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  "body": "\n## Definition\n\nLet $X$ be a scheme that is [[proper|9537RKBZPP]] over a field $k$, and let $\\mathcal{L}$ be an [[invertible sheaf|7JCF2TE3V5]] on $X$.\nFor a closed integral [[curve|XJM2MT4S3Y]] $C \\subseteq X$, proper over $k$, let $\\deg(\\mathcal{L}|_C)$ be the degree of the restriction of $\\mathcal{L}$ to $C$, also written $(\\mathcal{L} \\cdot C)$.\nThe sheaf $\\mathcal{L}$ is *nef* if $(\\mathcal{L} \\cdot C) \\ge 0$ for every closed integral curve $C \\subseteq X$.\n\n## Remarks\n\nA Cartier divisor $D$ is nef when $\\mathcal{O}_X(D)$ is nef, that is, $D \\cdot C \\ge 0$ for all curves $C$.\n\nExample: every [[ample|RRK8P6TXG2]] $\\mathcal{L}$ is nef.\nSome power $\\mathcal{L}^{\\otimes m}$ is very ample, so $m \\, (\\mathcal{L} \\cdot C)$ is the degree of $C$ in a projective embedding, which is positive.\nThe trivial bundle $\\mathcal{O}_X$ is nef, with degree $0$ on every curve, and it is not ample when $\\dim X \\ge 1$ and $X$ is projective.\n\nNon-example: on $\\mathbb{P}^1_k$, $\\mathcal{O}(-1)$ is not nef, since its degree on the curve $C = \\mathbb{P}^1_k$ itself is $-1$.\n\nOn a projective scheme, Kleiman's criterion says that $\\mathcal{L}$ is nef if and only if $(\\mathcal{L}^{\\dim V} \\cdot V) \\ge 0$ for every closed subvariety $V$.\nIt also says that the nef cone is the closure of the ample cone.\n",
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