# 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, ), 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 , , and also ).

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 ).

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

How to Cite This Entry:
Néron-Severi group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=N%C3%A9ron-Severi_group&oldid=53785
This article was adapted from an original article by V.A. Iskovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article