论文标题
傅立叶 - 穆凯合作伙伴和广义的kummer结构，订单的广义kummer表面$ 3 $
Fourier--Mukai partners and generalized Kummer structures on generalized Kummer surfaces of order $3$
论文作者
论文摘要
广义的kummer表面$ x $的订单$ 3 $是用订单$ 3 $ symblectic自称为订单的Abelian Surface $ a $的商的最低分辨率。我们研究了经典kummer表面的shioda问题的概括，这是为了了解$ x $由$ a $确定的，而相反。 Surface $ x $具有一个大的和nef divisor $ l_ {x} $，因此$ l_ {x}^{2} = 0 $或$ 2 $ mod $ 6 $。我们表明，对于$ l_ {x}^{2} = 6k $的表面，带有$ k \ neq0,6 \，mod \，9 $，表面$ x $确定$ a $ a $的先验lattice $ t（a）$ a $和$ t（a）$的hodge结构。相反，如果$ a $和$ b $是傅立叶 - 穆凯伙伴（即，如果其超验晶格的hodge结构是同构的），而$ y $是广义的kummer表面，这是$ 3 $ symplectic automorphism $ x $ x $ x $ x $ x $ y y y是$ y y is as is as y is as is as is as y is is as is as y是$ b $的$ b $的最低分辨率。这些结果也知道可以使用$ l_ {x}^{2} = 2 \，mod \，6 $中的表面保留。当$ k = 0 \ text {or} 6 \，mod \，9，$我们表明$ x $确定$ t（a）$及其hodge结构，但相反通常不存在。
A generalized Kummer surface $X$ of order $3$ is the minimal resolution of the quotient of an abelian surface $A$ by an order $3$ symplectic automorphism. We study a generalization of a problem of Shioda for classical Kummer surfaces, which is to understand how much $X$ is determined by $A$ and conversely. The surface $X$ posses a big and nef divisor $L_{X}$ such that $L_{X}^{2}=0$ or $2$ mod $6$. We show that for surfaces with $L_{X}^{2}=6k$ with $k\neq0,6\,mod\,9$, the surface $X$ determines the transcendental lattice $T(A)$ of $A$ and the Hodge structure on $T(A)$. Conversely if $A$ and $B$ are Fourier-Mukai partners (i.e. if the Hodge structures of their transcendental lattices are isomorphic) and $Y$ is the generalized Kummer surface which is the minimal resolution of the quotient of $B$ by an order $3$ symplectic automorphism, we obtain that $X$ and $Y$ are isomorphic. These results are also know to hold for surfaces with $L_{X}^{2}=2\,mod\,6$ from a previous work. When $k=0\text{ or }6\,mod\,9,$ we show that $X$ determines $T(A)$ and its Hodge structure, but the converse does not hold in general.
