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Let $D$ be a $\Bbb Q$-Cartier divisor on an algebraic variety $X$. We say $D$ is numerically effective if $D\cdot C\geq 0$ for every algebraic curve $C$ in $X$.

How should one interpret this geometrically?

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This is kind of a tricky question to give a single answer to. The characterization of nef divisors that I use most often is provided by Kleiman's Theorem, which says that the (closed) cone generated by classes of nef divisors (in the Neron-Severi space of $\mathbb Q$-divisors modulo numerical equivalence) is the closure of the cone of ample divisors, which is in turn the interior of the nef cone. So in some sense just as ampleness is a weakening of very ampleness that is technically easier to work with, nefness is a useful weakening of the concept of ampleness.

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