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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.

The previous example may be generalized to any algebraic plane curve in the following way. Let C be a plane curve defined by the equation 1=f(x,y) = 0, where f is a separable and irreducible polynomial in two indeterminates. If n is the degree of f in y, then the fiber consists of n distinct points, except for a finite number of values of x. Thus, this projection is a branched covering of degree n.

The exceptional values of x are the roots of the coefficient of in f, and the roots of the discriminant of f with respect to y. Over a root r of the discriminant, there is at least a ramified point, which is either a critical point or a singular point. If r is also a root of the coefficient of in f, then this ramified point is "at infinity". Over a root s of the coefficient of in f, the curve C has an infinite branch, and the fiber at s has less than n points. However, if one extends the projection to the projective completions of C and the x axis, and if s is not a root of the discriminant, the projection becomes a covering over a neighbourhood of s.

The fact that this projection is a branched covering of degree n may also be seen by considering the function fields. In fact, this projection corresponds to the field extension of degree n

We can also generalize branched coverings of the line with varying ramification degrees. Consider a polynomial of the form

as we choose different points , the fibers given by the vanishing locus of vary. At any point where the multiplicity of one of the linear terms in the factorization of increases by one, there is a ramification.

Branched coverings come with a symmetry group of transformations Since the symmetry group has stabilizers at the points of the ramification locus, branched coverings can be used to construct examples of orbifolds, or Deligne–Mumford stacks.