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| In [[algebraic geometry]], the '''Néron–Severi group''' of a [[algebraic variety|variety]] is
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| the group of divisors modulo [[algebraic equivalence]]; in other words it is the group of [[Component (group theory)|components]] of the [[Picard scheme]] of a variety. Its rank is called the [[Picard number]]. It is named after [[Francesco Severi]] and [[André Néron]]. | |
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| ==Definition==
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| In the cases of most importance to classical algebraic geometry, for a [[complete variety]] ''V'' that is [[non-singular]], the [[connected space|connected component]] of the Picard scheme is an [[abelian variety]] written
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| :Pic<sup>0</sup>(''V'') | |
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| and the quotient
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| :Pic(''V'')/Pic<sup>0</sup>(''V'')
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| is an abelian group NS(''V''), called the '''Néron–Severi group''' of ''V''. This is a [[finitely-generated abelian group]] by the Néron–Severi theorem, which was proved by Severi over the complex numbers and by Néron over more general fields.
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| In other words the Picard group fits into an [[exact sequence]]
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| :<math>1\to \mathrm{Pic}^0(V)\to\mathrm{Pic}(V)\to \mathrm{NS}(V)\to 0</math>
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| The fact that the rank is finite is [[Francesco Severi]]'s '''theorem of the base'''; the rank is the '''Picard number''' of ''V'', often denoted ρ(''V''). The elements of finite order are called Severi divisors, and form a finite group which is a birational invariant and whose order is called the '''Severi number'''. Geometrically NS(''V'') describes the [[algebraic equivalence]] classes of [[divisor (algebraic geometry)|divisors]] on ''V''; that is, using a stronger, non-linear equivalence relation in place of [[linear equivalence of divisors]], the classification becomes amenable to discrete invariants. Algebraic equivalence is closely related to [[numerical equivalence]], an essentially topological classification by [[intersection number]]s. | |
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| ==First Chern class and integral valued 2-cocycles==
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| The [[exponential sheaf sequence]]
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| :<math>0\to 2\pi i\mathbb Z \to \mathcal O_V\to\mathcal O_V^*\to 0</math>
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| gives rise to a long exact sequence featuring
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| :<math>\cdots \to H^1(V, \mathcal O_V^*)\to H^2(V, \mathbb Z)\to H^2(V,\mathcal O_V)\to \cdots.</math>
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| The first arrow is the [[first Chern class]] on the [[Picard group]]
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| :<math>c_1 : \mathrm {Pic}(V)\to H^2(V, \mathbb Z),</math>
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| and the second
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| :<math>\exp^* : H^2(V, 2i\pi \mathbb Z)\to H^2(V,\mathcal O_V).</math> | |
| The Neron-Severi group can be identified with the image of the first Chern class, or equivalently, by exactness, as the kernel of the second arrow exp*.
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| In the complex case, the Neron-Severi group is therefore the group of 2-cocycles whose [[Poincaré dual]] is represented by a complex hypersurface, that is, a [[Weil divisor]].
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| ==References==
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| *{{springer|id=N/n066300|title=Néron–Severi group|author=V.A. Iskovskikh}}
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| *A. Néron, ''Problèmes arithmétiques et géometriques attachée à la notion de rang d'une courbe algébrique dans un corps'' Bull. Soc. Math. France, 80 (1952) pp. 101–166
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| *A. Néron, ''La théorie de la base pour les diviseurs sur les variétés algébriques'', Coll. Géom. Alg. Liège, G. Thone (1952) pp. 119–126
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| * F. Severi, ''La base per le varietà algebriche di dimensione qualunque contenute in una data e la teoria generale delle corrispondénze fra i punti di due superficie algebriche'' Mem. Accad. Ital., 5 (1934) pp. 239–283
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| {{DEFAULTSORT:Neron-Severi Group}}
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| [[Category:Algebraic geometry]]
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