NeronSeveri

NeronSeveri is a concept in algebraic geometry that refers to the Néron–Severi group of an algebraic variety. The group is defined as the group of divisors modulo algebraic equivalence. It can be viewed as a finitely generated abelian group that captures the numerical properties of divisor classes on a variety.

The Néron–Severi group is a quotient of the Picard group, which classifies line bundles up to isomorphism. While the Picard group may contain torsion elements and infinite‐dimensional components, the Néron–Severi group is always finite rank. Its rank, known as the Picard number, measures the number of independent algebraic cycles of codimension one on the variety.

For smooth projective varieties over an algebraically closed field, the Néron–Severi group embeds into the second cohomology group via the first Chern class map. In characteristic zero this embedding respects the Hodge decomposition, making the Picard number equal to the dimension of the space of Hodge classes of type (1,1). The group also appears in the study of abelian varieties, where its rank is at most the dimension of the abelian variety.

Computing the Néron–Severi group is often challenging; however, for many special classes of varieties (for example, surfaces of general type, K3 surfaces, and abelian varieties) there are well‑known results and techniques. The Néron–Severi group plays a central role in the classification of algebraic surfaces, in the minimal model program, and in various moduli problems, as it controls the possible intersection behaviors of divisors.