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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Cremona group of the projective plane canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7997815 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Cremona group of the projective plane Context triple: [Noether’s AF+BG theorem, relatedTo, Cremona group of the projective plane]
-
A.
Clebsch–Aronhold invariants
The Clebsch–Aronhold invariants are classical algebraic invariants associated with binary forms, particularly quartic forms, that play a key role in invariant theory and the classification of algebraic curves.
-
B.
Veblen axioms for projective geometry
The Veblen axioms for projective geometry are a foundational set of incidence-based axioms introduced by Oswald Veblen to rigorously formalize the structure of projective spaces.
-
C.
Clebsch diagonal surfaces
Clebsch diagonal surfaces are classical 19th-century algebraic surfaces in projective three-space, famous as the first explicit smooth cubic surface with all 27 lines defined over the real numbers.
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D.
Erlangen Program
The Erlangen Program is Felix Klein’s influential 1872 framework that classifies and studies geometries based on their underlying symmetry groups and transformation properties.
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E.
The Real Projective Plane
The Real Projective Plane is a classic mathematical monograph by H. S. M. Coxeter that systematically develops the geometry and topology of the real projective plane, emphasizing its axiomatic foundations and non-Euclidean properties.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Cremona group of the projective plane Target entity description: 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.
-
A.
Clebsch–Aronhold invariants
The Clebsch–Aronhold invariants are classical algebraic invariants associated with binary forms, particularly quartic forms, that play a key role in invariant theory and the classification of algebraic curves.
-
B.
Veblen axioms for projective geometry
The Veblen axioms for projective geometry are a foundational set of incidence-based axioms introduced by Oswald Veblen to rigorously formalize the structure of projective spaces.
-
C.
Clebsch diagonal surfaces
Clebsch diagonal surfaces are classical 19th-century algebraic surfaces in projective three-space, famous as the first explicit smooth cubic surface with all 27 lines defined over the real numbers.
-
D.
Erlangen Program
The Erlangen Program is Felix Klein’s influential 1872 framework that classifies and studies geometries based on their underlying symmetry groups and transformation properties.
-
E.
The Real Projective Plane
The Real Projective Plane is a classic mathematical monograph by H. S. M. Coxeter that systematically develops the geometry and topology of the real projective plane, emphasizing its axiomatic foundations and non-Euclidean properties.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Cremona group of the projective plane Description of subject: 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.
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