• Source: Cremona group

In algebraic geometry, the Cremona group, introduced by Cremona (1863, 1865), is the group of birational automorphisms of the



n


{\displaystyle n}

-dimensional projective space over a field



k


{\displaystyle k}

. It is denoted by



C
r
(


P


n


(
k
)
)


{\displaystyle Cr(\mathbb {P} ^{n}(k))}


or



B
i
r
(


P


n


(
k
)
)


{\displaystyle Bir(\mathbb {P} ^{n}(k))}

or



C

r

n


(
k
)


{\displaystyle Cr_{n}(k)}

.
The Cremona group is naturally identified with the automorphism group





A
u
t


k


(
k
(

x

1


,
.
.
.
,

x

n


)
)


{\displaystyle \mathrm {Aut} _{k}(k(x_{1},...,x_{n}))}

of the field of the rational functions in



n


{\displaystyle n}

indeterminates over



k


{\displaystyle k}

, or in other words a pure transcendental extension of



k


{\displaystyle k}

, with transcendence degree



n


{\displaystyle n}

.
The projective general linear group of order



n
+
1


{\displaystyle n+1}

, of projective transformations, is contained in the Cremona group of order



n


{\displaystyle n}

. The two are equal only when



n
=
0


{\displaystyle n=0}

or



n
=
1


{\displaystyle n=1}

, in which case both the numerator and the denominator of a transformation must be linear.




The Cremona group in 2 dimensions


In two dimensions, Max Noether and Guido Castelnuovo showed that the complex Cremona group is generated by the standard quadratic transformation, along with




P
G
L

(
3
,
k
)


{\displaystyle \mathrm {PGL} (3,k)}

, though there was some controversy about whether their proofs were correct, and Gizatullin (1983) gave a complete set of relations for these generators. The structure of this group is still not well understood, though there has been a lot of work on finding elements or subgroups of it.

Cantat & Lamy (2010) showed that the Cremona group is not simple as an abstract group;
Blanc showed that it has no nontrivial normal subgroups that are also closed in a natural topology.
For the finite subgroups of the Cremona group see Dolgachev & Iskovskikh (2009).




The Cremona group in higher dimensions


There is little known about the structure of the Cremona group in three dimensions and higher though many elements of it have been described. Blanc (2010) showed that it is (linearly) connected, answering a question of Serre (2010). There is no easy analogue of the Noether–Castelnouvo theorem as Hudson (1927) showed that the Cremona group in dimension at least 3 is not generated by its elements of degree bounded by any fixed integer.




De Jonquières groups


A De Jonquières group is a subgroup of a Cremona group of the following form . Pick a transcendence basis





x

1


,
.
.
.
,

x

n




{\displaystyle x_{1},...,x_{n}}

for a field extension of



k


{\displaystyle k}

. Then a De Jonquières group is the subgroup of automorphisms of



k
(

x

1


,
.
.
.
,

x

n


)


{\displaystyle k(x_{1},...,x_{n})}

mapping the subfield



k
(

x

1


,
.
.
.
,

x

r


)


{\displaystyle k(x_{1},...,x_{r})}

into itself for some



r
≤
n


{\displaystyle r\leq n}

. It has a normal subgroup given by the Cremona group of automorphisms of



k
(

x

1


,
.
.
.
,

x

n


)


{\displaystyle k(x_{1},...,x_{n})}

over the field



k
(

x

1


,
.
.
.
,

x

r


)


{\displaystyle k(x_{1},...,x_{r})}

, and the quotient group is the Cremona group of



k
(

x

1


,
.
.
.
,

x

r


)


{\displaystyle k(x_{1},...,x_{r})}

over the field



k


{\displaystyle k}

. It can also be regarded as the group of birational automorphisms of the fiber bundle





P


r


×


P


n
−
r


→


P


r




{\displaystyle \mathbb {P} ^{r}\times \mathbb {P} ^{n-r}\to \mathbb {P} ^{r}}

.
When



n
=
2


{\displaystyle n=2}

and



r
=
1


{\displaystyle r=1}

the De Jonquières group is the group of Cremona transformations fixing a pencil of lines through a given point, and is the semidirect product of






P
G
L


2


(
k
)


{\displaystyle \mathrm {PGL} _{2}(k)}

and





P
G
L


2


(
k
(
t
)
)


{\displaystyle \mathrm {PGL} _{2}(k(t))}

.




