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.

Definition

Formal definition

Formally, a rational map between two varieties is an equivalence class is a morphism of varieties from a non-empty open set (this is, in particular, vacuously true 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 dominant in the equivalence class is a dominant morphism is said to be birational 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. By definition, a rational function is just a rational map whose range is the projective line. Composition of functions then allows us to "pull back induces a homomorphism. In particular, the following theorem is central: the functor from the category of projective varieties) 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.

Examples

Rational maps of projective spaces

-tuple by forgetting the last coordinates.

Inclusions of open subvarieties

.

Covering spaces on open subsets

Covering spaces on open subsets of a variety give ample examples of rational maps which are not birational. For example, Belyi's theorem which defines a dominant rational morphism which is not birational. Another class of examples come from hyperelliptic curves. This gives a ramified cover. For example, the cubic surface

Resolution of singularities

One of the canonical examples of a birational map is the resolution of singularities. Over a field of characteristic since topologically it is an elliptic curve with one of the circles contracted. Then, the birational map is given by normalization.

Birational equivalence

Two varieties are said to be birationally equivalent if there exists a birational map between them; this theorem states that birational equivalence of varieties is identical to isomorphism of their function fields as extensions of the base field. This is somewhat more liberal than the notion of isomorphism of varieties (which requires a globally defined morphism to witness the isomorphism, not merely a rational map), in that there exist varieties which are birational but not isomorphic.

is

. And the field of fractions. Note that at no time did we actually produce a rational map, though tracing through the proof of the theorem it is possible to do so.

See also

• Birational geometry

• Blowing up

• Function field of an algebraic variety

• Resolution of singularities

• Minimal model program

• Log structure

References

• , section I.4.