Picard group
E860119
algebraic invariant
birational invariant (for smooth projective varieties over a field)
functor
group
The Picard group is an algebraic invariant of a variety or scheme that classifies line bundles (or divisor classes) up to isomorphism, playing a central role in algebraic geometry.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Picard group canonical | 4 |
| Pic^0(C) | 1 |
| Picard group Pic(X) | 1 |
| Picard group of the curve | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10389151 — 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: Picard group Context triple: [Weil divisor, relatedTo, Picard group]
-
A.
Grothendieck group
The Grothendieck group is an algebraic construction that formally turns a commutative monoid (often arising from isomorphism classes of objects like vector bundles or modules) into an abelian group, playing a central role in K-theory and modern algebraic geometry.
-
B.
Brauer group
The Brauer group is an algebraic structure that classifies equivalence classes of central simple algebras over a field (or more general schemes), playing a key role in number theory, algebraic geometry, and cohomology.
-
C.
Weil group
The Weil group is an extension of the absolute Galois group introduced by André Weil to refine class field theory and play a central role in the formulation of the local and global Langlands correspondences.
-
D.
Pontryagin classes
Pontryagin classes are characteristic classes associated with real vector bundles that capture topological information about the bundle’s curvature and play a central role in differential topology and geometry.
-
E.
Alexander–Spanier cohomology
Alexander–Spanier cohomology is a cohomology theory in algebraic topology defined using cochains on all finite subsets of a space, notable for its generality and close relationship to Čech and singular cohomology.
- 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: Picard group Target entity description: The Picard group is an algebraic invariant of a variety or scheme that classifies line bundles (or divisor classes) up to isomorphism, playing a central role in algebraic geometry.
-
A.
Grothendieck group
The Grothendieck group is an algebraic construction that formally turns a commutative monoid (often arising from isomorphism classes of objects like vector bundles or modules) into an abelian group, playing a central role in K-theory and modern algebraic geometry.
-
B.
Brauer group
The Brauer group is an algebraic structure that classifies equivalence classes of central simple algebras over a field (or more general schemes), playing a key role in number theory, algebraic geometry, and cohomology.
-
C.
Weil group
The Weil group is an extension of the absolute Galois group introduced by André Weil to refine class field theory and play a central role in the formulation of the local and global Langlands correspondences.
-
D.
Pontryagin classes
Pontryagin classes are characteristic classes associated with real vector bundles that capture topological information about the bundle’s curvature and play a central role in differential topology and geometry.
-
E.
Alexander–Spanier cohomology
Alexander–Spanier cohomology is a cohomology theory in algebraic topology defined using cochains on all finite subsets of a space, notable for its generality and close relationship to Čech and singular cohomology.
- F. None of above. chosen
Statements (51)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic invariant
ⓘ
birational invariant (for smooth projective varieties over a field) ⓘ functor ⓘ group ⓘ |
| affineExample | Pic(A^n_k) = 0 for affine n-space over a field k ⓘ |
| appearsIn | Grothendieck’s formulation of algebraic geometry ⓘ |
| captures | algebraic equivalence classes of divisors (via Pic^0 and NS) ⓘ |
| classifies |
Cartier divisors modulo linear equivalence (under suitable hypotheses)
ⓘ
invertible sheaves ⓘ isomorphism classes of line bundles ⓘ |
| cohomologicalDescription | Pic(X) ≅ H^1(X, O_X^×) in the Zariski or étale topology ⓘ |
| coincidesWith |
divisor class group for a smooth projective variety over a field
ⓘ
group of Cartier divisors modulo linear equivalence for a regular integral scheme ⓘ |
| construction | group of isomorphism classes of line bundles on a scheme X with tensor product as operation ⓘ |
| curveExample | for a smooth projective curve C over an algebraically closed field, Pic^0(C) is isomorphic to the Jacobian of C ⓘ |
| decomposition | for a smooth projective variety over an algebraically closed field, Pic(X) has a connected component Pic^0(X) and a discrete part NS(X) ⓘ |
| definedBy | Émile Picard (historical origin of the concept) ⓘ |
| degreeMap | for a smooth projective curve C, there is a degree homomorphism deg: Pic(C) → Z ⓘ |
| field | algebraic geometry ⓘ |
| functoriality | contravariant in the scheme: a morphism f:Y→X induces f* : Pic(X) → Pic(Y) ⓘ |
| generalizationOf | ideal class group of a Dedekind domain (via Spec of the ring) ⓘ |
| groupOperation | tensor product of line bundles ⓘ |
| hasSubgroup | Néron–Severi group NS(X) as image of Pic(X) in numerical equivalence classes ⓘ |
| identityElement | class of the trivial line bundle O_X ⓘ |
| inverseElement | dual line bundle L^∨ ⓘ |
| isDefinedFor |
ringed spaces (in general form)
ⓘ
schemes ⓘ varieties ⓘ |
| kernelOfDegreeMap | Pic^0(C) for a smooth projective curve C ⓘ |
| namedAfter | Émile Picard ⓘ |
| notation |
Pic
ⓘ
Pic(X) ⓘ |
| Pic0Component | Pic^0(X) is an abelian variety for smooth projective X over an algebraically closed field ⓘ |
| projectiveLineExample | Pic(P^1_k) ≅ Z ⓘ |
| projectiveSpaceExample | Pic(P^n_k) ≅ Z for n ≥ 1 ⓘ |
| relatedConcept |
Brauer group
ⓘ
Cartier divisor ⓘ Jacobian variety ⓘ
linked to:
Jacobian varieties
Néron–Severi group ⓘ Picard scheme ⓘ Weil divisor ⓘ divisor class group ⓘ |
| relatedTo | class field theory via line bundles and divisors on curves ⓘ |
| topologicalAnalogue | for a complex manifold X, Pic(X) relates to H^2(X, Z) via the exponential sequence ⓘ |
| torsionSubgroup | classes of line bundles of finite order under tensor product ⓘ |
| usedIn |
classification of line bundles on algebraic varieties
ⓘ
intersection theory ⓘ moduli problems in algebraic geometry ⓘ study of ampleness and positivity of line bundles ⓘ study of divisors and linear systems ⓘ |
| zeroPicardGroupExample | Pic(Spec k) = 0 for a field k ⓘ |
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: Picard group Description of subject: The Picard group is an algebraic invariant of a variety or scheme that classifies line bundles (or divisor classes) up to isomorphism, playing a central role in algebraic geometry.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
Jacobian varieties
linked to: Picard group
subject linked to:
Jacobian varieties
linked to: Picard group
linked to: Picard group