Cayley surface
E930419
The Cayley surface is a classical cubic ruled surface in projective three-dimensional space, studied in algebraic geometry and named after the mathematician Arthur Cayley.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Cayley surface canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T11507795 — 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: Cayley surface Context triple: [Arthur Cayley, notableWork, Cayley surface]
-
A.
Fermat surface
A Fermat surface is an algebraic surface in projective space defined by a homogeneous equation where each variable appears with the same exponent, generalizing the notion of Fermat curves to higher dimensions.
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B.
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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C.
Clebsch
Clebsch is a German surname most notably associated with mathematician Alfred Clebsch, known for his contributions to algebraic geometry and invariant theory.
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D.
Klein quartic
The Klein quartic is a highly symmetric algebraic curve of genus 3 that plays a central role in complex geometry, group theory, and the study of Riemann surfaces.
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E.
Kummer surfaces
Kummer surfaces are special quartic algebraic surfaces in projective three-space characterized by having 16 ordinary double points, extensively studied in the context of complex geometry and abelian varieties.
- 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: Cayley surface Target entity description: The Cayley surface is a classical cubic ruled surface in projective three-dimensional space, studied in algebraic geometry and named after the mathematician Arthur Cayley.
-
A.
Fermat surface
A Fermat surface is an algebraic surface in projective space defined by a homogeneous equation where each variable appears with the same exponent, generalizing the notion of Fermat curves to higher dimensions.
-
B.
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.
-
C.
Clebsch
Clebsch is a German surname most notably associated with mathematician Alfred Clebsch, known for his contributions to algebraic geometry and invariant theory.
-
D.
Klein quartic
The Klein quartic is a highly symmetric algebraic curve of genus 3 that plays a central role in complex geometry, group theory, and the study of Riemann surfaces.
-
E.
Kummer surfaces
Kummer surfaces are special quartic algebraic surfaces in projective three-space characterized by having 16 ordinary double points, extensively studied in the context of complex geometry and abelian varieties.
- F. None of above. chosen
Statements (32)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic surface
ⓘ
cubic surface ⓘ projective variety ⓘ ruled surface ⓘ |
| ambientSpace |
P^3
ⓘ
projective 3-space ⓘ |
| contains |
a one-parameter family of lines
ⓘ
infinitely many lines ⓘ |
| degree | 3 ⓘ |
| dimension | 2 ⓘ |
| fieldOfDefinition | complex numbers ⓘ |
| hasCoordinateDescription | can be given by a cubic polynomial in homogeneous coordinates [x:y:z:w] on P^3 ⓘ |
| hasProperty |
classical surface
ⓘ
irreducible surface ⓘ non-smooth surface ⓘ projective surface ⓘ rational surface ⓘ ruled by lines ⓘ singular surface ⓘ |
| hasSingularity |
ordinary double point
ⓘ
rational double point ⓘ |
| isClassicalExampleOf |
ruled cubic surface
ⓘ
singular cubic surface ⓘ |
| isDefinedBy | homogeneous cubic equation in four variables ⓘ |
| isRelatedTo |
Fano varieties
ⓘ
birational geometry ⓘ projective geometry ⓘ |
| isUsedAs |
example in classification of algebraic surfaces
ⓘ
example in the study of cubic hypersurfaces ⓘ example in the study of ruled surfaces ⓘ |
| namedAfter | Arthur Cayley ⓘ |
| studiedIn | algebraic geometry ⓘ |
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: Cayley surface Description of subject: The Cayley surface is a classical cubic ruled surface in projective three-dimensional space, studied in algebraic geometry and named after the mathematician Arthur Cayley.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
Cayley