Jacobians: Difference between revisions
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The Jacobian variety of a smooth [[algebraic curve]] C is the variety of degree 0 divisors of C, up to [[ | {{subpages}} | ||
The Jacobian variety of a smooth [[algebraic curve]] C is the variety of degree 0 divisors of C, up to [[rational equivalence]]; i.e. it is the kernel of the degree map from Pic(C) to the integers; sometimes also denoted as Pic<sup>0</sup>. It is an principally polarized [[Abelian variety]] of dimension g. | |||
Principal polarization: | Principal polarization: | ||
The | The principal polarization of the Jacobian variety is given by the theta divisor: some shift from Pic<sup>g-1</sup> to Jacobian of the image of Sym<sup>g-1</sup>C in | ||
Pic<sup>g-1</sup>. | Pic<sup>g-1</sup>. | ||
Examples: | Examples: | ||
* A genus 1 curve is naturally | * A genus 1 curve is naturally isomorphic to the variety of degree 1 divisors, and therefore to is isomorphic to it's Jacobian. | ||
Related theorems and problems: | Related theorems and problems: | ||
* [[Abels theorem]] states that the map <math>\mathcal{M}_g\to\mathcal{A}_g</math>, which takes a curve to it's | * [[Abels theorem]] states that the map <math>\mathcal{M}_g\to\mathcal{A}_g</math>, which takes a curve to it's Jacobian is an injection. | ||
* The [[ | * The [[Schottky problem]] calls for the classification of the map above. | ||
Latest revision as of 00:39, 27 October 2013
The Jacobian variety of a smooth algebraic curve C is the variety of degree 0 divisors of C, up to rational equivalence; i.e. it is the kernel of the degree map from Pic(C) to the integers; sometimes also denoted as Pic0. It is an principally polarized Abelian variety of dimension g.
Principal polarization: The principal polarization of the Jacobian variety is given by the theta divisor: some shift from Picg-1 to Jacobian of the image of Symg-1C in Picg-1.
Examples:
- A genus 1 curve is naturally isomorphic to the variety of degree 1 divisors, and therefore to is isomorphic to it's Jacobian.
Related theorems and problems:
- Abels theorem states that the map , which takes a curve to it's Jacobian is an injection.
- The Schottky problem calls for the classification of the map above.
