Rational mapping

A generalization of the concept of a rational function on an algebraic variety. Namely, a rational mapping from an irreducible algebraic variety to an algebraic variety (both defined over a field ) is an equivalent class of pairs , where is a non-empty open subset of and is a morphism from to . Two pairs and are said to be equivalent if and coincide on . In particular, a rational mapping from a variety to an affine line is a rational function on . For every rational mapping there is a pair such that for all equivalent pairs and is the restriction of to . The open subset is called the domain of regularity of the rational mapping , and is the image of the variety (written ) under .
If is a rational mapping of algebraic varieties and is dense in , then determines an imbedding of fields, . Conversely, an imbedding of the fields of rational functions determines a rational mapping from to . If induces an isomorphism of the fields and of rational functions, then is called a birational mapping.
The set of points of at which the rational mapping is not regular has codimension 1, in general. But if is complete and is smooth and irreducible, then this set has codimension at least 2. If and are complete irreducible varieties over an algebraically closed field of characteristic 0, then the rational mapping can be included in a commutative diagram (see ): (*)
where , are morphisms of an algebraic variety and is a composite of monoidal transformations (cf. Monoidal transformation). If is a birational mapping of complete non-singular surfaces, then there exists a diagram (*) in which both and are composites of monoidal transformations with non-singular centres (Zariski's theorem), that is, every birational mapping of complete non-singular surface decomposes into monoidal transformations with non-singular centres and their inverses. In the case when , the question of whether every birational mapping can be decomposed in this way is open (1990).