Let $X$ be a complex smooth cubic threefold and $C$ be a smooth twisted cubic, then $C\subset Y\subset X$ for a unique cubic surface $Y$ in $X$ (or equivalently a hyperplane section of $X$). When $Y$ is smooth, we have $27$ lines on $Y$.


What can we say about the position of the a line $L$ and the twisted cubic $C$ (or say $\mathcal{O}\_L(C)$)?


I will ask a question about the singular $Y$ separately.



twisted cubic in a smooth hyperplane section of a cubic threefold
Although this is not necessarily, let me assume that $Y$ is a smooth cubic surface. Then the linear system of a twisted cubic curve $C \subset Y$ generates a linear system that induces a morphism
$$
\pi \colon Y \to \mathbb{P}^2
$$
which is a blowup of 6 points, say $P\_1,\dots,P\_6$, and so that $\mathcal{O}\_Y(C)$ is the pullback of $\mathcal{O}\_{\mathbb{P}^2}(C)$. In other words, $C$ is the preimage of a line under $\pi$.


If $C$ is irreducible, the corresponding line does not pass through the points $P\_i$. Therefore, it doesn't intersect 6 lines on $Y$ (the exceptional divisors over the $P\_i$), intersect ones 15 lines (the strict transforms of lines through $P\_i$ and $P\_j$, $i \ne j$), and intersect twice 6 lines (the strict transforms of conics).


