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| In [[mathematics]], the '''Jacobian variety''' ''J(C)'' of a non-singular [[algebraic curve]] ''C'' of [[genus (mathematics)|genus]] ''g'' is the [[moduli space]] of degree 0 [[line bundle]]s. It is the connected component of the identity in the [[Picard group]] of ''C'', hence an [[abelian variety]].
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| ==Introduction==
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| The Jacobian variety is named after [[Carl Gustav Jacobi]], who proved the complete version [[Abel-Jacobi theorem]], making the injectivity statement of [[Niels Abel]] into an isomorphism. It is a principally polarized [[abelian variety]], of [[dimension]] ''g'', and hence, over the complex numbers, it is a [[complex torus]]. If ''p'' is a point of ''C'', then the curve ''C'' can be mapped to a [[subvariety]] of ''J'' with the given point ''p'' mapping to the identity of ''J'', and ''C'' generates ''J'' as a [[Group (mathematics)|group]].
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| ==Construction for complex curves==
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| Over the complex numbers, the Jacobian variety can be realized as the [[quotient space]] ''V''/''L'', where ''V'' is the dual of the [[vector space]] of all global holomorphic differentials on ''C'' and ''L'' is the [[lattice (group)|lattice]] of all elements of ''V'' of the form
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| :<math>
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| \omega \mapsto \int_{\gamma} \omega
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| </math>
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| where ''γ'' is a closed [[path (topology)|path]] in ''C''.
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| The Jacobian of a curve over an arbitrary field was constructed by {{harvtxt|Weil|1948}} as part of his proof of the Riemann hypothesis for curves over a finite field.
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| The [[Abel-Jacobi theorem]] states that the torus thus built is a variety, the classical Jacobian of a curve, that indeed parametrizes the degree 0 line bundles, that is, it can be identified with its [[Picard variety]] of degree 0 divisors modulo linear equivalence.
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| ==Further notions==
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| [[Torelli's theorem]] states that a complex curve is determined by its Jacobian (with its polarization).
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| The [[Schottky problem]] asks which principally polarized abelian varieties are the Jacobians of curves.
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| The [[Picard variety]], the [[Albanese variety]], and [[intermediate Jacobian]]s are generalizations of the Jacobian for higher dimensional varieties. For varieties of higher dimension the construction of the Jacobian variety as a quotient of the space of holomorphic 1-forms generalizes to give the [[Albanese variety]], but in general this need not be isomorphic to the Picard variety.
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| ==References==
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| * {{cite book | author=P. Griffiths | authorlink=Phillip Griffiths | coauthors=[[Joe Harris (mathematician)|J. Harris]] | title=Principles of Algebraic Geometry | series=Wiley Classics Library | publisher=Wiley Interscience | year=1994 | isbn=0-471-05059-8 | pages=333–363}}
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| *{{cite conference | author=J.S. Milne | title=Jacobian Varieties | booktitle=Arithmetic Geometry |publisher=Springer-Verlag|location=New York| year=1986 | pages=167–212|isbn=0-387-96311-1}}
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| *{{Cite book | last1=Mumford | first1=David | author1-link=David Mumford | title=Curves and their Jacobians | publisher=The University of Michigan Press, Ann Arbor, Mich. | id={{MathSciNet | id = 0419430}} | year=1975}}
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| *{{eom|id=J/j054140|first=V.V. |last=Shokurov|title=Jacobi variety}}
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| *{{Cite book | last1=Weil | first1=André | author1-link=André Weil | title=Variétés abéliennes et courbes algébriques | publisher=Hermann | location=Paris | oclc=826112 | id={{MathSciNet | id = 0029522}} | year=1948}}
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| *{{Cite book | last1=Hartshorne | first1=Robin | author1-link=Robin Hartshorne | title=Algebraic Geometry | publisher=Springer | location=New York | isbn=0-387-90244-9}}
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| *[[Montserrat Teixidor i Bigas]] ''On the number of parameters for curves whose Jacobians possess non-trivial endomorphisms.'';<ref>[http://130.203.133.150/showciting;jsessionid=262D99DFBDE685219DD4BDB77FB91E04?cid=1654190&sort=recent On the number of parameters for curves whose Jacobians possess non-trivial endomorphisms]</ref> ''Theta Divisors for vector bundles'' in ''Curves, Jacobians, and Abelian Varieties''<ref>[http://www.ams.org/books/conm/136/ Curves, Jacobians, and Abelian Varieties]</ref>
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| {{reflist}}
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| {{Algebraic curves navbox}}
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| [[Category:Abelian varieties]]
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| [[Category:Algebraic curves]]
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| [[Category:Geometry of divisors]]
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| [[Category:Moduli theory]]
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