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abelian variety
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(Definition)
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This extremely terse definition needs some further explanation.
This implies that for every ring  , the  -points of an abelian variety form an abelian group.
Proposition 2 An abelian variety is projective.
If  is a curve, then the Jacobian of  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  is an elliptic curve, then  is an abelian variety (and in fact  is naturally isomorphic to its Jacobian).
See Mumford's excellent book Abelian Varieties. The bibliography for algebraic geometry has details and other books.
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"abelian variety" is owned by archibal.
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(view preamble)
Cross-references: bibliography for algebraic geometry, abelian, isomorphic, elliptic curve, development, Jacobian, curve, abelian group, ring, implies, commutative, group, variety, group scheme, field
There are 7 references to this entry.
This is version 3 of abelian variety, born on 2004-04-05, modified 2004-04-06.
Object id is 5742, canonical name is AbelianVariety.
Accessed 3248 times total.
Classification:
| AMS MSC: | 14K99 (Algebraic geometry :: Abelian varieties and schemes :: Miscellaneous) |
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Pending Errata and Addenda
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