Fermat surface
E530319
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.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Fermat cubic surface (n = 3) | 1 |
| Fermat hypersurface | 1 |
| Fermat quartic surface (n = 4) | 1 |
| Fermat quintic surface (n = 5) | 1 |
| Fermat surface canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5570775 — 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: Fermat surface Context triple: [Fermat curve, relatedObject, Fermat surface]
-
A.
Fermat curve
A Fermat curve is an algebraic curve defined by an equation of the form \(x^n + y^n = 1\), studied in number theory and algebraic geometry for its rich arithmetic and geometric properties.
-
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.
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.
-
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.
Fermat polygonal number theorem
The Fermat polygonal number theorem is a result in number theory stating that every positive integer can be expressed as a sum of a fixed number of polygonal numbers of a given order.
- 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: Fermat surface Target entity description: 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.
-
A.
Fermat curve
A Fermat curve is an algebraic curve defined by an equation of the form \(x^n + y^n = 1\), studied in number theory and algebraic geometry for its rich arithmetic and geometric properties.
-
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.
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.
-
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.
Fermat polygonal number theorem
The Fermat polygonal number theorem is a result in number theory stating that every positive integer can be expressed as a sum of a fixed number of polygonal numbers of a given order.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic surface
ⓘ
complex surface ⓘ hypersurface ⓘ projective variety ⓘ smooth surface ⓘ |
| appearsIn | classification of algebraic surfaces ⓘ |
| definedIn | projective 3-space ⓘ |
| definedOver |
algebraically closed field
ⓘ
complex numbers ⓘ |
| embeddedIn | projective space P^3 ⓘ |
| generalizes | Fermat curve ⓘ |
| hasAutomorphismGroupContaining |
(μ_n)^4 / μ_n
ⓘ
symmetric group S_4 ⓘ |
| hasBettiNumber | b_2 depending explicitly on n ⓘ |
| hasCanonicalBundle | O(n−4) ⓘ |
| hasDefiningEquation | x^n + y^n + z^n + w^n = 0 ⓘ |
| hasDegree | n ⓘ |
| hasDimension | 2 ⓘ |
| hasEulerCharacteristic | topological Euler characteristic depending polynomially on n ⓘ |
| hasHodgeStructure | pure Hodge structure of weight 2 on H^2 ⓘ |
| hasKodairaDimension |
0 for n = 4 (K3 case)
ⓘ
2 for n ≥ 5 ⓘ −∞ for n = 3 (cubic surface) ⓘ |
| hasLFunction | expressible in terms of Jacobi sums over finite fields ⓘ |
| hasModuli | discrete for fixed n up to projective equivalence ⓘ |
| hasNeronSeveriGroup | generated by explicit algebraic cycles for many n ⓘ |
| hasParameter | degree n ≥ 3 ⓘ |
| hasPicardNumber | often large compared to generic surface of same degree ⓘ |
| hasProperty |
Kähler surface
ⓘ
minimal surface of general type for n ≥ 5 ⓘ simply connected (over C) ⓘ |
| hasSpecialCase |
Fermat cubic surface (n = 3)
ⓘ
linked to:
Fermat surface
Fermat quartic surface (n = 4) ⓘ
linked to:
Fermat surface
Fermat quintic surface (n = 5) ⓘ
linked to:
Fermat surface
|
| hasSymmetryGroup |
(μ_n)^4 / μ_n
ⓘ
S_4 ⓘ |
| is | two-dimensional projective variety ⓘ |
| isSingularIf | characteristic of base field divides n ⓘ |
| isSmoothIf | characteristic of base field does not divide n ⓘ |
| isSpecialCaseOf |
Fermat hypersurface
ⓘ
linked to:
Fermat surface
|
| liesIn | projective 3-space P^3(k) ⓘ |
| namedAfter | Pierre de Fermat ⓘ |
| relatedTo |
Jacobi sums
ⓘ
cyclotomic fields ⓘ |
| studiedIn |
Hodge theory
ⓘ
algebraic geometry ⓘ complex geometry ⓘ |
| usedToStudy | zeta functions of varieties over finite fields ⓘ |
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: Fermat surface Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Fermat surface
linked to: Fermat surface
linked to: Fermat surface
linked to: Fermat surface