Obstructions to Semiorthogonal Decompositions for Singular Threefolds I: K-Theory
Obstructions to Semiorthogonal Decompositions for Singular Threefolds I: K-Theory
复制标题
奇异三重半正交分解的障碍 I：K 理论
DOI：
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发表时间：
2019
影响因子：
0.8
通讯作者：
E. Shinder
中科院分区：
文献类型：
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作者：
Martin Kalck;Nebojsa Pavic;E. Shinder
本文研究了Gorenstein投影簇允许Kawamata引入的半正交分解的必要条件，重点研究了具有孤立复合$A_n$奇点的三重投影簇。我们引入来自代数$\mathrm{K}$-理论的障碍，并将其转化为极大非因式性的概念。
利用这些障碍，我们表明，许多类的节点三倍不承认川俣型半正交分解。这些包括节点超曲面和双固体，除了一个节点二次曲面，和德尔佩佐threefolds度$1 \le d \le 4$与最大类组秩。
我们还调查了当一个光滑的三重奇异曲线的爆破承认一个Kawamata型半正交分解，我们给这个问题一个完整的答案时，曲线是节点，只有合理的组件。
We investigate necessary conditions for Gorenstein projective varieties to admit semiorthogonal decompositions introduced by Kawamata, with main emphasis on threefolds with isolated compound $A_n$ singularities. We introduce obstructions coming from Algebraic $\mathrm{K}$-theory and translate them into the concept of maximal nonfactoriality.
Using these obstructions we show that many classes of nodal threefolds do not admit Kawamata type semiorthogonal decompositions. These include nodal hypersurfaces and double solids, with the exception of a nodal quadric, and del Pezzo threefolds of degrees $1 \le d \le 4$ with maximal class group rank.
We also investigate when does a blow up of a smooth threefold in a singular curve admit a Kawamata type semiorthogonal decomposition and we give a complete answer to this question when the curve is nodal and has only rational components.