Does there exists a smooth projective surface $X$ which contains a projective line $L$ and a smooth conic $C$ such that $L\cap C=$empty?



Intersection of a line and a conic in a surface
Yes. This holds for any cubic surface over an algebraically closed field $k$.


Let $S$ be such a surface. Let $L'$ be a line. The pencil of hyperplanes containing $L'$ forms a conic bundle $\pi: S \to \mathbb{P}^1$. This conic bundle has $5$ singular fibres, which are each a pair of lines meeting in a point. Take $C$ a smooth fibre and $L$ one of the lines in a singular fibre. Then $L \cap C = \emptyset$.


