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.