Branched covering
Encyclopedia
In mathematics
, branched covering is a term mainly used in algebraic geometry
, to describe morphism
s f from an algebraic variety
V to another one W, the two dimension
s being the same, and the typical fibre of f being of dimension 0.
In that case, there will be an open set W′ of W (for the Zariski topology
) that is dense in W, such that the restriction of f to W′ (from V′ = f−1(W′) to W′, that is) is unramified. Depending on the context, we can take this as local homeomorphism
for the strong topology
, over the complex number
s, 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 V and W are both Riemann surface
s, we require only that f is holomorphic and not constant, and then there is a finite set of points P of W, outside of which we do find an honest covering
The set of exceptional points on W is called the ramification locus (i.e. this is the complement of the largest possible open set W′); see ramification
. In general monodromy
occurs according to the fundamental group
of W′ 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. by extracting roots of functions in the function field
. The hyperelliptic curves are prototypic examples.
An unramified covering then is the occurrence of an empty ramification locus.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, branched covering is a term mainly used in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, to describe morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
s f from an algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
V to another one W, the two dimension
Dimension of an algebraic variety
In mathematics, the dimension of an algebraic variety V in algebraic geometry is defined, informally speaking, as the number of independent rational functions that exist on V.For example, an algebraic curve has by definition dimension 1...
s being the same, and the typical fibre of f being of dimension 0.
In that case, there will be an open set W′ of W (for the Zariski topology
Zariski topology
In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...
) that is dense in W, such that the restriction of f to W′ (from V′ = f−1(W′) to W′, that is) is unramified. Depending on the context, we can take this as local homeomorphism
Local homeomorphism
In mathematics, more specifically topology, a local homeomorphism is intuitively a function, f, between topological spaces that preserves local structure. Equivalently, one can cover the domain of this function by open sets, such that f restricted to each such open set is a homeomorphism onto its...
for the strong topology
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...
, over the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s, or as an étale morphism
Étale morphism
In algebraic geometry, a field of mathematics, an étale morphism is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy the hypotheses of the implicit function theorem, but because open sets in the Zariski topology are so large, they are not...
in general (under some slightly stronger hypotheses, on flatness
Flatness
Flatness may refer to:*Flatness *Flatness *Flatness *Flatness *Flatness , a geometrical tolerance required in certain manufacturing situations*Flatness...
and separability
Separability
In mathematics and in physics, separability may refer to properties of:* Separable differential equations** Separable ordinary differential equation, a class of equations that can be separated into a pair of integrals...
). Generically, then, such a morphism resembles a covering space in the topological sense. For example if V and W are both Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
s, we require only that f is holomorphic and not constant, and then there is a finite set of points P of W, outside of which we do find an honest covering
- V′ → W′.
The set of exceptional points on W is called the ramification locus (i.e. this is the complement of the largest possible open set W′); see ramification
Ramification
In mathematics, ramification is a geometric term used for 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign...
. In general monodromy
Monodromy
In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and algebraic and differential geometry behave as they 'run round' a singularity. As the name implies, the fundamental meaning of monodromy comes from 'running round singly'...
occurs according to the fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
of W′ 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. by extracting roots of functions in the function field
Function field
Function field may refer to:*Function field of an algebraic variety*Function field...
. The hyperelliptic curves are prototypic examples.
An unramified covering then is the occurrence of an empty ramification locus.