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The divisor class group under algebraic equivalence on a non-singular projective variety.
 
The divisor class group under algebraic equivalence on a non-singular projective variety.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066300/n0663001.png" /> be a non-singular projective variety of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066300/n0663002.png" /> defined over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066300/n0663003.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066300/n0663004.png" /> be the group of divisors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066300/n0663005.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066300/n0663006.png" /> be the subgroup of divisors that are algebraically equivalent to zero. The quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066300/n0663007.png" /> is called the Néron–Severi group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066300/n0663008.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066300/n0663009.png" />. The Néron–Severi theorem asserts that the Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066300/n06630010.png" /> is finitely generated.
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Let $X$ be a non-singular projective variety of dimension $\ge 2$ defined over an [[algebraically closed field]] $k$, let $D(X)$ be the group of divisors of $X$ and let $D_{\mathrm{a}}(X)$ be the subgroup of divisors that are algebraically equivalent to zero. The quotient group $D(X)/D_{\mathrm{a}}(X)$ is called the Néron–Severi group of $X$ and is denoted by $\mathrm{NS}(X)$. The Néron–Severi theorem asserts that the Abelian group $\mathrm{NS}(X)$ is finitely generated.
  
In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066300/n06630011.png" />, F. Severi presented, in a series of papers on the theory of the base (see, for example, [[#References|[1]]]), a proof of this theorem using topological and transcendental tools. The first abstract proof (valid for a field of arbitrary characteristic) is due to A. Néron (see [[#References|[2]]], [[#References|[3]]], and also [[#References|[4]]]).
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In the case $k = \mathbf{C}$, F. Severi presented, in a series of papers on the theory of the base (see, for example, [[#References|[1]]]), a proof of this theorem using topological and transcendental tools. The first abstract proof (valid for a field of arbitrary characteristic) is due to A. Néron (see [[#References|[2]]], [[#References|[3]]], and also [[#References|[4]]]).
  
The rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066300/n06630012.png" /> is the algebraic [[Betti number|Betti number]] of the group of divisors on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066300/n06630013.png" />, that is, the algebraic rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066300/n06630014.png" />. This is also called the Picard number of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066300/n06630015.png" />. The elements of the finite torsion subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066300/n06630016.png" /> are called Severi divisors, and the order of this subgroup is called the Severi number; the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066300/n06630017.png" /> is a birational invariant (see [[#References|[6]]]).
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The rank of $\mathrm{NS}(X)$ is the algebraic [[Betti number|Betti number]] of the group of divisors on $X$, that is, the algebraic rank of $X$. This is also called the ''Picard number'' of the variety $X$. The elements of the finite torsion subgroup $\mathrm{NS}_{\mathrm{tors}}(X)$ are called ''Severi divisors'', and the order of this subgroup is called the ''Severi number''; the group $\mathrm{NS}_{\mathrm{tors}}(X)$ is a birational invariant (see [[#References|[6]]]).
  
 
There are generalizations of the Néron–Severi theorem to other groups of classes of algebraic cycles (see [[#References|[1]]] (classical theory) and [[#References|[7]]] (modern theory)).
 
There are generalizations of the Néron–Severi theorem to other groups of classes of algebraic cycles (see [[#References|[1]]] (classical theory) and [[#References|[7]]] (modern theory)).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Néron,  "La théorie de la base pour les diviseurs sur les variétés algébriques" , ''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066300/n06630018.png" /> Coll. Géom. Alg. Liège'' , G. Thone  (1952)  pp. 119–126</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Lang,  A. Néron,  "Rational points of abelian varieties over function fields"  ''Amer. J. Math.'' , '''81'''  (1959)  pp. 95–118</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  R. Hartshorne,  "Algebraic geometry" , Springer  (1977)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  M. Baldassarri,  "Algebraic varieties" , Springer  (1956)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  I.V. Dolgachev,  V.A. Iskovskikh,  "Geometry of algebraic varieties"  ''J. Soviet Math.'' , '''5''' :  6  (1976)  pp. 803–864  ''Itogi Nauk. i Tekhn. Algebra. Topol. Geom.'' , '''12'''  (1974)  pp. 77–170</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  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</TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top">  A. Néron,  "La théorie de la base pour les diviseurs sur les variétés algébriques" , ''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066300/n06630018.png" /> Coll. Géom. Alg. Liège'' , G. Thone  (1952)  pp. 119–126</TD></TR>
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<TR><TD valign="top">[4]</TD> <TD valign="top">  S. Lang,  A. Néron,  "Rational points of abelian varieties over function fields"  ''Amer. J. Math.'' , '''81'''  (1959)  pp. 95–118</TD></TR>
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<TR><TD valign="top">[5]</TD> <TD valign="top">  R. Hartshorne,  "Algebraic geometry" , Springer  (1977)</TD></TR>
 +
<TR><TD valign="top">[6]</TD> <TD valign="top">  M. Baldassarri,  "Algebraic varieties" , Springer  (1956)</TD></TR>
 +
<TR><TD valign="top">[7]</TD> <TD valign="top">  I.V. Dolgachev,  V.A. Iskovskikh,  "Geometry of algebraic varieties"  ''J. Soviet Math.'' , '''5''' :  6  (1976)  pp. 803–864  ''Itogi Nauk. i Tekhn. Algebra. Topol. Geom.'' , '''12'''  (1974)  pp. 77–170</TD></TR>
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</table>
  
  
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A study of the Picard number of a certain number of algebraic varieties has been made by T. Shioda.
 
A study of the Picard number of a certain number of algebraic varieties has been made by T. Shioda.
  
