---
tag: XMZ09SBM0X
kind: concept
name: Néron–Severi group
msc: [14C22, 14C20, 14C25]
sources:
  - url: https://en.wikipedia.org/wiki/N%C3%A9ron%E2%80%93Severi_group
same_as:
  wikidata: Q7071450
  wikipedia: Néron–Severi group
---

## Summary

The Néron–Severi group of a smooth projective variety is the group of divisors modulo algebraic equivalence, equivalently the [[Picard group|3Z9DCWYXQZ]] modulo its identity component.
The theorem of the base, due to Severi over $\C$ and Néron in general, says that it is a finitely generated abelian group.
Its rank is the Picard number, and over $\C$ it is the group of integral cohomology classes of type $(1, 1)$.

## Explanations

### L1 human

The Picard group of a variety can be huge, because line bundles come in continuous families.
The Néron–Severi group forgets the continuous part and keeps only the discrete data.
On an elliptic curve, line bundles are classified by their degree and a point of the curve; the Néron–Severi group keeps the degree only and is $\Z$.
On a product of two curves, it also records how a divisor sits with respect to both factors.