Rational mapping
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In mathematics, in particular the subfield of algebraic geometry, a rational map is a kind of partial function between algebraic varieties. In this article we use the convention that varieties are irreducible.
DefinitionA first attemptSuppose we take the prescription literally that a rational map is to be a partial function between two varieties. Then we must make the (ultimately incorrect) definition:
to Failed to parse (Missing texvc executable; please see math/README to configure.): W is a pair Failed to parse (Missing texvc executable; please see math/README to configure.): F = (U, f) , where Failed to parse (Missing texvc executable; please see math/README to configure.): U \subset V is an open set and Failed to parse (Missing texvc executable; please see math/README to configure.): f \colon U \to W is a morphism of varieties. To see how far we can get with this definition, we will try to define the composition of two rational maps Failed to parse (Missing texvc executable; please see math/README to configure.): F = (U,f), G = (T, g) with Failed to parse (Missing texvc executable; please see math/README to configure.): U \subset V, T \subset W . This should just be the composition Failed to parse (Missing texvc executable; please see math/README to configure.): g \circ f , defined on at least a subset of Failed to parse (Missing texvc executable; please see math/README to configure.): U , but this is meaningless unless the range of Failed to parse (Missing texvc executable; please see math/README to configure.): f is inside Failed to parse (Missing texvc executable; please see math/README to configure.): T . Nonetheless, we have at least a partially defined composition law, whenever Failed to parse (Missing texvc executable; please see math/README to configure.): f(U) \subset T . We might try to extend the definition by saying that the composition Failed to parse (Missing texvc executable; please see math/README to configure.): G \circ F is obtained by extending Failed to parse (Missing texvc executable; please see math/README to configure.): g onto a larger open set containing Failed to parse (Missing texvc executable; please see math/README to configure.): f(U) (if possible) and then composing with Failed to parse (Missing texvc executable; please see math/README to configure.): f . This is easily seen not to depend on the extension we take. With this definition, when Failed to parse (Missing texvc executable; please see math/README to configure.): g is the identity map on Failed to parse (Missing texvc executable; please see math/README to configure.): T then Failed to parse (Missing texvc executable; please see math/README to configure.): g \circ f = f . Likewise, if Failed to parse (Missing texvc executable; please see math/README to configure.): f is the identity map on Failed to parse (Missing texvc executable; please see math/README to configure.): U , then Failed to parse (Missing texvc executable; please see math/README to configure.): g \circ f = g . This means that we have a different rational map which is the identity for every open subset of every variety, which is at least philosophically offensive. However, we have a sort of built-in correction if we take the extended definition of composition, in that if we have identities defined on nested open sets Failed to parse (Missing texvc executable; please see math/README to configure.): T \subset T' , they compose equally with every Failed to parse (Missing texvc executable; please see math/README to configure.): (U, f) with Failed to parse (Missing texvc executable; please see math/README to configure.): f(U) \subset T . Since we would like, if possible, to have only one identity map, we can try to identify Failed to parse (Missing texvc executable; please see math/README to configure.): (T, {\rm id}_T), (T', {\rm id}_{T'}) in this case. Since of course the identity map makes sense on any domain, and since every open set is contained in the entire variety Failed to parse (Missing texvc executable; please see math/README to configure.): W , this means that we have at least succeeded in identifying all prospective identity maps. The result is the following:
is the collection of all pairs Failed to parse (Missing texvc executable; please see math/README to configure.): (T, {\rm id}_T)
with Failed to parse (Missing texvc executable; please see math/README to configure.): T \subset W
an open subset.
