Branched covering on arvau

Wikipedia / 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 from an algebraic variety, the two dimensions being of dimension 0.

(for the Zariski topology) that is dense, 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 are both compact Riemann surfaces, outside of which we do find an honest covering

Ramification locus

is called the ramification locus). In general monodromy occurs according to the fundamental group 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.

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.

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 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 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 in , then this ramified point is "at infinity".

Over a root 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

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

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.

Hyperelliptic curve

A hyperelliptic curve by a polynomial of the form

Higher Degree Coverings of the Affine Line

We can generalize the previous example by taking the morphism

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

-module isomorphism of the target with

Superelliptic Curves

Superelliptic curves from polynomials of the form

has no repeated roots.

Ramified Coverings of Projective Space

with ramification locus

by considering the morphism of projective schemes

Applications

. 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.