For some reason, I'm convinced the answer to the following question should be (obviously) negative, but I can't come up with a good reason.
Does there exist a number field $K$, a smooth projective geometrically connected curve $X/K$ of genus $g\geq 2$ with a $K$-rational point $x$ such that, for any number field $L/K$, $x$ does not intersect $ X(L)$?
Let me make the last part of the question more precise. Firstly, let $\mathcal X$ be the minimal regular model of $X$ over $O_K$. When I say that $x$ does not intersect $X(L)$ I mean that the intersection product $(x,y)_{\mathcal X}$ on $\mathcal X$ equals zero for all $y\in X(L)-\{x\}$. (Here $x$ and $y$ also denote their Zariski closures in $\mathcal X$.)
I think it could happen that some $K$-rational point does not intersect any other $K$-rational point. Take for example a curve with $X(K) = \{pt\}$. For some reason I do think that there should always be some (other) $L$-rational point which intersects this $K$-rational point.