Considering that this was motivated by the extended definition of composition, we are led to make a similar definition for all rational maps. Formal definitionFormally, a rational map Failed to parse (Missing texvc executable; please see math/README to configure.): f \colon V \to W between two varieties is an equivalence class of pairs Failed to parse (Missing texvc executable; please see math/README to configure.): (f_U, U) in which Failed to parse (Missing texvc executable; please see math/README to configure.): f_U is a morphism of varieties from Failed to parse (Missing texvc executable; please see math/README to configure.): U to Failed to parse (Missing texvc executable; please see math/README to configure.): W , and two such pairs Failed to parse (Missing texvc executable; please see math/README to configure.): (f_U, U) and Failed to parse (Missing texvc executable; please see math/README to configure.): (f_V, V) are considered equivalent if Failed to parse (Missing texvc executable; please see math/README to configure.): f_U and Failed to parse (Missing texvc executable; please see math/README to configure.): f_V coincide on the intersection Failed to parse (Missing texvc executable; please see math/README to configure.): U \cap V (this is, in particular, vacuously true if the intersection is empty). The proof that this defines an equivalence relation relies on the following lemma:
Given such an equivalence class, Failed to parse (Missing texvc executable; please see math/README to configure.): f itself is supposed to be interpreted as the function obtained by gluing together the partial functions Failed to parse (Missing texvc executable; please see math/README to configure.): f_U to obtain a function on the union of all Failed to parse (Missing texvc executable; please see math/README to configure.): U which appear. This itself may be a partial function if the Failed to parse (Missing texvc executable; please see math/README to configure.): U do not form an open cover of Failed to parse (Missing texvc executable; please see math/README to configure.): V . We return to the problem of composition. Two arbitrary rational maps Failed to parse (Missing texvc executable; please see math/README to configure.): f \colon V \to W, g \colon W \to X can not necessarily be composed, but if Failed to parse (Missing texvc executable; please see math/README to configure.): f has dense image then they can: pick any representative pairs Failed to parse (Missing texvc executable; please see math/README to configure.): (f_U, U), (g_V, V) and note that since Failed to parse (Missing texvc executable; please see math/README to configure.): f_U(U) is dense in Failed to parse (Missing texvc executable; please see math/README to configure.): W it must intersect Failed to parse (Missing texvc executable; please see math/README to configure.): V , and so Failed to parse (Missing texvc executable; please see math/README to configure.): f_U^{-1}(V)
is a nonempty open subset of Failed to parse (Missing texvc executable; please see math/README to configure.): V
(as a morphism is necessarily continuous). Then Failed to parse (Missing texvc executable; please see math/README to configure.): g_V \circ f_U
is defined on this set, so we take Failed to parse (Missing texvc executable; please see math/README to configure.): g \circ f
to be the equivalence class containing the pair Failed to parse (Missing texvc executable; please see math/README to configure.): (g_V \circ f_U, f_U^{-1}(V))
. Dominant and birational mapsWe encountered the following condition in considering composition: Failed to parse (Missing texvc executable; please see math/README to configure.): f is said to be dominant if the range of any of the Failed to parse (Missing texvc executable; please see math/README to configure.): f_U is dense in Failed to parse (Missing texvc executable; please see math/README to configure.): W (this then implies that the range of any of them is dense, since open sets are dense in a variety). Since dominant rational maps can be composed, we can go further and talk about their inverses: a dominant rational map Failed to parse (Missing texvc executable; please see math/README to configure.): f is said to be birational if there exists a rational map Failed to parse (Missing texvc executable; please see math/README to configure.): g \colon W \to V 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 Failed to parse (Missing texvc executable; please see math/README to configure.): V and Failed to parse (Missing texvc executable; please see math/README to configure.): W . 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 (or codomain, more categorically precisely) 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 Failed to parse (Missing texvc executable; please see math/README to configure.): f \colon V \to W induces a homomorphism of fields Failed to parse (Missing texvc executable; please see math/README to configure.): K(W) \to K(V) . 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{C} ) to the category of 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. An example of birational equivalenceTwo 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{P}^2_k
is biratonal to the variety Failed to parse (Missing texvc executable; please see math/README to configure.): X
contained in Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{P}^3_k
consisting of the set of projective points Failed to parse (Missing texvc executable; please see math/README to configure.): [w : x : y : z]
such that Failed to parse (Missing texvc executable; please see math/README to configure.): xy - wz = 0
, but not isomorphic. Indeed, any two lines in Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{P}^2_k intersect, but the lines in Failed to parse (Missing texvc executable; please see math/README to configure.): X defined by Failed to parse (Missing texvc executable; please see math/README to configure.): w = x = 0 and Failed to parse (Missing texvc executable; please see math/README to configure.): y = z = 0 cannot intersect since their intersection would have all coordinates zero. To compute the function field of Failed to parse (Missing texvc executable; please see math/README to configure.): X 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 Failed to parse (Missing texvc executable; please see math/README to configure.): w \neq 0
and therefore identify this subset with the affine Failed to parse (Missing texvc executable; please see math/README to configure.): xyz -plane. There, the coordinate ring of Failed to parse (Missing texvc executable; please see math/README to configure.): X is
via the map Failed to parse (Missing texvc executable; please see math/README to configure.): p(x,y,z) \mapsto p(x,y,xy) . And the field of fractions of the latter is just Failed to parse (Missing texvc executable; please see math/README to configure.): k(x,y) , isomorphic to that of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{P}^2_k . 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 alsoReferences
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