Let $Y_d$ be a Fano threefold of Picard rank $1$ and index $2$ (eg cubic 3fold). There is a natural map $Y_d\to \mathbb{P}^{d+1}$.

  Smooth sections of its anticanonical bundle are $K3$ surfaces.

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

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 $\mathbb{P}^{d+1}$?

Let $Y_d$ be a Fano threefold of Picard rank $1$ and index $2$ (eg cubic 3fold). There is a natural map $Y_d\to \mathbb{P}^{d+1}$.

Smooth sections of its anticanonical bundles are $K3$ surfaces. My question is: does such a general $K3$ surface contain lines in $\mathbb{P}^{d+1}$?

Thank you!

Let $Y_d$ be a Fano threefold of Picard rank $1$ and index $2$ (eg cubic 3fold). There is a natural map $Y_d\to \mathbb{P}^{d+1}$.

Smooth sections of its anticanonical bundle are $K3$ surfaces.

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