Jacobian varieties
E860099
Jacobian varieties are complex algebraic varieties associated to algebraic curves that parametrize degree-zero line bundles (or divisor classes) on the curve and carry a natural structure of principally polarized abelian varieties.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Jacobian of a curve | 1 |
| Jacobian varieties canonical | 1 |
| Picard variety | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10388947 — 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: Jacobian varieties Context triple: [Sur les courbes algébriques et les variétés qui s’en déduisent, topic, Jacobian varieties]
-
A.
Shimura varieties
Shimura varieties are higher-dimensional algebraic varieties that generalize modular curves and play a central role in the Langlands program by connecting number theory, automorphic forms, and arithmetic geometry.
-
B.
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.
-
C.
Jacobi's inversion problem
Jacobi's inversion problem is a fundamental question in algebraic geometry and the theory of Abelian functions, concerning the inversion of Abelian integrals and the characterization of their multi-valued inverses.
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D.
Brill–Noether theory
Brill–Noether theory is a branch of algebraic geometry that studies linear series on algebraic curves, particularly the existence and dimension of spaces of special divisors and maps to projective spaces.
-
E.
Hodge–Riemann bilinear relations
The Hodge–Riemann bilinear relations are fundamental positivity and orthogonality conditions on the intersection form in Hodge theory that underpin results such as the hard Lefschetz theorem and the Hodge index theorem.
- 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: Jacobian varieties Target entity description: Jacobian varieties are complex algebraic varieties associated to algebraic curves that parametrize degree-zero line bundles (or divisor classes) on the curve and carry a natural structure of principally polarized abelian varieties.
-
A.
Shimura varieties
Shimura varieties are higher-dimensional algebraic varieties that generalize modular curves and play a central role in the Langlands program by connecting number theory, automorphic forms, and arithmetic geometry.
-
B.
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.
-
C.
Jacobi's inversion problem
Jacobi's inversion problem is a fundamental question in algebraic geometry and the theory of Abelian functions, concerning the inversion of Abelian integrals and the characterization of their multi-valued inverses.
-
D.
Brill–Noether theory
Brill–Noether theory is a branch of algebraic geometry that studies linear series on algebraic curves, particularly the existence and dimension of spaces of special divisors and maps to projective spaces.
-
E.
Hodge–Riemann bilinear relations
The Hodge–Riemann bilinear relations are fundamental positivity and orthogonality conditions on the intersection form in Hodge theory that underpin results such as the hard Lefschetz theorem and the Hodge index theorem.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
abelian variety
ⓘ
algebraic variety ⓘ complex torus ⓘ principally polarized abelian variety ⓘ |
| associatedTo |
algebraic curve
ⓘ
compact Riemann surface ⓘ smooth projective curve ⓘ |
| carriesStructure |
group variety
ⓘ
principally polarization ⓘ |
| constructedAs |
H^0(C,Ω^1)^∨ / H_1(C,ℤ)
ⓘ
quotient of C^g by a lattice ⓘ |
| contains | theta divisor ⓘ |
| definedOver |
base field of the curve
ⓘ
ℂ for complex curves ⓘ |
| dependsOn | isomorphism class of the curve ⓘ |
| determines | curve up to isomorphism for genus ≥ 2 (via Torelli theorem) ⓘ |
| dimension | genus of the curve ⓘ |
| functorialIn | morphisms of curves ⓘ |
| generalizes | elliptic curve ⓘ |
| groupLaw | addition of divisor classes ⓘ |
| hasEndomorphismRing | End(J(C)) ⓘ |
| hasInvariant | Néron–Tate height (in arithmetic setting) NERFINISHED ⓘ |
| hasLattice | period lattice ⓘ |
| hasPoint | origin corresponding to trivial line bundle ⓘ |
| hasPolarization | theta divisor ⓘ |
| isCommutative | true ⓘ |
| isomorphicTo |
Pic^0(C)
NERFINISHED
ⓘ
Picard variety of degree zero NERFINISHED ⓘ |
| isPrincipallyPolarized | true ⓘ |
| isProjective | true ⓘ |
| mayHaveProperty | complex multiplication ⓘ |
| parametrizes |
degree-zero line bundles on a curve
ⓘ
divisor classes of degree zero on a curve ⓘ |
| relatedTo |
Abel map
NERFINISHED
ⓘ
Abel–Jacobi map NERFINISHED ⓘ Albanese variety of the curve ⓘ Picard group of the curve NERFINISHED ⓘ Riemann theta function NERFINISHED ⓘ Torelli theorem NERFINISHED ⓘ period matrix of a curve ⓘ |
| specialCase | elliptic curve when genus equals 1 ⓘ |
| universalProperty |
universal abelian variety receiving a map from the curve
ⓘ
universal regular quotient of degree-zero divisors ⓘ |
| usedIn |
algebraic geometry
ⓘ
arithmetic geometry ⓘ number theory ⓘ theory of moduli of curves ⓘ |
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: Jacobian varieties Description of subject: Jacobian varieties are complex algebraic varieties associated to algebraic curves that parametrize degree-zero line bundles (or divisor classes) on the curve and carry a natural structure of principally polarized abelian varieties.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Picard variety
this entity surface form:
Jacobian of a curve