Abelian Scheme
One can also define abelian varieties scheme-theoretically and relative to a base. This allows for a uniform treatment of phenomena such as reduction mod p of abelian varieties (see Arithmetic of abelian varieties), and parameter-families of abelian varieties. An abelian scheme over a base scheme S of relative dimension g is a proper, smooth group scheme over S whose geometric fibers are connected and of dimension g. The fibers of an abelian scheme are abelian varieties, so one could think of an abelian scheme over S as being a family of abelian varieties parametrised by S.
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Famous quotes containing the word scheme:
“We doubt not the destiny of our country—that she is to accomplish great things for human nature, and be the mother of a nobler race than the world has yet known. But she has been so false to the scheme made out at her nativity, that it is now hard to say which way that destiny points.”
—Margaret Fuller (1810–1850)