---
tag: BB0X043XD1
kind: definition
name: Néron–Severi group
msc: [14C22, 14C20, 14C25]
defines: XMZ09SBM0X
uses: [A4H4CZ2V35, 3Z9DCWYXQZ, 7JCF2TE3V5, 02SJG1HC32]
sources:
  - url: https://en.wikipedia.org/wiki/N%C3%A9ron%E2%80%93Severi_group
---

## Definition

Let $X$ be a smooth [[projective variety|A4H4CZ2V35]] over an [[algebraically closed field|02SJG1HC32]] $k$, with [[Picard group|3Z9DCWYXQZ]] $\operatorname{Pic}(X)$ of isomorphism classes of [[invertible sheaves|7JCF2TE3V5]].
Consider a connected variety $T$ over $k$, points $t_0, t_1 \in T(k)$ and an invertible sheaf $M$ on $X \times T$.
An invertible sheaf $L$ on $X$ is *algebraically equivalent to $0$* if such data exist with:

1. $M|_{X \times \set{t_0}} \cong \mathcal{O}_X$;
2. $M|_{X \times \set{t_1}} \cong L$.

Let $\operatorname{Pic}^0(X) \subset \operatorname{Pic}(X)$ be the subgroup of classes algebraically equivalent to $0$.
The *Néron–Severi group* of $X$ is
$$\operatorname{NS}(X) = \operatorname{Pic}(X) / \operatorname{Pic}^0(X).$$

## Remarks

The theorem of the base says that $\operatorname{NS}(X)$ is finitely generated; its rank $\rho(X)$ is the *Picard number*.
Example: for $X = \mathbb{P}^n$, $\operatorname{Pic}(X) \cong \Z$, generated by $\mathcal{O}(1)$, and $\operatorname{Pic}^0(X) = 0$, so $\operatorname{NS}(\mathbb{P}^n) \cong \Z$.
For an elliptic curve $E$, $\operatorname{Pic}(E) \cong \Z \oplus E(k)$, with $\operatorname{Pic}^0(E) \cong E(k)$ the classes of degree $0$, so $\operatorname{NS}(E) \cong \Z$ through the degree.
Non-example: $\operatorname{Pic}(X)$ itself is usually not a candidate for this role: for an elliptic curve over $\C$, $\operatorname{Pic}(E)$ contains $E(\C)$, which is not finitely generated.
Over $\C$, the first Chern class identifies $\operatorname{NS}(X)$ with the image of $\operatorname{Pic}(X) \to H^2(X(\C), \Z)$, which is the group of integral classes of Hodge type $(1, 1)$ by the Lefschetz theorem on $(1, 1)$-classes.