Branched covering

In mathematics, a branched covering is 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 formula_1 from an algebraic variety formula_2 to another one formula_3, the two dimensions being the same, and the typical fibre of formula_1 being of dimension 0.
In that case, there will be an open set formula_5 of formula_3 (for the Zariski topology) that is dense in formula_3, such that the restriction of formula_1 to formula_5 (from formula_10 to formula_5, 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 formula_2 and formula_3 are both Riemann surfaces, we require only that formula_1 is holomorphic and not constant, and then there is a finite set of points formula_15 of formula_3, outside of which we do find an honest covering
Ramification locus.
The set of exceptional points on formula_3 is called the ramification locus (i.e. this is the complement of the largest possible open set formula_5). In general monodromy occurs according to the fundamental group of formula_5 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.
Examples.
Elliptic curve.
Morphisms of curves provide many examples of ramified coverings. For example, let be the elliptic curve of equation
The projection of onto the -axis is a ramified cover with ramification locus given by
This is because for these three values of the fiber is the double point formula_23 while for any other value of , the fiber consists of two distinct points (over an algebraically closed field).
This projection induces an algebraic extension of degree two of the function fields:
Also, if we take the fraction fields of the underlying commutative rings, we get the morphism
Hence this projection is a degree 2 branched covering. This can be homogenized to construct a degree 2 branched covering of the corresponding projective elliptic curve to the projective line.
Plane algebraic curve.
The previous example may be generalized to any algebraic plane curve in the following way.
Let be a plane curve defined by the equation , where is a separable and irreducible polynomial in two indeterminates. If is the degree of in , then the fiber consists of distinct points, except for a finite number of values of . Thus, this projection is a branched covering of degree .
The exceptional values of are the roots of the coefficient of formula_25 in , and the roots of the discriminant of with respect to .
Over a root of the discriminant, there is at least a ramified point, which is either a critical point or a singular point. If is also a root of the coefficient of formula_25 in , then this ramified point is "at infinity".
Over a root of the coefficient of formula_25 in , the curve has an infinite branch, and the fiber at has less than points. However, if one extends the projection to the projective completions of and the -axis, and if is not a root of the discriminant, the projection becomes a covering over a neighbourhood of .
The fact that this projection is a branched covering of degree may also be seen by considering the function fields. In fact, this projection corresponds to the field extension of degree 
Varying Ramifications.
We can also generalize branched coverings of the line with varying ramification degrees. Consider a polynomial of the form
as we choose different points formula_30, the fibers given by the vanishing locus of formula_31 vary. At any point where the multiplicity of one of the linear terms in the factorization of formula_31 increases by one, there is a ramification.
Scheme Theoretic Examples.
Elliptic Curves.
Morphisms of curves provide many examples of ramified coverings of schemes. For example, the morphism from an affine elliptic curve to a line
is a ramified cover with ramification locus given by
This is because at any point of formula_35 in formula_36 the fiber is the scheme
Also, if we take the fraction fields of the underlying commutative rings, we get the field homomorphism
which is an algebraic extension of degree two;
hence we got a degree 2 branched covering of an elliptic curve to the affine line. This can be homogenized to construct a morphism of a projective elliptic curve to formula_39.
Hyperelliptic curve.
A hyperelliptic curve provides a generalization of the above degree formula_40 cover of the affine line, by considering the affine scheme defined over formula_41 by a polynomial of the form
Higher Degree Coverings of the Affine Line.
We can generalize the previous example by taking the morphism
where formula_46 has no repeated roots. Then the ramification locus is given by
where the fibers are given by
Then, we get an induced morphism of fraction fields
There is an formula_50-module isomorphism of the target with
Hence the cover is of degree formula_52.
Superelliptic Curves.
Superelliptic curves are a generalization of hyperelliptic curves and a specialization of the previous family of examples since they are given by affine schemes formula_53 from polynomials of the form
Ramified Coverings of Projective Space.
Another useful class of examples come from ramified coverings of projective space. Given a homogeneous polynomial formula_57 we can construct a ramified covering of formula_58 with ramification locus
by considering the morphism of projective schemes
Again, this will be a covering of degree formula_1.
Applications.
Branched coverings formula_62 come with a symmetry group of transformations formula_63. 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.