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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]]. | |||
==Introduction== | |||
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]]. | |||
==Construction for complex curves== | |||
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 | |||
:<math> | |||
\omega \mapsto \int_{\gamma} \omega | |||
</math> | |||
where ''γ'' is a closed [[path (topology)|path]] in ''C''. | |||
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. | |||
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. | |||
==Further notions== | |||
[[Torelli's theorem]] states that a complex curve is determined by its Jacobian (with its polarization). | |||
The [[Schottky problem]] asks which principally polarized abelian varieties are the Jacobians of curves. | |||
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. | |||
==References== | |||
* {{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}} | |||
*{{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}} | |||
*{{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}} | |||
*{{eom|id=J/j054140|first=V.V. |last=Shokurov|title=Jacobi variety}} | |||
*{{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}} | |||
*{{Cite book | last1=Hartshorne | first1=Robin | author1-link=Robin Hartshorne | title=Algebraic Geometry | publisher=Springer | location=New York | isbn=0-387-90244-9}} | |||
*[[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> | |||
{{reflist}} | |||
{{Algebraic curves navbox}} | |||
[[Category:Abelian varieties]] | |||
[[Category:Algebraic curves]] | |||
[[Category:Geometry of divisors]] | |||
[[Category:Moduli theory]] | |||
Revision as of 06:23, 27 November 2013
In mathematics, the Jacobian variety J(C) of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of C, hence an abelian variety.
Introduction
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.
Construction for complex curves
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 of all elements of V of the form
where γ is a closed path in C.
The Jacobian of a curve over an arbitrary field was constructed by Template:Harvtxt as part of his proof of the Riemann hypothesis for curves over a finite field.
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.
Further notions
Torelli's theorem states that a complex curve is determined by its Jacobian (with its polarization).
The Schottky problem asks which principally polarized abelian varieties are the Jacobians of curves.
The Picard variety, the Albanese variety, and intermediate Jacobians 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.
References
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- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - Montserrat Teixidor i Bigas On the number of parameters for curves whose Jacobians possess non-trivial endomorphisms.;[1] Theta Divisors for vector bundles in Curves, Jacobians, and Abelian Varieties[2]
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