Formally, a rational map between two varieties is an equivalence class of pairs in which is a morphism of varieties from a non-emptyopen set to, and two such pairs and are considered equivalent if and coincide on the intersection . 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 of projective varieties with dominant rational maps 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 maps 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.
Inclusions of open subvarieties
On a connected variety, the inclusion of any open subvariety is a birational equivalence since the two varieties have equivalent function fields. That is, every rational function can be restricted to a rational function and conversely, a rational function defines a rational equivalence class on. An excellent example of this phenomena is the birational equivalence of and, hence.
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 states that every algebraic curve admits a map which ramifies at three points. Then, there is an associated covering space which defines a dominant rational morphism which is not birational. Another class of examples come fromHyperelliptic curves which are double covers of ramified at a finite number of points. Another class of examples are given by a taking a hypersurface and restricting a rational map to. This gives a ramified cover. For example, the Cubic surface given by the vanishing locus has a rational map to sending. This rational map can be expressed as the degree field extension
One of the canonical examples of a birational map is the Resolution of singularities. Over a field of characteristic 0, every singular variety has an associated nonsingular variety with a birational map. This map has the property that it is an isomorphism on and the fiber over is a normal crossing divisor. For example, a nodal curve such as is birational to 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, in that there exist varieties which are birational but not isomorphic. The usual example is that is birational to the variety contained in consisting of the set of projective points such that, but not isomorphic. Indeed, any two lines in intersect, but the lines in defined by and cannot intersect since their intersection would have all coordinates zero. To compute the function field of we pass to an affine subset in which ; in projective space this means we may take and therefore identify this subset with the affine -plane. There, the coordinate ring of is via the map. And the field of fractions of the latter is just, isomorphic to that of. 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.
