Are there any ?
Namely let $X$ be a smooth del Pezzo surface defined over $\mathbb{Q}$ that has rational points and such that the degree of the del Pezzo is small, say $d=3$ or $4$. Is it possible that all of its $\mathbb{Q}$ points lie on some exceptional curve ( i.e. a line, equivalently a $-1$ curve) which is possibly defined over $\overline{\mathbb{Q}}$ but not over $\mathbb{Q}$ ?
In the case $d=3$ Kollar (based on work of Segre) has proved that if $X(\mathbb{Q})\neq 0$ and $X$ smooth then it is Zariski dense in $X$. Does this mean that $X(\mathbb{Q})$ is not a subset of the $27$ lines of $X$ ?
Any example would be appreciated (or ruling out of such an instance) would be appreciated.