Difference between revisions of "Rational mapping"
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− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <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> |
Revision as of 21:55, 30 March 2012
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 [2]):
![]() | (*) |
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).
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 |
Rational mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rational_mapping&oldid=23950