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?



Does ampleness descend along finite maps?
There is a non-quasi-affine variety $X$ with quasi-affine normalization. See [Tag 0271](https://stacks.math.columbia.edu/tag/0271). Then $\mathcal{O}\_X$ is a counter example.


