Néron-Severi group

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)