geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
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
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 $\phi$.
By the Riemann existence theorem, every connected compact Riemann surface admits the structure of a branched cover of the Riemann sphere. (MO discussion)
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 20:34:32. See the history of this page for a list of all contributions to it.