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 |
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.