Branched covering

In mathematics, branched covering is a term used to describe a map that is almost a covering map, except on a small set.

In topology

In topology, a map is a branched covering if it is a covering map everywhere except for a nowhere-dense set known as the branch set. Examples include the map from a wedge of circles to a single circle, where the map is a homeomorphism on each circle.

In algebraic geometry

In algebraic geometry, the term branched covering is used to describe morphisms f from an algebraic variety V to another one W, the two dimensions being the same, and the typical fibre of f being of dimension 0.

In that case, there will be an open set W′ of W (for the Zariski topology) that is dense in W, such that the restriction of f to W′ (from V′ = f⁻¹(W′) to W′, that is) is unramified. Depending on the context, we can take this as local homeomorphism for the strong topology, over the complex numbers, or as an étale morphism in general (under some slightly stronger hypotheses, on flatness and separability). Generically, then, such a morphism resembles a covering space in the topological sense. For example, if V and W are both Riemann surfaces, we require only that f is holomorphic and not constant, and then there is a finite set of points P of W, outside of which we do find an honest covering

V′ → W′.

Ramification locus

The set of exceptional points on W is called the ramification locus (i.e. this is the complement of the largest possible open set W′). In general monodromy occurs according to the fundamental group of W′ acting on the sheets of the covering (this topological picture can be made precise also in the case of a general base field).

Kummer extensions

Branched coverings are easily constructed as Kummer extensions, i.e. as algebraic extension of the function field. The hyperelliptic curves are prototypic examples.

Unramified covering

An unramified covering then is the occurrence of an empty ramification locus.