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



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> **Question:** does such a general $K3$ surface contain lines in $\mathbb{P}^{d+1}$?
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Lines on an anticanonical K3 on a Fano 3-fold
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.


This is explained (for instance) at p. 797 of


C. F. Doran, A. Harder, A. Y. Novoseltsev, A. Thompson: [Calabi–Yau threefolds fibred by high rank lattice polarized K3 surfaces](https://link.springer.com/article/10.1007/s00209-019-02279-9), *Mathematische Zeitschrift* **294** (2020).


