Fano visitor problem for K3 surfaces

Let $X$ be a K3 surface with Picard number 1 and genus $g$, such that $g\not\equiv 3 \mod 4$. In this paper, we show that $X$ is a Fano visitor, i.e., there is a smooth Fano variety $Y$ and an embedding $D^b(X)\hookrightarrow D^b(Y)$ given by a fully faithful functor. If $g\equiv 3\mod 4$, we construct a smooth weak Fano variety $Y$. Our proof is based on several results concerning a sequence of flips associated with a K3 surface and an ample line bundle. This sequence is constructed by using the work of Bayer and Macr\`i on the description of the birational geometry of a moduli space of sheaves on a K3 surface through Bridgeland stability conditions, and the study of the fixed locus of antisymplectic involutions on hyperk\"ahler manifolds by Sacc\`a, Macr\`i, O'Grady, and Flapan.