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A generalization of the concept of a [[Rational function|rational function]] on an [[Algebraic variety|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 [[#References|[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|Monoidal transformation]]). If  $  \phi :  X \rightarrow Y $
is a [[Birational mapping|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> 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 {{MR|0199184}} {{ZBL|0122.38603}} </TD></TR></table>

[[Category:Algebraic geometry]]