The phrase  "theory of the basetheory of the base"  is a somewhat old-fashioned one and refers to the considerations involved in proving that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066300/n06630019.png" /> is a finitely-generated Abelian group and indicating an explicit minimal set of generators (a minimal base in the terminology of Severi), cf. e.g. [[#References|[a4]]], Sect. V.7 (for the case of surfaces).
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The phrase  "theory of the base"  is a somewhat old-fashioned one and refers to the considerations involved in proving that $\mathrm{NS}(X)$  is a finitely-generated Abelian group and indicating an explicit minimal set of generators (a minimal base in the terminology of Severi), cf. e.g. [[#References|[a4]]], Sect. V.7 (for the case of surfaces).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T. Shioda,  "On the Picard number of a complex projective variety"  ''Ann. Sci. Ecole Norm. Sup.'' , '''14'''  (1981)  pp. 303–321</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T. Shioda,  "On the Picard number of a Fermat surface"  ''J. Fac. Sci. Univ. Tokyo'' , '''28'''  (1982)  pp. 724–734</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  T. Shioda,  "An explicit algorithm for computing the Picard number of certain algebraic surfaces"  ''Amer. J. Math.'' , '''108'''  (1986)  pp. 415–432</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  O. Zariski,  "Algebraic surfaces" , Springer  (1935)</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  T. Shioda,  "On the Picard number of a complex projective variety"  ''Ann. Sci. Ecole Norm. Sup.'' , '''14'''  (1981)  pp. 303–321</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  T. Shioda,  "On the Picard number of a Fermat surface"  ''J. Fac. Sci. Univ. Tokyo'' , '''28'''  (1982)  pp. 724–734</TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top">  T. Shioda,  "An explicit algorithm for computing the Picard number of certain algebraic surfaces"  ''Amer. J. Math.'' , '''108'''  (1986)  pp. 415–432</TD></TR>
 +
<TR><TD valign="top">[a4]</TD> <TD valign="top">  O. Zariski,  "Algebraic surfaces" , Springer  (1935)</TD></TR>
 +
</table>
 +
 
 +
{{TEX|done}}

Latest revision as of 05:47, 15 April 2023

The divisor class group under algebraic equivalence on a non-singular projective variety.

Let $X$ be a non-singular projective variety of dimension $\ge 2$ defined over an algebraically closed field $k$, let $D(X)$ be the group of divisors of $X$ and let $D_{\mathrm{a}}(X)$ be the subgroup of divisors that are algebraically equivalent to zero. The quotient group $D(X)/D_{\mathrm{a}}(X)$ is called the Néron–Severi group of $X$ and is denoted by $\mathrm{NS}(X)$. The Néron–Severi theorem asserts that the Abelian group $\mathrm{NS}(X)$ is finitely generated.

In the case $k = \mathbf{C}$, F. Severi presented, in a series of papers on the theory of the base (see, for example, [1]), a proof of this theorem using topological and transcendental tools. The first abstract proof (valid for a field of arbitrary characteristic) is due to A. Néron (see [2], [3], and also [4]).

The rank of $\mathrm{NS}(X)$ is the algebraic Betti number of the group of divisors on $X$, that is, the algebraic rank of $X$. This is also called the Picard number of the variety $X$. The elements of the finite torsion subgroup $\mathrm{NS}_{\mathrm{tors}}(X)$ are called Severi divisors, and the order of this subgroup is called the Severi number; the group $\mathrm{NS}_{\mathrm{tors}}(X)$ is a birational invariant (see [6]).

There are generalizations of the Néron–Severi theorem to other groups of classes of algebraic cycles (see [1] (classical theory) and [7] (modern theory)).

References

[1] 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
[2] 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
[3] A. Néron, "La théorie de la base pour les diviseurs sur les variétés algébriques" , Coll. Géom. Alg. Liège , G. Thone (1952) pp. 119–126
[4] S. Lang, A. Néron, "Rational points of abelian varieties over function fields" Amer. J. Math. , 81 (1959) pp. 95–118
[5] R. Hartshorne, "Algebraic geometry" , Springer (1977)
[6] M. Baldassarri, "Algebraic varieties" , Springer (1956)
[7] I.V. Dolgachev, V.A. Iskovskikh, "Geometry of algebraic varieties" J. Soviet Math. , 5 : 6 (1976) pp. 803–864 Itogi Nauk. i Tekhn. Algebra. Topol. Geom. , 12 (1974) pp. 77–170


Comments

A study of the Picard number of a certain number of algebraic varieties has been made by T. Shioda.

The phrase "theory of the base" is a somewhat old-fashioned one and refers to the considerations involved in proving that $\mathrm{NS}(X)$ is a finitely-generated Abelian group and indicating an explicit minimal set of generators (a minimal base in the terminology of Severi), cf. e.g. [a4], Sect. V.7 (for the case of surfaces).

References

[a1] T. Shioda, "On the Picard number of a complex projective variety" Ann. Sci. Ecole Norm. Sup. , 14 (1981) pp. 303–321
[a2] T. Shioda, "On the Picard number of a Fermat surface" J. Fac. Sci. Univ. Tokyo , 28 (1982) pp. 724–734
[a3] T. Shioda, "An explicit algorithm for computing the Picard number of certain algebraic surfaces" Amer. J. Math. , 108 (1986) pp. 415–432
[a4] O. Zariski, "Algebraic surfaces" , Springer (1935)
How to Cite This Entry:
Néron-Severi group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=N%C3%A9ron-Severi_group&oldid=17702
This article was adapted from an original article by V.A. Iskovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article