# Lines on an anticanonical K3 on a Fano 3-fold

Let $$Y_d$$ be a Fano threefold of Picard rank $$1$$ and index $$2$$ (eg cubic 3fold). There is a natural anticanonical map $$Y_d\to \mathbb{P}^{d+1}$$. Smooth sections of the anticanonical bundle are $$K3$$ surfaces, so we can ask the following

Question: does such a general $$K3$$ surface contain lines in $$\mathbb{P}^{d+1}$$?

If $$Y$$ is a Fano threefold with Picard rank $$1$$, then its general anticanonical element $$X \in |-K_Y|$$ is a $$K3$$ surface with Picard rank $$1$$. In particular, $$X$$ contains no lines.