References


Alberich-Carramiñana, Maria (2002), Geometry of the plane Cremona maps, Lecture Notes in Mathematics, vol. 1769, Berlin, New York: Springer-Verlag, doi:10.1007/b82933, ISBN 978-3-540-42816-9, MR 1874328
Blanc, Jérémy (2010), "Groupes de Cremona, connexité et simplicité", Annales Scientifiques de l'École Normale Supérieure, Série 4, 43 (2): 357–364, arXiv:0903.2489, doi:10.24033/asens.2123, ISSN 0012-9593, MR 2662668
Cantat, Serge; Lamy, Stéphane (2010). "Normal subgroups in the Cremona group". Acta Mathematica. 210 (2013): 31–94. arXiv:1007.0895. Bibcode:2010arXiv1007.0895C. doi:10.1007/s11511-013-0090-1. S2CID 55261367.
Coolidge, Julian Lowell (1931), A treatise on algebraic plane curves, Oxford University Press, ISBN 978-0-486-49576-7, MR 0120551
Cremona, L. (1863), "Sulle trasformazioni geometriche delle figure piane (nota 1)", Giornale di Matematiche di Battaglini, 1: 305–311
Cremona, L. (1865), "Sulle trasformazioni geometriche delle figure piane (nota 2)", Giornale di Matematiche di Battaglini, 3: 269–280, 363–376
Demazure, Michel (1970), "Sous-groupes algébriques de rang maximum du groupe de Cremona", Annales Scientifiques de l'École Normale Supérieure, Série 4, 3 (4): 507–588, doi:10.24033/asens.1201, ISSN 0012-9593, MR 0284446
Dolgachev, Igor V. (2012), Classical Algebraic Geometry: a modern view (PDF), Cambridge University Press, ISBN 978-1-107-01765-8, archived from the original (PDF) on 2012-03-11, retrieved 2012-04-18
Dolgachev, Igor V.; Iskovskikh, Vasily A. (2009), "Finite subgroups of the plane Cremona group", Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, Progr. Math., vol. 269, Boston, MA: Birkhäuser Boston, pp. 443–548, arXiv:math/0610595, doi:10.1007/978-0-8176-4745-2_11, ISBN 978-0-8176-4744-5, MR 2641179, S2CID 2188718
Gizatullin, M. Kh. (1983), "Defining relations for the Cremona group of the plane", Mathematics of the USSR-Izvestiya, 21 (2): 211–268, Bibcode:1983IzMat..21..211G, doi:10.1070/IM1983v021n02ABEH001789, ISSN 0373-2436, MR 0675525
Godeaux, Lucien (1927), Les transformations birationelles du plan, Mémorial des sciences mathématiques, vol. 22, Gauthier-Villars et Cie, JFM 53.0595.02
"Cremona group", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
"Cremona transformation", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Hudson, Hilda Phoebe (1927), Cremona transformations in plane and space, Cambridge University Press, ISBN 978-0-521-35882-8, Reprinted 2012
Semple, J. G.; Roth, L. (1985), Introduction to algebraic geometry, Oxford Science Publications, The Clarendon Press Oxford University Press, ISBN 978-0-19-853363-4, MR 0814690
Serre, Jean-Pierre (2009), "A Minkowski-style bound for the orders of the finite subgroups of the Cremona group of rank 2 over an arbitrary field", Moscow Mathematical Journal, 9 (1): 193–208, doi:10.17323/1609-4514-2009-9-1-183-198, ISSN 1609-3321, MR 2567402, S2CID 13589478
Serre, Jean-Pierre (2010), "Le groupe de Cremona et ses sous-groupes finis" (PDF), Astérisque, Seminaire Bourbaki 1000 (332): 75–100, ISBN 978-2-85629-291-4, ISSN 0303-1179, MR 2648675

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