Let $k$ be a number field and let $X \subset \mathbb{P}^n$ be a non-singular hypersurface of degree $n-1$. Let $F(X)$ denote the Fano variety of lines of $X$. Then it is known that for general $X$ the dimension of $F(X)$ is $n-2$. In particular, over an algebraic closure $\overline{k}$ of $k$, every point lies on a line (as $\dim X = n-1$) and so the lines are Zariski dense over $\overline{k}$. My question is about what happens over $k$.
Does there exist such an $X$ such that the rational points on $F(X)$ are Zariski dense over $k$? i.e. the collection of lines defined over $k$ in $X$ are Zariski dense?
Some remarks:
I said above that it is known that $\dim F(X) = n -2$ for general $X$. One might be worried as a geometer's general might not include any examples defined over a countable field like $k$. Thankfully I don't think this is a concern as it is a conjecture of Debarre and de Jong that we always have $\dim F(X) = n -2$, although I'm not sure what the current state of affairs is.
Here is a non-example. Take $n=4$, i.e. $X$ is a cubic threefold. Then it is known that $F(X)$ is a surface of general type. The Bombieri-Lang conjecture therefore predicts that the rational points on $F(X)$ are not Zariski dense. In fact this conjecture is known in this case by a theorem of Faltings, as $F(X)$ embeds inside its Albanese variety.
