---
tag: YARENZRXTD
kind: concept
name: Branched covering
msc: [57M12, 30F10]
sources:
  - url: https://en.wikipedia.org/wiki/Branched_covering
same_as:
  wikidata: Q4956410
  wikipedia: Branched covering
---

## Summary

A branched covering is a map that is a [[covering map|3XZRRE165N]] except over a discrete set of branch values, where several sheets come together.
For [[Riemann surfaces|5K0N8QMV6Q]], every non-constant proper [[holomorphic map|T19P7J53A1]] is a branched covering of finite degree.
Branched coverings of the [[Riemann sphere|0CDMXGPSES]] describe compact Riemann surfaces as the surfaces of multivalued algebraic functions, and the Riemann–Hurwitz formula computes their genus.

## Explanations

### L1 human

The map $z \mapsto z^2$ on the complex plane is two-to-one away from $0$: each nonzero $w$ has two square roots.
Over $w = 0$ the two sheets meet in the single point $z = 0$, a [[branch point|XP5JC1VXKZ]].
Away from such special points the map is an honest covering, like a stack of identical sheets.
A branched covering is a map that behaves like this: a covering, except at isolated points where sheets are glued together like $z \mapsto z^k$.