Cremona group of the projective plane
E702288
birational transformation group
group under composition
infinite group
mathematical group
non‑linear group
The Cremona group of the projective plane is the group of all birational self-maps of the complex projective plane, serving as a fundamental object in algebraic geometry and the study of plane transformations.
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
birational transformation group
ⓘ
group under composition ⓘ infinite group ⓘ mathematical group ⓘ non‑linear group ⓘ |
| actsOn |
P^2(C)
ⓘ
complex projective plane ⓘ |
| alsoKnownAs |
Cremona group of P^2
NERFINISHED
ⓘ
Cremona group of the complex projective plane NERFINISHED ⓘ plane Cremona group NERFINISHED ⓘ |
| baseSpace | projective plane over C ⓘ |
| centralConceptIn |
classical algebraic geometry
ⓘ
modern birational dynamics ⓘ |
| contains |
Jonquières transformations
NERFINISHED
ⓘ
automorphism group of P^2(C) ⓘ projective linear group PGL(3,C) ⓘ standard quadratic Cremona involution NERFINISHED ⓘ |
| definedOver | complex numbers ⓘ |
| fieldOfStudy |
algebraic geometry
ⓘ
birational geometry ⓘ group theory ⓘ |
| generatedBy | PGL(3,C) and a standard quadratic transformation (Noether–Castelnuovo theorem) ⓘ |
| hasElementType | birational self‑map of P^2(C) ⓘ |
| hasProperty |
acts birationally on rational surfaces obtained by blow‑ups of P^2
ⓘ
contains elements of arbitrarily large finite order ⓘ contains elements of infinite order ⓘ contains free subgroups on two generators ⓘ contains many involutions ⓘ highly non‑amenable ⓘ not finitely generated as an abstract group ⓘ not linear over any field ⓘ uncountable ⓘ |
| hasSubgroup |
de Jonquières subgroup
NERFINISHED
ⓘ
group of automorphisms of P^2(C) ⓘ group of birational maps preserving a pencil of lines ⓘ |
| hasTypicalElement | rational map given by homogeneous polynomials of the same degree in three variables ⓘ |
| isSubgroupOf | Cremona group of the projective space of any higher dimension (via embeddings) NERFINISHED ⓘ |
| namedAfter | Luigi Cremona NERFINISHED ⓘ |
| operation | composition of rational maps ⓘ |
| relatedTo |
birational classification of surfaces
ⓘ
minimal model program ⓘ plane algebraic curves ⓘ rational surfaces ⓘ |
| studiedSince | 19th century ⓘ |
| symbol |
Bir(P^2)
NERFINISHED
ⓘ
Bir(P^2_C) NERFINISHED ⓘ Cr_2(C) NERFINISHED ⓘ |
| usedFor |
classifying birational maps of the plane
ⓘ
studying birational rigidity of surfaces ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.
Noether’s theorem in algebraic geometry (Noether’s AF+BG theorem)
→
relatedTo
→
Cremona group of the projective plane
ⓘ
subject surface form:
Noether’s AF+BG theorem