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.
In algebraic geometry, the term branched covering is used to describe morphisms
[clarification needed] 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.
is holomorphic and not constant, and then there is a finite set of points
, 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).
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).
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.
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 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 fact, this projection corresponds to the field extension of degree n We can also generalize branched coverings of the line with varying ramification degrees.
At any point where the multiplicity of one of the linear terms in the factorization of
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
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
A hyperelliptic curve provides a generalization of the above degree
by a polynomial of the form We can generalize the previous example by taking the morphism where
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
from polynomials of the form Another useful class of examples come from ramified coverings of projective space.
with ramification locus by considering the morphism of projective schemes Again, this will be a covering of degree
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.