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.

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, Mathematische Zeitschrift 294 (2020).

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