# 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{* }

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 |

**How to Cite This Entry:**

Rational mapping.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Rational_mapping&oldid=48439