In
mathematics, a branched covering is a map that is almost a
covering map A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...
, 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
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
on each circle.
In algebraic geometry
In
algebraic geometry, the term branched covering is used to describe
morphisms
from an
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
to another one
, the two
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
s 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
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
) that is
dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
in
, such that the restriction of
to
(from
to
, that is) is
unramified. Depending on the context, we can take this as
local homeomorphism
In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure.
If f : X \to Y is a local homeomorphism, X is said to be an ...
for the
strong topology In mathematics, a strong topology is a topology which is stronger than some other "default" topology. This term is used to describe different topologies depending on context, and it may refer to:
* the final topology on the disjoint union
* the to ...
, over the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, or as an
étale morphism
In algebraic geometry, an étale morphism () is a morphism of schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy t ...
in general (under some slightly stronger hypotheses, on
flatness and
separability). Generically, then, such a morphism resembles a
covering space A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...
in the topological sense. For example, if
and
are both
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
s, 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
:
.
Ramification locus
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).
Kummer extensions
Branched coverings are easily constructed as
Kummer extension In abstract algebra and number theory, Kummer theory provides a description of certain types of field extensions involving the adjunction of ''n''th roots of elements of the base field. The theory was originally developed by Ernst Eduard Kummer a ...
s, i.e. as
algebraic extension
In mathematics, an algebraic extension is a field extension such that every element of the larger field is algebraic over the smaller field ; that is, if every element of is a root of a non-zero polynomial with coefficients in . A field ext ...
of the
function field. The
hyperelliptic curve
In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus ''g'' > 1, given by an equation of the form
y^2 + h(x)y = f(x)
where ''f''(''x'') is a polynomial of degree ''n'' = 2''g'' + 1 > 4 or ''n'' = 2''g'' + 2 > 4 with ''n'' dis ...
s 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
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
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
while for any other value of , the fiber consists of two distinct points (over an
algebraically closed field).
This projection induces an
algebraic extension
In mathematics, an algebraic extension is a field extension such that every element of the larger field is algebraic over the smaller field ; that is, if every element of is a root of a non-zero polynomial with coefficients in . A field ext ...
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 mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
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
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
in , then this ramified point is "
at infinity".
Over a root of the coefficient of
in , the curve has an infinite branch, and the fiber at has less than points. However, if one extends the projection to the
projective completion
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables wi ...
s 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
, 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.
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
in
the fiber is the scheme
:
Also, if we take the fraction fields of the underlying commutative rings, we get the
field homomorphism
Field theory is the branch of mathematics in which fields are studied. This is a glossary of some terms of the subject. (See field theory (physics) for the unrelated field theories in physics.)
Definition of a field
A field is a commutative ri ...
:
which is an
algebraic extension
In mathematics, an algebraic extension is a field extension such that every element of the larger field is algebraic over the smaller field ; that is, if every element of is a root of a non-zero polynomial with coefficients in . A field ext ...
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
.
Hyperelliptic curve
A
hyperelliptic curve
In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus ''g'' > 1, given by an equation of the form
y^2 + h(x)y = f(x)
where ''f''(''x'') is a polynomial of degree ''n'' = 2''g'' + 1 > 4 or ''n'' = 2''g'' + 2 > 4 with ''n'' dis ...
provides a generalization of the above degree
cover of the affine line, by considering the affine scheme defined over
by a polynomial of the form
:
where
for
Higher Degree Coverings of the Affine Line
We can generalize the previous example by taking the morphism
:
where
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
-module isomorphism of the target with
:
Hence the cover is of degree
.
Superelliptic Curves
Superelliptic curve In mathematics, a superelliptic curve is an algebraic curve defined by an equation of the form
:y^m = f(x),
where m \geq 2 is an integer and ''f'' is a polynomial of degree d\geq 3 with coefficients in a field k; more precisely, it is the smooth pro ...
s are a generalization of hyperelliptic curves and a specialization of the previous family of examples since they are given by affine schemes
from polynomials of the form
:
where
and
has no repeated roots.
Ramified Coverings of Projective Space
Another useful class of examples come from ramified coverings of projective space. Given a homogeneous polynomial