Intersection of a line and a conic in a surface

Question

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?

Answer 1

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 $L \cap C = \emptyset$.

Answer 2

Consider a smooth cubic surface $X$, namely the blow up of $\mathbb{P}^2_k$ at six distinct points $\{p_1, \ldots, p_6 \}$ in general position.

Then you can take as $C$ the strict transform in $X$ of a conic through $4$ of the points $p_i$ and as $L$ the exceptional divisor at one of the remaining points.

Answer 3

Yet another example is a cubic scroll. Let $L,C \subset \mathbb{P}^4$ be a line and a conic, whose linear span does not intersect the line. Choose an isomorphism $L \cong \mathbb{P}^1 \cong C$ and let $S$ be the union of lines in $\mathbb{P}^4$ joining points in $L$ with their images in $C$. Alternatively, $$ S \cong \mathbb{P}_{\mathbb{P}^1}(O(1) \oplus O(2)). $$