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