Derived categories of singular surfaces
Derived categories of singular surfaces
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DOI：
10.4171/jems/1106
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发表时间：
2018-09
影响因子：
2.6
通讯作者：
J. Karmazyn;A. Kuznetsov;E. Shinder
中科院分区：
文献类型：
--
作者：
J. Karmazyn;A. Kuznetsov;E. Shinder
我们开发了一种方法，允许构造具有循环商奇点的曲面的派生范畴的半正交分解，其分量等价于局部有限维代数的派生范畴。我们首先解释了如何从曲面分辨率的半正交分解导出具有有理奇异点的曲面X的半正交分解。在X具有环商奇点的情况下，我们引入了分辨率的半正交分解的分量的附合条件，使得用局部有限维代数的派生范畴识别诱导分解的分量成为可能。进一步，我们给出了X的Brauer群中存在这种半正交分解的一个障碍，并证明了在存在阻碍的情况下，对依附条件的适当修正给出了x的扭曲派生范畴的半正交分解。我们通过展示任意法向射影环面的非扭曲或扭曲派生范畴的半正交分解来说明这一理论，这取决于其Weil因子类群是否无扭转。对于加权投影平面，我们显式地计算了分量的生成器，并将我们的结果与基于秩为1的自反束的迭代扩展的Kawamata的结果联系起来。
We develop an approach that allows to construct semiorthogonal decompositions of derived categories of surfaces with cyclic quotient singularities whose components are equivalent to derived categories of local finite dimensional algebras. We first explain how to induce a semiorthogonal decomposition of a surface X with rational singularities from a semiorthogonal decomposition of its resolution. In the case when X has cyclic quotient singularities, we introduce the condition of adherence for the components of the semiorthogonal decomposition of the resolution that allows to identify the components of the induced decomposition with derived categories of local finite dimensional algebras. Further, we present an obstruction in the Brauer group of X to the existence of such semiorthogonal decomposition, and show that in the presence of the obstruction a suitable modification of the adherence condition gives a semiorthogonal decomposition of the twisted derived category of X. We illustrate the theory by exhibiting a semiorthogonal decomposition for the untwisted or twisted derived category of any normal projective toric surface depending on whether its Weil divisor class group is torsion-free or not. For weighted projective planes we compute the generators of the components explicitly and relate our results to the results of Kawamata based on iterated extensions of reflexive sheaves of rank 1.