Rational mapping
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 formula_1 between two varieties is an equivalence class of pairs formula_2 in which formula_3 is a morphism of varieties from a non-empty open set formula_4 to formula_5, and two such pairs formula_2 and formula_7 are considered equivalent if formula_3 and formula_9 coincide on the intersection formula_10 (this is, in particular, vacuously true if the intersection is empty , but since formula_11 is assumed irreducible, this is impossible). The proof that this defines an equivalence relation relies on the following lemma: formula_12 is said to be birational if there exists a rational map formula_13 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 formula_14 and formula_5. 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 formula_16 induces a homomorphism of fields formula_17. 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 formula_18) 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. There is a rational map formula_19 sending a ratio formula_20. Since the point formula_21 cannot have an image, this map is only rational, and not a morphism of varieties. More generally, there are rational maps formula_22 sending for formula_23 sending an formula_24-tuple to an formula_25-tuple by forgetting the last coordinates. Inclusions of open subvarieties. On a connected variety formula_26, the inclusion of any open subvariety formula_27 is a birational equivalence since the two varieties have equivalent function fields. That is, every rational function formula_28can be  restricted to a rational function formula_29 and conversely, a rational function formula_29 defines a rational equivalence class formula_31 on formula_26. An excellent example of this phenomenon is the birational equivalence of formula_33 and formula_34, hence formula_35. 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 formula_36 admits a map formula_37 which ramifies at three points. Then, there is an associated covering space formula_38 which defines a dominant rational morphism which is not birational. Another class of  examples come from Hyperelliptic curves which are double covers of formula_39 ramified at a finite number of points. Another class of examples are given by a taking a hypersurface formula_40 and restricting a rational map formula_41 to formula_26. This gives a ramified cover. For example, the Cubic surface given by the vanishing locus formula_43 has a rational map to formula_44 sending formula_45. This rational map can be expressed as the degree formula_46 field extension formula_47 Resolution of singularities. One of the canonical examples of a birational map is the Resolution of singularities. Over a field of characteristic 0, every singular  variety formula_26 has an associated nonsingular variety formula_49 with a birational map formula_50. This map has the property that it is an isomorphism on formula_51 and the fiber over formula_52 is a normal crossing divisor. For example, a nodal curve such as formula_53 is birational to formula_39 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. The usual example is that formula_55 is birational to the variety formula_26 contained in formula_57 consisting of the set of projective points formula_58 such that formula_59, but not isomorphic. Indeed, any two lines in formula_55 intersect, but the lines in formula_26 defined by formula_62 and formula_63 cannot intersect since their intersection would have  all coordinates zero. To compute the function field of formula_26 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 formula_65; in projective space this means we may take formula_66 and therefore identify this subset with the affine formula_67-plane. There, the coordinate ring of formula_26 is via the map formula_70. And the field of fractions of the latter is just formula_71, isomorphic to that of formula_55. 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.