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.
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 NERFINISHED ⓘ |
| notation |
Pic
NERFINISHED
ⓘ
Pic(X) NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
Cartier divisor NERFINISHED ⓘ Jacobian variety NERFINISHED ⓘ Néron–Severi group NERFINISHED ⓘ Picard scheme NERFINISHED ⓘ 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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.