In
mathematics, in particular the subfield of
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrica ...
, a rational map or rational mapping is a kind of
partial function
In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is ...
between
algebraic varieties
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. ...
. This article uses the convention that varieties are
irreducible.
Definition
Formal definition
Formally, a rational map
between two varieties is an
equivalence class of pairs
in which
is a
morphism of varieties In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regula ...
from a
non-empty open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are ...
to
, and two such pairs
and
are considered equivalent if
and
coincide on the intersection
(this is, in particular,
vacuously true if the intersection is empty, but since
is assumed irreducible, this is impossible). The proof that this defines an
equivalence relation relies on the following lemma:
* If two morphisms of varieties are equal on some non-empty open set, then they are equal.
is said to be birational if there exists a rational map
which is its inverse, where the composition is taken in the above sense.
The importance of rational maps to algebraic geometry is in the connection between such maps and maps between the
function fields of
and
. Even a cursory examination of the definitions reveals a similarity between that of rational map and that of rational function; in fact, a rational function is just a rational map whose range is the projective line. Composition of functions then allows us to "pull back" rational functions along a rational map, so that a single rational map
induces a
homomorphism of fields
. In particular, the following theorem is central: the
functor from the
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
*C ...
of
projective varieties
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 ...
with dominant rational maps (over a fixed base field, for example
) to the category of finitely generated
field extensions of the base field with reverse inclusion of extensions as morphisms, which associates each variety to its function field and each map to the associated map of function fields, is an
equivalence of categories
In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences fr ...
.
Examples
Rational maps of projective spaces
There is a rational map
sending a ratio