View source for Cremona transformation

A
[[Birational transformation|birational transformation]] of a
projective space $\def\P{\mathbb{P}} \P_k^n$, $n\ge 2$, over a field $k$. 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|Cremona group]], and is denoted by $\def\Cr{\rm{Cr}}\Cr(\P_k^n)$.

The simplest examples of Cremona transformations which are not
projective transformations are quadratic birational transformations of
the plane. In non-homogeneous coordinates $(x,y)$ they may be expressed as
linear-fractional transformations 
$$x\mapsto \frac{a_1x+b_1y+c_1}{a_2x+b_2y+c_2}, \quad y\mapsto \frac{a_3x+b_3y+c_3}{a_4x+b_4y+c_4}.$$
Among these transformations,
special consideration is given to the standard quadratic
transformation $\tau$: 
$$(x,y)\mapsto \left( } \frac{1}{x},\frac{1}{y} } \right),$$
or, in homogeneous coordinates, 
$$(x_0,x_1,x_2) \mapsto(x_1x_2,x_0x_2,x_0x_1). $$
This
transformation is an isomorphism off the coordinate axes: 
$$\tau:\P_k^2\setminus \{x_0x_1x_2 = 0\} \tilde\to \P_k^2 \setminus \{x_0x_1x_2 = 0 \}, $$
it has
three fundamental points (points at which is it undefined) $(0,0,1)$, $(0,1,0)$
and $(1,0,0)$, and maps each coordinate axis onto the unique fundamental
point not contained in that axis.

By Noether's theorem (see
[[Cremona group|Cremona group]]), if $k$ is an algebraically closed
field, each Cremona transformation of the plane $\P_k^2$ 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
[[#References|[1]]]). A Geiser involution $\alpha : \P_k^2 \to \P_k^2$ 
is defined by a linear
system of curves of degree 8 on $\P_k^2$, which pass with multiplicity 3
through 7 points in general position. A Bertini involution 
$\beta : \P_k^2 \to \P_k^2$ is
defined by a linear system of curves of degree 17 on $\P_k^2$, which pass
with multiplicity 6 through 8 points in general position.

A Cremona transformation of the form 
$$x\mapsto x,$$

$$y\mapsto \frac{P(x)y+Q(x)}{R(x)y+S(x)},\quad P,Q,R,S\in k[x],$$
is called a de Jonquières transformation. De Jonquières
transformations are most naturally interpreted as birational
transformations of the quadric $\P_k^1\times \P_k^1$ which preserve projection onto one
of the factors. One can then restate Noether's theorem as follows: The
group ${\rm Bir}(P^1\times P^1)$ of birational automorphisms of the quadric is generated by
an involution $\sigma$ and by the de Jonquières transformations, where $\sigma\in {\rm Aut}(P^1\times P^1)$
is the automorphism defined by permutation of factors.

Any biregular automorphism of the affine space $\def\A{\mathbb{A}}\A_k^n$ in $\P_k^n$ may be
extended to a Cremona transformation of $\P_k^n$, so that ${\rm
Aut}(\P^1\times \P^1) \subset {\rm Cr}(\P_k^n)$. When $n=2$ the
group ${\rm Aut}(\A_k^2)$ is generated by the subgroup of affine transformations and
the subgroup of transformations of the form 
$$x\mapsto ax+b,\quad y\mapsto cy+Q(x),$$

$$a\ne 0,\quad c\ne 0,\quad a,b\in k,\; Q(x)\in k[x],$$
moreover, it is the amalgamated product of these subgroups
[[#References|[5]]]. The structure of the group ${\rm Aut}(\A_k^n)$, $n\ge 3$, is not
known. In general, up to the present time (1987) no significant
results have been obtained concerning Cremona transformations for
dimensions $n\ge 3$.

====References====
<table><TR><TD valign="top">[1]</TD> <TD
valign="top"> H.P. Hudson, "Cremona transformations in plane and
space" , Cambridge Univ. Press (1927) {{MR|1521296}} {{ZBL|53.0595.01}} </TD></TR><TR><TD
valign="top">[2]</TD> <TD valign="top"> L. Godeaux, "Les transformations birationelles du plan" , Gauthier-Villars (1927)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">
A.B. Coble, "Algebraic geometry and theta functions" ,Amer. Math. Soc. (1929) {{MR|0733252}} {{MR|0123958}} {{ZBL|55.0808.02}} </TD></TR><TR><TD valign="top">[4]</TD> <TD
valign="top"> M. Nagata, "On rational surfaces II" ''Mem. Coll. Sci. Univ. Kyoto'' , '''33''' (1960)
pp. 271–393 {{MR|0126444}} {{ZBL|0100.16801}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">
I.R. Shafarevich, "On some infinitedimensional groups" ''Rend. di Math'' , '''25''' (1966) pp. 208–212 {{MR|485898}} {{ZBL|}} </TD></TR></table>



====Comments====
The fact that ${\rm Aut}(\P_k^2)$ is the amalgamated product of the
subgroup of affine transformations (cf. [[Affine transformation|Affine transformation]]) with that of the
transformations (*) was first proved (for ${\rm char}\; k = 0$) by H.W.E. Jung
[[#References|[a1]]]; the case of arbitrary ground field was proved by
W. van der Kulk
[[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD
valign="top"> H.W.E. Jung, "Ueber ganze birationale Transformationen der Ebene" ''J. Reine Angew. Math.'' , '''184''' (1942) pp. 161–174 {{MR|}} {{ZBL|0027.08503}} {{ZBL|68.0382.01}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">
W. van der Kulk, "On polynomial rings in two variables" ''Nieuw Arch. Wiskunde'' , '''1''' (1953) pp. 33–41 {{MR|}} {{ZBL|0050.26002}} </TD></TR></table>