A
birational transformation
of a projective space
,
,
over a field
.
Birational transformations of the plane and
of three-dimensional space were systematically studied (from
1863
on)
by
L. Cremona.
The group of Cremona transformations is also named after him — the
Cremona group,
and is denoted by
.
The simplest examples of Cremona transformations which are
not projective transformations are quadratic birational transformations
of the plane. In non-homogeneous coordinates
they may be expressed as linear-fractional transformations
Among these transformations, special consideration is given to the
standard quadratic transformation
:
or, in homogeneous coordinates,
This transformation is an isomorphism off the coordinate axes:
it has three fundamental points (points at which is it undefined)
,
and
,
and maps each coordinate axis onto the unique
fundamental point not contained in that axis.
By Noether's theorem (see
Cremona group),
if
is an algebraically closed field, each Cremona transformation of the plane
can be expressed as a composition of quadratic transformations.
An important place in the theory of Cremona transformations is
occupied by certain special classes of transformations, in
particular — Geiser involutions and Bertini involutions (see
[1]).
A
Geiser involution
is defined by a linear system of curves of degree 8 on
,
which pass with multiplicity 3 through 7 points in general position. A
Bertini involution
is defined by a linear system of curves of degree 17 on
,
which pass with multiplicity 6 through 8 points in general position.
A Cremona transformation of the form
is called a
de Jonquières transformation.
De Jonquières transformations are most naturally interpreted
as birational transformations of the quadric
which preserve projection onto one of the factors. One
can then restate Noether's theorem as follows: The group
of birational automorphisms of the quadric is generated by an involution
and by the de Jonquières transformations, where
is the automorphism defined by permutation of factors.
Any biregular automorphism of the affine space
in
may be extended to a Cremona transformation of
,
so that
.
When
the group
is generated by the subgroup of affine transformations
and the subgroup of transformations of the form
moreover, it is the amalgamated product of these subgroups
[5].
The structure of the group
,
,
is not known. In general, up to the present time
(
1987)
no significant results have been obtained
concerning Cremona transformations for dimensions
.