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单词 AbelianVariety
释义

abelian variety


Definition 1.

An abelian varietyMathworldPlanetmath over a field k is a proper group scheme over Speck that is a varietyPlanetmathPlanetmath.

This extremely terse definition needs some further explanation.

Proposition 1.

The group law on an abelian variety is commutativePlanetmathPlanetmathPlanetmath.

This implies that for every ring R, the R-points of an abelian variety form an abelian group.

Proposition 2.

An abelian variety is projective.

If C is a curve, then the Jacobian of C is an abelian variety. This example motivated the development of the theory of abelian varieties, and many properties of curves are best understood by looking at the Jacobian.

If E is an elliptic curveMathworldPlanetmath, then E is an abelian variety (and in fact E is naturally isomorphicPlanetmathPlanetmathPlanetmathPlanetmath to its Jacobian).

See Mumford’s excellent book Abelian Varieties. The bibliography for algebraic geometry has details and other books.

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更新时间:2025/5/4 8:06:43