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Difference between revisions of "Abelian scheme"

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A smooth [[Group scheme|group scheme]] over a base scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010250/a0102501.png" />, the fibres of which are Abelian varieties (cf. [[Abelian variety|Abelian variety]]). The following is an equivalent definition: An Abelian scheme over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010250/a0102502.png" />, or an Abelian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010250/a0102504.png" />-scheme, is a proper smooth group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010250/a0102505.png" />-scheme all fibres of which are geometrically connected. Intuitively, an Abelian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010250/a0102506.png" />-scheme may be understood as a family of Abelian varieties parametrized by the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010250/a0102507.png" />. A number of fundamental properties of Abelian varieties carry over to Abelian schemes. For instance, an Abelian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010250/a0102508.png" />-scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010250/a0102509.png" /> is a commutative group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010250/a01025010.png" />-scheme [[#References|[1]]], and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010250/a01025011.png" /> is a normal scheme, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010250/a01025012.png" /> is projective over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010250/a01025013.png" />, [[#References|[2]]].
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A smooth [[Group scheme|group scheme]] over a base scheme $S$, the fibres of which are Abelian varieties (cf. [[Abelian variety|Abelian variety]]). The following is an equivalent definition: An Abelian scheme over $S$, or an Abelian $S$-scheme, is a proper smooth group $S$-scheme all fibres of which are geometrically connected. Intuitively, an Abelian $S$-scheme may be understood as a family of Abelian varieties parametrized by the scheme $S$. A number of fundamental properties of Abelian varieties carry over to Abelian schemes. For instance, an Abelian $S$-scheme $A$ is a commutative group $S$-scheme [[#References|[1]]], and if $S$ is a normal scheme, $A$ is projective over $S$, [[#References|[2]]].
  
 
Abelian schemes are used in the context of moduli schemes of Abelian varieties with various auxiliary structures, and also in the theory of reduction of Abelian varieties (cf. [[Néron model|Néron model]]).
 
Abelian schemes are used in the context of moduli schemes of Abelian varieties with various auxiliary structures, and also in the theory of reduction of Abelian varieties (cf. [[Néron model|Néron model]]).

Latest revision as of 15:13, 23 April 2014

A smooth group scheme over a base scheme $S$, the fibres of which are Abelian varieties (cf. Abelian variety). The following is an equivalent definition: An Abelian scheme over $S$, or an Abelian $S$-scheme, is a proper smooth group $S$-scheme all fibres of which are geometrically connected. Intuitively, an Abelian $S$-scheme may be understood as a family of Abelian varieties parametrized by the scheme $S$. A number of fundamental properties of Abelian varieties carry over to Abelian schemes. For instance, an Abelian $S$-scheme $A$ is a commutative group $S$-scheme [1], and if $S$ is a normal scheme, $A$ is projective over $S$, [2].

Abelian schemes are used in the context of moduli schemes of Abelian varieties with various auxiliary structures, and also in the theory of reduction of Abelian varieties (cf. Néron model).

References

[1] D. Mumford, "Geometric invariant theory" , Springer (1965)
[2] M. Raynaud, "Faisceaux amples sur les schémas en groupes et les espaces homogénes" , Springer (1970)
How to Cite This Entry:
Abelian scheme. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abelian_scheme&oldid=11728
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article