Comparing ample and nef line bundles

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 $$C \subset X$$.

• 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. – Bort Apr 29 at 14:05