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.
between two varieties is an equivalence class of pairs
is a morphism of varieties from a non-empty open set
(this is, in particular, vacuously true if the intersection is empty, but since
The proof that this defines an equivalence relation relies on the following lemma:
in the equivalence class is a dominant morphism, i.e. has a dense image.
is said to be birational if there exists a rational map
In particular, the following theorem is central: the functor from the category 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.
cannot have an image, this map is only rational, and not a morphism of varieties.
defines a rational equivalence class
An excellent example of this phenomenon is the birational equivalence of
Covering spaces on open subsets of a variety give ample examples of rational maps which are not birational.
For example, Belyi's theorem states that every algebraic curve
which defines a dominant rational morphism which is not birational.
Another class of examples come from hyperelliptic curves which are double covers of
ramified at a finite number of points.
Another class of examples are given by a taking a hypersurface
For example, the cubic surface given by the vanishing locus
This rational map can be expressed as the degree
One of the canonical examples of a birational map is the resolution of singularities.
Over a field of characteristic 0, every singular variety
since topologically it is an elliptic curve with one of the circles contracted.
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.
consisting of the set of projective points
we pass to an affine subset (which does not change the field, a manifestation of the fact that a rational map depends only on its behavior in any open subset of its domain) in which
; in projective space this means we may take
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.