法诺访客，法诺维度和orbifold法诺主机
Fano visitors, Fano dimension and orbifold Fano hosts
摘要：
在arXiv：1503.00125中，作者证明了每个完全交光滑投射簇$Y$是Fano访问者，即它的导出范畴$D^b(Y)$等价于光滑Fano簇$X$的导出范畴$D^b(X)$的完全三角化子范畴，称为$Y$的Fano宿主。他们还引入了$Y$的Fano维数作为Fano宿主$X$的最小维数的概念，并得到了每个完全交簇的Fano维数的上界。在本文中，我们提供了一个Hodge理论的Fano主机的存在性，使我们能够确定Fano维数精确的许多有趣的例子，如低亏格曲线，五次Calabi-Yau 3-folds和一般的完全交叉Calabi-Yau品种。接下来，我们开始系统地搜索更多的法诺游客。推广了arXiv：1503.00125的方法，证明了亏格不超过4的平滑曲线都是Fano访问者，亏格不超过9的一般曲线都是Fano访问者.对于表面和更高维度的品种，我们发现更多的例子，法诺游客和提出自然的问题。推广了Bondal问题，研究了Fano orbifolds的导出范畴的三角化子范畴.证明了存在Fano orbifolds，其导出范畴包含与加权射影空间中的拟光滑完全交、曲线的Jacobian、一般Enriques曲面、某些Kummer曲面族、双椭圆曲面、具有$κ=1,$经典Godeaux曲面的曲面、乘积商曲面、全纯辛簇等相关的orbifolds的导出范畴.我们发现了Fano orbifolds，其导出范畴包含准幻影范畴或幻影范畴。
英文摘要：
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 $κ=1,$ 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.
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