nLab
branched cover of the Riemann sphere

Contents

Definition

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

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

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

Properties

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).

Revised on July 22, 2014 04:03:02 by Toby Bartels (173.190.166.59)