The length classification of threefold flops via noncommutative algebras
The length classification of threefold flops via noncommutative algebras
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DOI：
10.1016/j.aim.2018.11.023
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发表时间：
2017-09
影响因子：
1.7
通讯作者：
J. Karmazyn
中科院分区：
文献类型：
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作者：
J. Karmazyn
具有不可约中心的平滑三重触发器通过长度不变量来分类，其取值为1、2、3、4、5或6。Katz和莫里森的分类确定了Kleinian奇点的6种可能的部分分解，这些部分分解可以作为一般的超平面截面出现，并且与这种部分分解相关联的同时分解产生长度为l的普适翻转。我们引入了长度为l的泛触发代数，从这个泛触发代数中可以通过模构造来恢复长度为l的泛触发，并且我们将这些代数中的每一个都表示为具有关系的图的路代数。这种明确的实现，然后可以用来构建NCCR相关的任何长度的三重触发器的例子，如超势定义的关系，恢复由Curto和莫里森所描述的通用触发器的矩阵分解描述，并实现压缩代数的例子。
Smooth threefold flops with irreducible centres are classified by the length invariant, which takes values 1, 2, 3, 4, 5 or 6. This classification by Katz and Morrison identifies 6 possible partial resolutions of Kleinian singularities that can occur as generic hyperplane sections, and the simultaneous resolutions associated to such a partial resolution produce theuniversal flop of length l.In this paper we translate these ideas into noncommutative algebra. We introduce theuniversal flopping algebra of length lfrom which the universal flop of lengthlcan be recovered by a moduli construction, and we present each of these algebras as the path algebra of a quiver with relations. This explicit realisation can then be used to construct examples of NCCRs associated threefold flops of any length as quiver with relations defined by superpotentials, to recover the matrix factorisation description of the universal flop conjectured by Curto and Morrison, and to realise examples of contraction algebras.