Cartier divisor
E860118
A Cartier divisor is a type of divisor on an algebraic variety defined locally by a single rational function, corresponding to an invertible sheaf and generalizing the notion of a principal divisor.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Cartier divisor canonical | 4 |
How this entity was disambiguated
This entity first appeared as the object of triple T10389148 — 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: Cartier divisor Context triple: [Weil divisor, relatedTo, Cartier divisor]
-
A.
Weil divisor
A Weil divisor is a formal integer linear combination of irreducible subvarieties of codimension one on an algebraic variety, used to study its geometric and arithmetic properties.
-
B.
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.
-
C.
Lefschetz pencil
A Lefschetz pencil is a geometric structure on an algebraic variety given by a one-parameter family of hyperplane sections with only isolated, well-controlled singularities, fundamental in the study of its topology and geometry.
-
D.
Chern classes
Chern classes are fundamental topological invariants in differential and algebraic geometry that classify complex vector bundles and capture their curvature and twisting properties.
-
E.
Plücker formulas
Plücker formulas are classical algebraic geometry relations that connect the degree and singularities of plane algebraic curves with the invariants of their dual curves.
- 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: Cartier divisor Target entity description: A Cartier divisor is a type of divisor on an algebraic variety defined locally by a single rational function, corresponding to an invertible sheaf and generalizing the notion of a principal divisor.
-
A.
Weil divisor
A Weil divisor is a formal integer linear combination of irreducible subvarieties of codimension one on an algebraic variety, used to study its geometric and arithmetic properties.
-
B.
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.
-
C.
Lefschetz pencil
A Lefschetz pencil is a geometric structure on an algebraic variety given by a one-parameter family of hyperplane sections with only isolated, well-controlled singularities, fundamental in the study of its topology and geometry.
-
D.
Chern classes
Chern classes are fundamental topological invariants in differential and algebraic geometry that classify complex vector bundles and capture their curvature and twisting properties.
-
E.
Plücker formulas
Plücker formulas are classical algebraic geometry relations that connect the degree and singularities of plane algebraic curves with the invariants of their dual curves.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
Weil divisor generalization
ⓘ
divisor in algebraic geometry ⓘ geometric object ⓘ |
| appearsIn |
EGA II
ⓘ
Hartshorne Algebraic Geometry ⓘ |
| associatedTo | rational map to projective space via complete linear system ⓘ |
| canBeEffective | yes ⓘ |
| canBePulledBackAlong | morphism of schemes ⓘ |
| canBeViewedAs | global section of the sheaf of Cartier divisors ⓘ |
| correspondsTo |
invertible sheaf
ⓘ
line bundle ⓘ |
| definedLocallyBy | single rational function ⓘ |
| definedOn |
algebraic variety
ⓘ
scheme ⓘ |
| defines | Cartier class in Picard group ⓘ |
| determines | invertible sheaf O_X(D) ⓘ |
| effectiveIf | locally given by regular function ⓘ |
| forms | abelian group under addition ⓘ |
| generalizes | principal divisor ⓘ |
| groupDenotedBy | Div(X) ⓘ |
| hasLocalEquation | nonzero rational function ⓘ |
| hasOperation | linear equivalence ⓘ |
| hasPoleLocus | subscheme defined by poles of local equations ⓘ |
| hasSheafTheoreticDescription | invertible subsheaf of K_X ⓘ |
| hasSubgroup | group of principal divisors ⓘ |
| hasZeroLocus | subscheme defined by vanishing of local equations ⓘ |
| isDeterminedBy | invertible sheaf up to linear equivalence ⓘ |
| isEquivalentTo | invertible subsheaf of the sheaf of total quotient rings ⓘ |
| isLocallyPrincipal | yes ⓘ |
| isReflexive | under dualization of associated line bundle ⓘ |
| isSectionOf | sheaf of total quotient rings modulo units ⓘ |
| linearlyEquivalentIf | difference is principal divisor ⓘ |
| lineBundleAssociation | O_X(D) ⓘ |
| mayNotCoincideWith | Weil divisor on singular variety ⓘ |
| namedAfter | Pierre Cartier ⓘ |
| onNormalVariety | determines Weil divisor ⓘ |
| onRegularScheme | equivalent to Weil divisor ⓘ |
| onSmoothVariety | same as Weil divisor ⓘ |
| principalSubgroupDenotedBy | Prin(X) ⓘ |
| pullbackWellDefinedIf | morphism is flat or divisor is Cartier ⓘ |
| quotientByPrincipalDivisors | Picard group ⓘ |
| relatedTo |
Picard group Pic(X)
ⓘ
linked to:
Picard group
|
| restrictsTo | Cartier divisor on open subscheme ⓘ |
| supports | closed subset of codimension at least 1 ⓘ |
| usedFor |
construction of linear systems
ⓘ
definition of canonical divisor ⓘ definition of line bundles ⓘ intersection theory ⓘ study of projective embeddings ⓘ |
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: Cartier divisor Description of subject: A Cartier divisor is a type of divisor on an algebraic variety defined locally by a single rational function, corresponding to an invertible sheaf and generalizing the notion of a principal divisor.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.