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A [[Birational transformation|birational transformation]] of a projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c0270501.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c0270502.png" />, over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c0270503.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c0270504.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c0270505.png" /> they may be expressed as linear-fractional transformations
+
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 $\tau$:
 +
$$(x,y)\mapsto (\frac{1}{x},\frac{1}{y}),$$
 +
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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c0270506.png" /></td> </tr></table>
+
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.
  
Among these transformations, special consideration is given to the standard quadratic transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c0270507.png" />:
+
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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c0270508.png" /></td> </tr></table>
+
A Cremona transformation of the form
 +
$$x\mapsto x,$$
  
or, in homogeneous coordinates,
+
$$y\mapsto \frac{P(x)y+Q(x)}{R(x)y+S(x)},\quad P,Q,R,S\in k[x],$$
 +
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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c0270509.png" /></td> </tr></table>
+
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
 +
$$x\mapsto ax+b,\quad y\mapsto cy+Q(x),$$
  
This transformation is an isomorphism off the coordinate axes:
+
$$a\ne 0,\quad c\ne 0,\quad a,b\in k,\; Q(x)\in k[x],$$
 
+
moreover, it is the amalgamated product of these subgroups
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c02705010.png" /></td> </tr></table>
+
[[#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
it has three fundamental points (points at which is it undefined) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c02705011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c02705012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c02705013.png" />, and maps each coordinate axis onto the unique fundamental point not contained in that axis.
+
results have been obtained concerning Cremona transformations for
 
+
dimensions $n\ge 3$.
By Noether's theorem (see [[Cremona group|Cremona group]]), if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c02705014.png" /> is an algebraically closed field, each Cremona transformation of the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c02705015.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c02705016.png" /> is defined by a linear system of curves of degree 8 on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c02705017.png" />, which pass with multiplicity 3 through 7 points in general position. A Bertini involution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c02705018.png" /> is defined by a linear system of curves of degree 17 on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c02705019.png" />, which pass with multiplicity 6 through 8 points in general position.
 
 
 
A Cremona transformation of the form
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c02705020.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c02705021.png" /></td> </tr></table>
 
 
 
is called a de Jonquières transformation. De Jonquières transformations are most naturally interpreted as birational transformations of the quadric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c02705022.png" /> which preserve projection onto one of the factors. One can then restate Noether's theorem as follows: The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c02705023.png" /> of birational automorphisms of the quadric is generated by an involution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c02705024.png" /> and by the de Jonquières transformations, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c02705025.png" /> is the automorphism defined by permutation of factors.
 
 
 
Any biregular automorphism of the affine space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c02705026.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c02705027.png" /> may be extended to a Cremona transformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c02705028.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c02705029.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c02705030.png" /> the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c02705031.png" /> is generated by the subgroup of affine transformations and the subgroup of transformations of the form
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c02705032.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c02705033.png" /></td> </tr></table>
 
 
 
moreover, it is the amalgamated product of these subgroups [[#References|[5]]]. The structure of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c02705034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c02705035.png" />, is not known. In general, up to the present time (1987) no significant results have been obtained concerning Cremona transformations for dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c02705036.png" />.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H.P. Hudson,   "Cremona transformations in plane and space" , Cambridge Univ. Press (1927)</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)</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</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</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD
 +
valign="top"> H.P. Hudson, "Cremona transformations in plane and
 +
space" , Cambridge Univ. Press (1927)</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)</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</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</TD></TR></table>
  
  
  
 
====Comments====
 
====Comments====
The fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c02705037.png" /> is the amalgamated product of the subgroup of affine transformations (cf. [[Affine transformation|Affine transformation]]) with that of the transformations (*) was first proved (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027050/c02705038.png" />) by H.W.E. Jung [[#References|[a1]]]; the case of arbitrary ground field was proved by W. van der Kulk [[#References|[a2]]].
+
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====
 
====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</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</TD></TR></table>
+
<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</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</TD></TR></table>

Revision as of 19:12, 11 November 2011

A 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, 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 $\tau$: $$(x,y)\mapsto (\frac{1}{x},\frac{1}{y}),$$ 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), 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 [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 [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

[1] H.P. Hudson, "Cremona transformations in plane and space" , Cambridge Univ. Press (1927)
[2] L. Godeaux, "Les transformations birationelles du plan" , Gauthier-Villars (1927)
[3] A.B. Coble, "Algebraic geometry and theta functions" ,Amer. Math. Soc. (1929)
[4] M. Nagata, "On rational surfaces II" Mem. Coll. Sci. Univ. Kyoto , 33 (1960) pp. 271–393
[5] I.R. Shafarevich, "On some infinitedimensional groups" Rend. di Math , 25 (1966) pp. 208–212


Comments

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

References

[a1] H.W.E. Jung, "Ueber ganze birationale Transformationen der Ebene" J. Reine Angew. Math. , 184 (1942) pp. 161–174
[a2] W. van der Kulk, "On polynomial rings in two variables" Nieuw Arch. Wiskunde , 1 (1953) pp. 33–41
How to Cite This Entry:
Cremona transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cremona_transformation&oldid=19605
This article was adapted from an original article by V.A. Iskovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article