nLab branched cover of the Riemann sphere

Definition

A branched cover of the Riemann sphere $\mathbb{C}P^1$ is a compact connected Riemann surface $\Sigma$ equipped with a non-constant holomorphic function

$$\phi: \Sigma \longrightarrow \mathbb{C}P^1 \,.$$

If we think of $\Sigma$ as retrievable from its field of meromorphic functions (see for example Mumford), then each such map $\phi$ is dual to a field extension $\mathbb{C}(t) \hookrightarrow Func(\Sigma)$; this is an algebraic field extension. The dual map $\phi$ is surjective, and restricts to a covering space projection after removing all the (finitely many) ramification points? of $\phi$.

Properties

Riemann surfaces

By the Riemann existence theorem, every connected compact Riemann surface admits the structure of a branched cover of the Riemann sphere. (MO discussion)

References

• Brian Ossermann, Branched covers of the Riemann sphere (pdf)

• David Mumford, Curves and their Jacobians, U. Michigan Press (1977).