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.
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).
