Let $X$ be a smooth projective variety and $A$ an ample $\mathbb{Q}$-divisor on $X$.


Is the round up $\ulcorner A\urcorner$ of $A$ ample? 


I think it's true. But I do not know how to arrange an argument.



Is the round up of ample Q-divisor ample?
This doesn't seem right. Let $\pi : X \to \mathbb P^2$ be the blow-up of $\mathbb P^2$ at a point, and let $A = L + 2/3 E$, where $E$ is the exceptional divisor, and $L$ is the strict transform of a line through the point. Take $H$ to be the pullback of a line in $\mathbb P^2$. Then $A$ is linearly equivalent to $(H-E)+2/3 E = H - 1/3 E$, which is ample. But the round-up of $A$ is $L+E$, linearly equivalent to $H$, which isn't ample.


