Clebsch diagonal surfaces
E262450
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Clebsch cubic surface | 2 |
| Clebsch diagonal surfaces canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2408489 — 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: Clebsch diagonal surfaces Context triple: [Alfred Clebsch, notableWork, Clebsch diagonal surfaces]
-
A.
Sur les courbes algébriques et les variétés qui s’en déduisent
Sur les courbes algébriques et les variétés qui s’en déduisent is a foundational 1948 monograph by André Weil that helped establish modern algebraic geometry and introduced key ideas leading to the Weil conjectures.
-
B.
Hilbert’s fourteenth problem
Hilbert’s fourteenth problem is one of David Hilbert’s famous list of 23 problems, concerning the finite generation of certain algebras of invariants in algebraic geometry and invariant theory.
-
C.
Über die Bildung des Formensystems der ternären biquadratischen Form
"Über die Bildung des Formensystems der ternären biquadratischen Form" is the 1907 doctoral dissertation of mathematician Emmy Noether, in which she investigates the invariant theory of certain higher-degree algebraic forms.
-
D.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
-
E.
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.
- 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: Clebsch diagonal surfaces Target entity description: 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.
-
A.
Sur les courbes algébriques et les variétés qui s’en déduisent
Sur les courbes algébriques et les variétés qui s’en déduisent is a foundational 1948 monograph by André Weil that helped establish modern algebraic geometry and introduced key ideas leading to the Weil conjectures.
-
B.
Hilbert’s fourteenth problem
Hilbert’s fourteenth problem is one of David Hilbert’s famous list of 23 problems, concerning the finite generation of certain algebras of invariants in algebraic geometry and invariant theory.
-
C.
Über die Bildung des Formensystems der ternären biquadratischen Form
"Über die Bildung des Formensystems der ternären biquadratischen Form" is the 1907 doctoral dissertation of mathematician Emmy Noether, in which she investigates the invariant theory of certain higher-degree algebraic forms.
-
D.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
-
E.
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.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
Fano surface
ⓘ
algebraic surface ⓘ cubic surface ⓘ projective variety ⓘ smooth cubic surface ⓘ |
| allLinesDefinedOver | real numbers ⓘ |
| appearsIn | classical classification of cubic surfaces ⓘ |
| canBeEmbeddedIn | P^3 via linear projection from P^4 ⓘ |
| canBeRealizedAs | intersection of a cubic hypersurface and a hyperplane in P^4 ⓘ |
| definedOver |
complex numbers
ⓘ
real numbers ⓘ |
| degree | 3 ⓘ |
| dimension | 2 ⓘ |
| discoveredInCentury | 19th century ⓘ |
| embeddedIn |
P^3
ⓘ
projective 3-space ⓘ |
| fieldOfStudy |
algebraic geometry
ⓘ
classical algebraic geometry ⓘ |
| hasAnticanonicalEmbedding | into P^3 as a cubic surface ⓘ |
| hasAutomorphismGroup | symmetric group S5 ⓘ |
| hasBettiNumber | b2 = 7 ⓘ |
| hasCanonicalBundle | anti-ample ⓘ |
| hasEquationForm | sum x_i = 0 and sum x_i^3 = 0 in P^4 ⓘ |
| hasEulerCharacteristic | 3 ⓘ |
| hasHodgeNumbers |
h^{1,0} = 0
ⓘ
h^{1,1} = 7 ⓘ h^{2,0} = 0 ⓘ |
| hasLinesConfiguration | 27 lines in classical cubic surface configuration ⓘ |
| hasNumberOfLines | 27 ⓘ |
| hasPicardNumber | 7 ⓘ |
| hasProperty |
all 27 lines are pairwise skew or intersect according to cubic surface incidence rules
ⓘ
all 27 lines are real ⓘ highly symmetric cubic surface ⓘ |
| hasRealForm | unique up to projective equivalence with all 27 lines real ⓘ |
| hasRealStructure | yes ⓘ |
| hasSymmetryGroup | S5 ⓘ |
| hasType | smooth projective rational surface ⓘ |
| isBirationalTo | blow-up of P^2 in six points ⓘ |
| isClassicalObjectIn | 19th-century projective geometry ⓘ |
| isExampleOf | del Pezzo surface of degree 3 ⓘ |
| isOftenDefinedBy | homogeneous coordinates satisfying x0 + x1 + x2 + x3 + x4 = 0 and x0^3 + x1^3 + x2^3 + x3^3 + x4^3 = 0 ⓘ |
| isRationalSurface | yes ⓘ |
| namedAfter | Alfred Clebsch ⓘ |
| notableFor | being the first explicit smooth cubic surface with all 27 lines real ⓘ |
| relatedTo |
Weyl group
ⓘ
surface form:
Weyl group of type E6 via lines configuration
|
| usedAs | standard example in the theory of cubic 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: Clebsch diagonal surfaces Description of subject: 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.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Clebsch cubic surface
this entity surface form:
Clebsch cubic surface