Rational mapping

A generalization of the concept of a rational function on an algebraic variety. Namely, a rational mapping from an irreducible algebraic variety $ X $ to an algebraic variety $ Y $( both defined over a field $ k $) is an equivalent class of pairs $ ( U , \phi _ {U} ) $, where $ U $ is a non-empty open subset of $ X $ and $ \phi _ {U} $ is a morphism from $ U $ to $ Y $. Two pairs $ ( U , \phi _ {U} ) $ and $ ( V , \psi _ {V} ) $ are said to be equivalent if $ \phi _ {U} $ and $ \psi _ {V} $ coincide on $ U \cap V $. In particular, a rational mapping from a variety $ X $ to an affine line is a rational function on $ X $. For every rational mapping $ \phi : X \rightarrow Y $ there is a pair $ ( \widetilde{U} , \phi _ {\widetilde{U} } ) $ such that $ U \subseteq \widetilde{U} $ for all equivalent pairs $ ( U , \phi _ {U} ) $ and $ \phi _ {U} $ is the restriction of $ \phi _ {\widetilde{U} } $ to $ U $. The open subset $ \widetilde{U} $ is called the domain of regularity of the rational mapping $ \phi $, and $ \phi ( \widetilde{U} ) $ is the image of the variety $ X $( written $ \phi ( X) $) under $ \phi $.

If $ \phi : X \rightarrow Y $ is a rational mapping of algebraic varieties and $ \phi ( X) $ is dense in $ Y $, then $ \phi $ determines an imbedding of fields, $ \phi ^ {*} : k ( Y) \rightarrow k ( Y) $. Conversely, an imbedding of the fields of rational functions $ \phi ^ {*} : k ( Y) \rightarrow k ( Y) $ determines a rational mapping from $ X $ to $ Y $. If $ \phi $ induces an isomorphism of the fields $ k ( X) $ and $ k ( Y) $ of rational functions, then $ \phi $ is called a birational mapping.

The set of points of $ X $ at which the rational mapping $ \phi : X \rightarrow Y $ is not regular has codimension 1, in general. But if $ Y $ is complete and $ X $ is smooth and irreducible, then this set has codimension at least 2. If $ X $ and $ Y $ are complete irreducible varieties over an algebraically closed field of characteristic 0, then the rational mapping $ \phi : X \rightarrow Y $ can be included in a commutative diagram (see [2]):

$$ \tag{* } \begin{array}{ccc} {} & Z &{} \\ {} _ \eta \swarrow &{} &\searrow _ {f} \\ X & \mathop \rightarrow \limits _ \phi & Y \\ \end{array} $$

where $ \eta $, $ f $ are morphisms of an algebraic variety $ Z $ and $ \eta $ is a composite of monoidal transformations (cf. Monoidal transformation). If $ \phi : X \rightarrow Y $ is a birational mapping of complete non-singular surfaces, then there exists a diagram (*) in which both $ f $ and $ \eta $ 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 $ \mathop{\rm dim} X \geq 3 $, the question of whether every birational mapping can be decomposed in this way is open (1990).

References

[1]	I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
[2]	H. Hironaka, "Resolution of singularities of an algebraic variety over a field of characteristic zero I" Ann. of Math. , 79 : 1–2 (1964) pp. 109–326 MR0199184 Zbl 0122.38603