Kummer surfaces
E463785
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.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Kummer surface | 1 |
| Kummer surfaces canonical | 1 |
| Kummer surfaces in algebraic geometry | 1 |
| Kummer variety | 1 |
| Shioda–Inose structures | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4706377 — 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: Kummer surfaces Context triple: [Ernst Eduard Kummer, knownFor, Kummer surfaces]
-
A.
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.
-
B.
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.
-
C.
Theorie der algebraischen Kurven
"Theorie der algebraischen Kurven" is a foundational 19th-century mathematical treatise by Julius Plücker that systematically develops the geometry and classification of algebraic curves.
-
D.
Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
-
E.
Singular Points of Complex Hypersurfaces
"Singular Points of Complex Hypersurfaces" is a foundational monograph in singularity theory that systematically studies the local and topological properties of singularities arising in complex algebraic and analytic hypersurfaces.
- 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: Kummer surfaces Target entity description: 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.
-
A.
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.
-
B.
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.
-
C.
Theorie der algebraischen Kurven
"Theorie der algebraischen Kurven" is a foundational 19th-century mathematical treatise by Julius Plücker that systematically develops the geometry and classification of algebraic curves.
-
D.
Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
-
E.
Singular Points of Complex Hypersurfaces
"Singular Points of Complex Hypersurfaces" is a foundational monograph in singularity theory that systematically studies the local and topological properties of singularities arising in complex algebraic and analytic hypersurfaces.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
K3 surface
ⓘ
algebraic surface ⓘ complex surface ⓘ quartic surface ⓘ |
| ambientSpace |
P^3
ⓘ
projective 3-space ⓘ |
| constructedAs | minimal resolution of A modulo ±1 ⓘ |
| constructedFrom | abelian surface ⓘ |
| contains |
16 singular points
ⓘ
16 tropes ⓘ |
| definedOver | projective three-space ⓘ |
| degree | 4 ⓘ |
| dimension | 2 ⓘ |
| EulerCharacteristic | 24 ⓘ |
| generalizesTo | higher-dimensional Kummer varieties ⓘ |
| hasConfiguration | 16_6 configuration of nodes and tropes ⓘ |
| hasInvolution | x ↦ −x on the abelian surface ⓘ |
| hasModel | quartic equation in homogeneous coordinates on P^3 ⓘ |
| hasProperty |
H^1(O)=0 after minimal resolution
ⓘ
admits models over finite fields ⓘ can be defined over number fields ⓘ can be minimally resolved to a smooth K3 surface ⓘ maximal number of nodes for a quartic surface in P^3 ⓘ nodal surface ⓘ often has high Picard rank in arithmetic settings ⓘ rational double points ⓘ simply connected after minimal resolution ⓘ trivial canonical bundle after minimal resolution ⓘ |
| hasSingularitiesAt | 2-torsion points of the abelian surface ⓘ |
| hasSingularityType | ordinary double point ⓘ |
| hasSymmetry | large finite automorphism group ⓘ |
| historicallyStudiedBy |
Ernst Kummer
ⓘ
linked to:
Ernst Eduard Kummer
Felix Klein ⓘ |
| namedAfter |
Ernst Kummer
ⓘ
linked to:
Ernst Eduard Kummer
|
| numberOf2TorsionPoints | 16 ⓘ |
| numberOfSingularPoints | 16 ⓘ |
| PicardNumber | at least 17 ⓘ |
| quotientOf | abelian surface by the involution x ↦ −x ⓘ |
| relatedTo |
Enriques surfaces
ⓘ
Shioda–Inose structures ⓘ
linked to:
Kummer surfaces
Siegel modular forms ⓘ abelian surface ⓘ moduli of abelian surfaces ⓘ theta functions ⓘ |
| specialCaseOf |
Kummer variety
ⓘ
linked to:
Kummer surfaces
|
| studiedIn |
algebraic geometry
ⓘ
birational geometry ⓘ complex geometry ⓘ |
| usedIn | construction of certain singular K3 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: Kummer surfaces Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
Kummer
linked to: Kummer surfaces
subject linked to:
Kummer surfaces
linked to: Kummer surfaces
subject linked to:
Kummer surfaces
linked to: Kummer surfaces