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.
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
Weil divisor generalization
ⓘ
divisor in algebraic geometry ⓘ geometric object ⓘ |
| appearsIn |
EGA II
NERFINISHED
ⓘ
Hartshorne Algebraic Geometry NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| relatedTo | Picard group Pic(X) NERFINISHED ⓘ |
| 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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.