Néron–Severi group

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.