Nef line bundle

Remarks

A Cartier divisor $D$ is nef when $\mathcal{O}_X(D)$ is nef, that is, $D \cdot C \ge 0$ for all curves $C$.

Example: every ample $\mathcal{L}$ is nef. Some 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. The 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.

Non-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 $-1$.

On 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 $V$. It also says that the nef cone is the closure of the ample cone.