Regular map (algebraic geometry)
Encyclopedia
In algebraic geometry
, a regular map between affine varieties is a mapping which is given by polynomials. For example, if X and Y are subvarieties of An resp. Am, then a regular map from X to Y is given by m polynomials in the n coordinates of An.
More generally, a map ƒ:X→Y between two varieties
is regular at a point x if there is a neighbourhood U of x and a neighbourhood V of ƒ(x) such that the restricted function ƒ:U→V is regular. Then ƒ is called regular, if it is regular at all points of X.
In the particular case that Y equals A1 the map ƒ:X→A1 is called a regular function
, and correspond to scalar functions in differential geometry. The ring of regular functions is a fundamental object in algebraic geometry.
Regular maps can be seen as the morphisms in the category of algebraic varieties.
A regular map whose inverse is also regular is called biregular, and are isomorphism
s in the category of algebraic varieties.
Because regular and biregular are very restrictive conditions – there are no non-constant regular functions on projective space – the weaker condition of a rational function
and birational maps are frequently used as well.
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...
, a regular map between affine varieties is a mapping which is given by polynomials. For example, if X and Y are subvarieties of An resp. Am, then a regular map from X to Y is given by m polynomials in the n coordinates of An.
More generally, a map ƒ:X→Y between two varieties
Abstract variety
In mathematics, in the field of algebraic geometry, the idea of abstract variety is to define a concept of algebraic variety in an intrinsic way. This followed the trend in the definition of manifold independent of any ambient space by some years, the first notions being those of Oscar Zariski and...
is regular at a point x if there is a neighbourhood U of x and a neighbourhood V of ƒ(x) such that the restricted function ƒ:U→V is regular. Then ƒ is called regular, if it is regular at all points of X.
In the particular case that Y equals A1 the map ƒ:X→A1 is called a regular function
Regular function
In mathematics, a regular function is a function that is analytic and single-valued in a given region. In complex analysis, any complex regular function is known as a holomorphic function...
, and correspond to scalar functions in differential geometry. The ring of regular functions is a fundamental object in algebraic geometry.
Regular maps can be seen as the morphisms in the category of algebraic varieties.
A regular map whose inverse is also regular is called biregular, and are isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
s in the category of algebraic varieties.
Because regular and biregular are very restrictive conditions – there are no non-constant regular functions on projective space – the weaker condition of a rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
and birational maps are frequently used as well.