Fano visitors, Fano dimension and orbifold Fano hosts

In arXiv:1503.00125, the authors proved that every complete intersection smooth projective variety $Y$ is a Fano visitor, i.e. its derived category $D^b(Y)$ is equivalent to a full triangulated subcategory of the derived category $D^b(X)$ of a smooth Fano variety $X$, called a Fano host of $Y$. They also introduced the notion of Fano dimension of $Y$ as the smallest dimension of a Fano host $X$ and obtained an upper bound for the Fano dimension of each complete intersection variety. In this paper, we provide a Hodge-theoretic criterion for the existence of a Fano host which enables us to determine the Fano dimensions precisely for many interesting examples, such as low genus curves, quintic Calabi-Yau 3-folds and general complete intersection Calabi-Yau varieties. Next we initiate a systematic search for more Fano visitors. We generalize the methods of arXiv:1503.00125 to prove that smooth curves of genus at most 4 are all Fano visitors and general curves of genus at most 9 are Fano visitors. For surfaces and higher dimensional varieties, we find more examples of Fano visitors and raise natural questions. We also generalize Bondal's question and study triangulated subcategories of derived categories of Fano orbifolds. We proved that there are Fano orbifolds whose derived categories contain derived categories of orbifolds associated to quasi-smooth complete intersections in weighted projective spaces, Jacobians of curves, generic Enriques surfaces, some families of Kummer surfaces, bielliptic surfaces, surfaces with $\kappa=1,$ classical Godeaux surfaces, product-quotient surfaces, holomorphic symplectic varieties, etc. From these constructions, we found Fano orbifolds whose derived categories contain quasi-phantom categories or phantom categories.