Weil divisor
E244844
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Weil divisor canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2228054 — 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: Weil divisor Context triple: [André Weil, notableConcept, Weil divisor]
-
A.
Riemann–Roch theorem
The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
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B.
Hasse–Weil zeta function
The Hasse–Weil zeta function is an analytic object in number theory that encodes arithmetic information about algebraic varieties over number fields, generalizing the Riemann zeta function and playing a central role in modern arithmetic geometry and conjectures like the Weil conjectures and the Birch–Swinnerton-Dyer conjecture.
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C.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
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D.
Zariski topology
The Zariski topology is a fundamental topology in algebraic geometry, defined on the spectrum of a ring or an algebraic variety, whose closed sets correspond to solution sets of polynomial equations.
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E.
Deuring reduction theorem
The Deuring reduction theorem is a result in number theory that relates the reduction of elliptic curves with complex multiplication modulo primes to the theory of quaternion algebras and endomorphism rings.
- 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: Weil divisor Target entity description: 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.
-
A.
Riemann–Roch theorem
The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
-
B.
Hasse–Weil zeta function
The Hasse–Weil zeta function is an analytic object in number theory that encodes arithmetic information about algebraic varieties over number fields, generalizing the Riemann zeta function and playing a central role in modern arithmetic geometry and conjectures like the Weil conjectures and the Birch–Swinnerton-Dyer conjecture.
-
C.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
-
D.
Zariski topology
The Zariski topology is a fundamental topology in algebraic geometry, defined on the spectrum of a ring or an algebraic variety, whose closed sets correspond to solution sets of polynomial equations.
-
E.
Deuring reduction theorem
The Deuring reduction theorem is a result in number theory that relates the reduction of elliptic curves with complex multiplication modulo primes to the theory of quaternion algebras and endomorphism rings.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
divisor in algebraic geometry
ⓘ
mathematical object ⓘ |
| alsoCalled | divisor ⓘ |
| appearsIn | Weil conjectures ⓘ |
| associatedTo | valuation of the function field ⓘ |
| canBePulledBackAlong | proper morphism under suitable conditions ⓘ |
| canBePushedForwardAlong | proper morphism ⓘ |
| canBeRestrictedTo | subvariety ⓘ |
| coincidesWithCartierDivisorOn | nonsingular variety ⓘ |
| definedAs | formal integer linear combination of irreducible subvarieties of codimension one ⓘ |
| definedOn | algebraic variety ⓘ |
| differsFromCartierDivisorOn | singular variety ⓘ |
| encodes | zeros and poles of rational functions ⓘ |
| generalizes | divisor on a smooth projective curve ⓘ |
| hasCoefficientType | integer ⓘ |
| hasComponentType | irreducible subvariety of codimension one ⓘ |
| hasConditionForEffectiveness | all coefficients are nonnegative integers ⓘ |
| hasEquivalenceRelation | linear equivalence of divisors ⓘ |
| hasGroupStructure | abelian group under addition ⓘ |
| hasNotation | Div(X) for group of Weil divisors on a variety X ⓘ |
| hasOperation |
addition
ⓘ
intersection with curves ⓘ linear equivalence ⓘ subtraction ⓘ |
| hasSubClass |
Cartier divisor
ⓘ
effective Weil divisor ⓘ principal divisor ⓘ |
| hasSupport | union of codimension-one subvarieties with nonzero coefficient ⓘ |
| isIntegralCombinationOf | prime divisors ⓘ |
| namedAfter | André Weil ⓘ |
| primeDivisorDefinedAs | irreducible reduced closed subscheme of codimension one ⓘ |
| quotientByLinearEquivalenceGives | divisor class group Cl(X) ⓘ |
| relatedTo |
Cartier divisor
ⓘ
Picard group ⓘ class group ⓘ line bundle ⓘ principal divisor ⓘ |
| usedIn |
algebraic geometry
ⓘ
arithmetic geometry ⓘ birational geometry ⓘ intersection theory ⓘ minimal model program ⓘ theory of linear systems on varieties ⓘ |
| usedToDefine |
Weil divisor class
ⓘ
divisor class group ⓘ |
| usedToStudy |
arithmetic properties of varieties
ⓘ
geometric properties of varieties ⓘ |
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: Weil divisor Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.