projection $(z, w) \mapsto z$ from the curve $w^2 = z^3 - z$ to $\hat{\mathbb{C}}$ has degree $2$ , branched at $z = -1, 0, 1, \infty$ .
A polynomial $p: \mathbb{C} \to \mathbb{C}$ of degree $n$ extends to a branched covering $\hat{\mathbb{C}} \to \hat{\mathbb{C}}$ with total branching $\sum (e_p - 1) = 2n - 2$ .

The Riemann-Hurwitz Formula

Theorem10.4Riemann-Hurwitz formula

Let $f: S \to T$ be a branched covering of degree $n$ between compact Riemann surfaces of genera $g_S$ and $g_T$ . Then

$2g_S - 2 = n(2g_T - 2) + \sum_{p \in S}(e_p - 1).$

Equivalently, the Euler characteristics satisfy $\chi(S) = n\chi(T) - \sum(e_p - 1)$ .

Proof

Triangulate $T$ with vertices at the branch values. Pull back to a triangulation of $S$ :

Faces: $n \cdot F_T$ (each face has $n$ preimages).
Edges: $n \cdot E_T$ (each edge has $n$ preimages).
Vertices: $n \cdot V_T - \sum(e_p - 1)$ (at branch points, preimages are identified).

Therefore $\chi(S) = n(V_T - E_T + F_T) - \sum(e_p - 1) = n\chi(T) - \sum(e_p-1)$ .

Since $\chi = 2 - 2g$ : $2-2g_S = n(2-2g_T) - \sum(e_p-1)$ . $\blacksquare$

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Applications

ExampleGenus of hyperelliptic curves

The hyperelliptic curve $w^2 = \prod_{i=1}^{2g+2}(z-a_i)$ defines a double cover ( $n=2$ ) of $\hat{\mathbb{C}}$ ( $g_T = 0$ ) branched at the $2g+2$ points $a_i$ (each with $e = 2$ ). The Riemann-Hurwitz formula gives:

$2g_S - 2 = 2(0-2) + (2g+2) \cdot 1 = -4 + 2g + 2 = 2g - 2.$

So $g_S = g$ , confirming the genus.

ExampleBranching of polynomial maps

A polynomial $p(z)$ of degree $n$ gives a map $\hat{\mathbb{C}} \to \hat{\mathbb{C}}$ with $g_S = g_T = 0$ . Riemann-Hurwitz: $-2 = n(-2) + R$ where $R = \sum(e_p - 1)$ . So $R = 2n - 2$ . For $p(z) = z^n$ : two branch points ( $0$ and $\infty$ ), each contributing $n - 1$ , totaling $2(n-1) = 2n-2$ .

The Genus Formula for Plane Curves

Theorem10.5Genus of a smooth plane curve

A smooth projective plane curve of degree $d$ (defined by a homogeneous polynomial of degree $d$ in $\mathbb{CP}^2$ ) has genus

$g = \frac{(d-1)(d-2)}{2}.$

RemarkProof via Riemann-Hurwitz

Project the curve onto a line in $\mathbb{CP}^2$ . This gives a branched covering of degree $d$ of $\hat{\mathbb{C}}$ . The number of branch points (counted with multiplicity) is computed from the discriminant, and the Riemann-Hurwitz formula yields the genus.