In Example 2.2.19 of


Lazarsfeld, Positivity in Algebraic Geometry I,


I found the following statement:


Let $D$ be a divisor on an irreducible projective variety $X$. Then $D$ is nef and big if and only if there exists an effective divisor $N$ such $D-\frac{1}{k}N$ is ample for $k\gg 0$.


It is clear to me that $D$ nef and big implies that there exists an effective divisor $N$ such $D-\frac{1}{k}N$ is ample for $k\gg 0$. It is quite clear that $D-\frac{1}{k}N$ ample implies $D$ big.


I can not understand why $D-\frac{1}{k}N$ ample implies $D$ nef. Perhaps I am just misunderstanding the staments. We may write $\epsilon = \frac{1}{k}$ and $D-\epsilon N$. Does the statement mean $D-\epsilon N$ is ample for some $\epsilon$ small enough or for any $0< \epsilon\ll 1$ ?



Big and Nef divisors
$D-\frac{1}{k}N$ ample for any $k\gg 0$, that is $D-\epsilon N$ ample for any $0<\epsilon\ll 1$ ample, implies that $D$ is nef. This is because $Nef(X)$ is the closure of $Amp(X)$. If $C$ is an irreducible effective curve then $(D-\epsilon N)\cdot C = D\cdot C-\epsilon N\cdot C >0$. Therefore $D\cdot C>\epsilon N\cdot C$ for any $0<\epsilon\ll 1$, that is $D\cdot C\geq 0$, and $D$ is nef.


On the other hand $D-\epsilon N$ for some $\epsilon >0$ implies that $D$ is big by Corollary 2.2.7 of Lazarsfeld, *Positivity in Algebraic Geometry I*. Nowever $D$ is not nef in general. For instance, let us consider $X = \mathbb{P}(\mathcal{O}\_{\mathbb{P}^1}\oplus \mathcal{O}\_{\mathbb{P}^1}(-3))$. The anti-canonical divisor is $-K\_{X}=2C\_0+(e+2)F$, where $C\_0$ is the section and $F$ is the fiber. Now, if $\frac{1}{3}<\epsilon < 2$ then 
$$D:=-K\_{X}-\epsilon C\_0 = (2-\epsilon)C\_0+(e+2)F$$
is ample. Indeed the cone of curves is generated by $C\_0$ and $F$. We have $D\cdot C\_0 = 3\epsilon -1 > 0$ because $\epsilon >\frac{1}{3}$, and $D\cdot F = 2-\epsilon >0$ because $\epsilon < 2$. 


However, $-K\_{X}\cdot C\_0 = -1$, and $-K\_X$ is not nef. By the way, this construction works with $X = \mathbb{P}(\mathcal{O}\_{\mathbb{P}^1}\oplus \mathcal{O}\_{\mathbb{P}^1}(-e))$ for any $e\geq 3$.


