Suppose $X$ is an algebraic variety over a field $F$ of characteristic 0. By resolution of singularities, there is a nonsingular variety $Y$ over $F$ with a proper birational morphism $Y \rightarrow X$.
It is true that a rational point on $Y$ induces one on $X$. How about the converse? Are there examples where $X$ admits a rational point but $Y$ doesn't?