论文标题
Kollár--shepherd-Barron的分类方面
Categorical aspects of the Kollár--Shepherd-Barron correspondence
论文作者
论文摘要
众所周知，$ 2 $维的循环商奇异性$ \ overline {w} $具有与Kalck和Karmazyn引入的有限尺寸的尺寸关联代数$ \ Ovilline {r} $相同的奇异性类别。我们研究了由表面$ \ overline {w} $的变形引起的代数$ \ operline {r} $的变形。我们证明它们是莫里塔（Morita） - 与距离代数$ \ hat ocyclic颤动的路径代数{r} $，用于在$ \ overline {w} $的versal变形空间的每个不可减少组件中的一般平滑（如Kollár和Shepherd-barron所述）。此外，只有$ \ hat {r} $是半简单的，并且只有当平滑为$ \ mathbb {q} $ - gorenstein时（一个方向是由于kawamata造成的）。我们提供许多应用程序。例如，我们描述了所有Dolgachev表面上的$ 10 $的强大杰出集合，并将Quivers Quivers类别的可允许类别的可接受嵌入到理性表面的派生类别中。
It is well known that a $2$-dimensional cyclic quotient singularity $\overline{W}$ has the same singularity category as a finite dimensional associative algebra $\overline{R}$ introduced by Kalck and Karmazyn. We study the deformations of the algebra $\overline{R}$ induced by the deformations of the surface $\overline{W}$ to a smooth surface. We show that they are Morita--equivalent to path algebras $\hat{R}$ of acyclic quivers for general smoothings within each irreducible component of the versal deformation space of $\overline{W}$ (as described by Kollár and Shepherd-Barron). Furthermore, $\hat{R}$ is semi-simple if and only if the smoothing is $\mathbb{Q}$-Gorenstein (one direction is due to Kawamata). We provide many applications. For example, we describe strong exceptional collections of length $10$ on all Dolgachev surfaces and classify admissible embeddings of derived categories of quivers into derived categories of rational surfaces.
