Contents

complex geometry

# Contents

## 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).

Last revised on January 6, 2019 at 15:34:32. See the history of this page for a list of all contributions to it.