First, let me emphasize that for $X$ a not-necessarily proper variety, we say that a line bundle $L$ on $X$ is ample, if for some positive integer $n$, $L^{\otimes n}$ arises as $j^*O(1)$ for some (not-necessarilty closed) immersion $j:X\rightarrow \mathbb{P}^n$.

Now, let $X$ be a variety and $L$ a line bundle on $X$, and let
$f:X^{nor}\rightarrow X$ be the normalization map. Suppose that $f^*L$ is ample. Is it true that $L$ is ample?

First, let me emphasize that for $X$ a not-necessarily proper variety, we say that a line bundle $L$ on $X$ is ample, if for some positive integer $n$, $L^{\otimes n}$ arises as $j^*O(1)$ for some (not-necessarily closed) immersion $j:X\rightarrow \mathbb{P}^n$.

Now, let $X$ be a variety and $L$ a line bundle on $X$, and let
$f:X^{nor}\rightarrow X$ be the normalization map. Suppose that $f^*L$ is ample. Is it true that $L$ is ample?