Algebraic variety
Jacobian variety and abelian variety

Let C be a smooth complete curve and Pic (C) {\displaystyle \operatorname {Pic} (C)} the Picard group of it; i.e., the group of isomorphism classes of line bundles on C. Since C is smooth, Pic (C) {\displaystyle \operatorname {Pic} (C)} can be identified as the divisor class group of C and thus there is the degree homomorphism deg: Pic (C) → Z {\displaystyle \operatorname {deg} :\operatorname {Pic} (C)\to \mathbb {Z} }. The Jacobian variety Jac (C) {\displaystyle \operatorname {Jac} (C)} of C is the kernel of this degree map; i.e., the group of the divisor classes on C of degree zero. A Jacobian variety is an example of an abelian variety, a complete variety with a compatible abelian group structure on it (the name "abelian" is however not because it is an abelian group). An abelian variety turns out to be projective (in short, algebraic theta functions give an embedding into a projective space. See equations defining abelian varieties); thus, Jac (C) {\displaystyle \operatorname {Jac} (C)} is a projective variety. The tangent space to Jac (C) {\displaystyle \operatorname {Jac} (C)} at the identity element is naturally isomorphic to H 1 (C, O C); {\displaystyle \operatorname {H} ^{1}(C,{\mathcal {O}}_{C});} hence, the dimension of Jac (C) {\displaystyle \operatorname {Jac} (C)} is the genus of C {\displaystyle C}.

Fix a point P 0 {\displaystyle P_{0}} on C {\displaystyle C}. For each integer n > 0 {\displaystyle n>0}, there is a natural morphism

where C n {\displaystyle C^{n}} is the product of n copies of C. For g = 1 {\displaystyle g=1} (i.e., C is an elliptic curve), the above morphism for n = 1 {\displaystyle n=1} turns out to be an isomorphism; in particular, an elliptic curve is an abelian variety.
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