I started reading Positivity in Algebraic Geometry I by Roberts Lazarsfeld, and he introduces nef (numerically effective) line bundles, after observing that for an ample line bundle $\mathscr{L}$, one has \begin{align} (\mathscr{L}^{\otimes k} \cdot V) > 0 && \text{for all subvarieties }V \subset X \text{ of dimension } k \end{align}

Following this, a line bundle $\mathscr{L}$ is defined to be nef, if \begin{align} (\mathscr{L} \cdot C) \geq 0 && \text{for all curves } C \subset X.\end{align} In particular we can reduce the question about nefness to computing the degree of $\mathscr{L}|_C$. Then Lazarsfeld proves a theorem by Kleiman, that $\mathscr{L}$ is nef if and only if \begin{align}(\mathscr{L}^{\otimes k} \cdot V) \geq 0 && \text{for all subvarieties } V \subset X \text{ of dimension } k\end{align} So nef is clearly a generalization of ample.

I wonder if there is an analogous statement for ampleness, i.e. $\mathscr{L}$ is ample if and only if $\deg{\mathscr{L}|_C} > 0$ for all curves $C \subset X$.

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    $\begingroup$ No, that is not true. There is a counterexample due to Mumford explained later in the same chapter you are reading: Example 1.5.2 on p.72. $\endgroup$ – Bort Apr 29 at 14:05

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