Jacobian varieties

E860099

abelian variety algebraic variety complex torus principally polarized abelian variety

Jacobian varieties are complex algebraic varieties associated to algebraic curves that parametrize degree-zero line bundles (or divisor classes) on the curve and carry a natural structure of principally polarized abelian varieties.

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Observed surface forms (3)

Surface form	Occurrences
Jacobian variety	0
Jacobian of a curve	1
Picard variety	1

Statements (47)

Predicate	Object
instanceOf	abelian variety ⓘ algebraic variety ⓘ complex torus ⓘ principally polarized abelian variety ⓘ
associatedTo	algebraic curve ⓘ compact Riemann surface ⓘ smooth projective curve ⓘ
carriesStructure	group variety ⓘ principally polarization ⓘ
constructedAs	H^0(C,Ω^1)^∨ / H_1(C,ℤ) ⓘ quotient of C^g by a lattice ⓘ
contains	theta divisor ⓘ
definedOver	base field of the curve ⓘ ℂ for complex curves ⓘ
dependsOn	isomorphism class of the curve ⓘ
determines	curve up to isomorphism for genus ≥ 2 (via Torelli theorem) ⓘ
dimension	genus of the curve ⓘ
functorialIn	morphisms of curves ⓘ
generalizes	elliptic curve ⓘ
groupLaw	addition of divisor classes ⓘ
hasEndomorphismRing	End(J(C)) ⓘ
hasInvariant	Néron–Tate height (in arithmetic setting) NERFINISHED ⓘ
hasLattice	period lattice ⓘ
hasPoint	origin corresponding to trivial line bundle ⓘ
hasPolarization	theta divisor ⓘ
isCommutative	true ⓘ
isomorphicTo	Pic^0(C) NERFINISHED ⓘ Picard variety of degree zero NERFINISHED ⓘ
isPrincipallyPolarized	true ⓘ
isProjective	true ⓘ
mayHaveProperty	complex multiplication ⓘ
parametrizes	degree-zero line bundles on a curve ⓘ divisor classes of degree zero on a curve ⓘ
relatedTo	Abel map NERFINISHED ⓘ Abel–Jacobi map NERFINISHED ⓘ Albanese variety of the curve ⓘ Picard group of the curve NERFINISHED ⓘ Riemann theta function NERFINISHED ⓘ Torelli theorem NERFINISHED ⓘ period matrix of a curve ⓘ
specialCase	elliptic curve when genus equals 1 ⓘ
universalProperty	universal abelian variety receiving a map from the curve ⓘ universal regular quotient of degree-zero divisors ⓘ
usedIn	algebraic geometry ⓘ arithmetic geometry ⓘ number theory ⓘ theory of moduli of curves ⓘ

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Sur les courbes algébriques et les variétés qui s’en déduisent → topic → Jacobian varieties ⓘ

Brill–Noether theory → usesConcept → Jacobian varieties ⓘ

this entity surface form: Picard variety

Brill–Noether theory → usesConcept → Jacobian varieties ⓘ

this entity surface form: Jacobian of a curve