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Concept
Rational mapping
Summary
In mathematics, in particular the subfield of algebraic geometry, a rational map or rational mapping is a kind of partial function between algebraic varieties. This article uses the convention that varieties are irreducible.
Formally, a rational map between two varieties is an equivalence class of pairs in which is a morphism of varieties from a non-empty open set 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 of projective varieties 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.
There is a rational map sending a ratio . Since the point cannot have an image, this map is only rational, and not a morphism of varieties. More generally, there are rational maps sending for sending an -tuple to an -tuple by forgetting the last coordinates.
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