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.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Kummer surface | 0 |
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
NERFINISHED
ⓘ
Felix Klein NERFINISHED ⓘ |
| namedAfter | Ernst Kummer NERFINISHED ⓘ |
| numberOf2TorsionPoints | 16 ⓘ |
| numberOfSingularPoints | 16 ⓘ |
| PicardNumber | at least 17 ⓘ |
| quotientOf | abelian surface by the involution x ↦ −x ⓘ |
| relatedTo |
Enriques surfaces
NERFINISHED
ⓘ
Shioda–Inose structures NERFINISHED ⓘ Siegel modular forms ⓘ abelian surface ⓘ moduli of abelian surfaces ⓘ theta functions ⓘ |
| specialCaseOf | Kummer variety NERFINISHED ⓘ |
| studiedIn |
algebraic geometry
ⓘ
birational geometry ⓘ complex geometry ⓘ |
| usedIn | construction of certain singular K3 surfaces ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.