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  "body": "\n## Definition\n\nLet $X$ 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]].\nConsider a connected variety $T$ over $k$, points $t_0, t_1 \\in T(k)$ and an invertible sheaf $M$ on $X \\times T$.\nAn invertible sheaf $L$ on $X$ is *algebraically equivalent to $0$* if such data exist with:\n\n1. $M|_{X \\times \\set{t_0}} \\cong \\mathcal{O}_X$;\n2. $M|_{X \\times \\set{t_1}} \\cong L$.\n\nLet $\\operatorname{Pic}^0(X) \\subset \\operatorname{Pic}(X)$ be the subgroup of classes algebraically equivalent to $0$.\nThe *Néron–Severi group* of $X$ is\n$$\\operatorname{NS}(X) = \\operatorname{Pic}(X) / \\operatorname{Pic}^0(X).$$\n\n## Remarks\n\nThe theorem of the base says that $\\operatorname{NS}(X)$ is finitely generated; its rank $\\rho(X)$ is the *Picard number*.\nExample: for $X = \\mathbb{P}^n$, $\\operatorname{Pic}(X) \\cong \\Z$, generated by $\\mathcal{O}(1)$, and $\\operatorname{Pic}^0(X) = 0$, so $\\operatorname{NS}(\\mathbb{P}^n) \\cong \\Z$.\nFor 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.\nNon-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.\nOver $\\C$, 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.\n",
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