Branched covering

In mathematics, a branched covering is a map that is almost a covering map, except on a small set. 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, the term branched covering is used to describe morphisms from an algebraic variety to another one , the two dimensions being the same, and the typical fibre of being of dimension 0. In that case, there will be an open set of (for the Zariski topology) that is dense in , such that the restriction of to (from to , 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 and are both compact Riemann surfaces, we require only that is holomorphic and not constant, and then there is a finite set of points of , outside of which we do find an honest covering The set of exceptional points on is called the ramification locus (i.e. this is the complement of the largest possible open set ). In general monodromy occurs according to the fundamental group of acting on the sheets of the covering (this topological picture can be made precise also in the case of a general base field). Branched coverings are easily constructed as Kummer extensions, i.e. as algebraic extension of the function field. The hyperelliptic curves are prototypic examples. An unramified covering then is the occurrence of an empty ramification locus. Morphisms of curves provide many examples of ramified coverings. For example, let C be the elliptic curve of equation The projection of C onto the x-axis is a ramified cover with ramification locus given by This is because for these three values of x the fiber is the double point while for any other value of x, the fiber consists of two distinct points (over an algebraically closed field).