To be precise, suppose $X$ is a smooth variety defined over the rational number field $Q$, fix a $Q$ model of it, and let $P$ be a $Q$ point of $X$ under this model. Then take $\pi:\widetilde{X}\to X$ to be the geometric blowing-up of $X$ at $P$. Can we find a $Q$ model of $\widetilde{X}$ such that under that model, the morphism $\pi$ is also defined over $Q$? If not, what is the best thing we can do?
Thanks