Twists and braids for general 3-fold flops
Twists and braids for general 3-fold flops
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DOI：
10.4171/jems/868
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发表时间：
2015-04
影响因子：
2.6
通讯作者：
W. Donovan;M. Wemyss
中科院分区：
文献类型：
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作者：
W. Donovan;M. Wemyss
给定仅具有 Gorenstein 终端奇点的拟射影 3 重 X，我们证明从 X 开始的随角异色函子满足更高阶的辫子关系，其组合由实超平面排列 H 控制。这导致了一个通用理论，结合了具有 3 阶辫子关系的已知特殊情况，其中我们表明，即使对于在一点相交的两条平滑有理曲线，也可以出现更高阶的关系。对于任何这样的允许单独可动曲线的三重，该理论产生了 H 的复数补集的基本群对 X 的派生范畴的作用。我们还在更一般的情况下构建了这样的动作，其中个别曲线可能会在分析上失败，但不会在代数上失败，此外，在 X 是 Q 阶乘的附加假设下，我们将动作提升为仿射纯辫子群的形式。在此过程中，我们产生了两种新类型的派生自等价性。一种使用扑动收缩的方案理论纤维的交换变形，另一种使用具有简化方案结构的纤维的非交换变形，概括了 Toda 和作者的结构，仅考虑扑动轨迹不可约的情况。对于不可约曲线的 A 型触发器，我们证明了两个自等价性是相关的，但在其他情况下它们非常不同，非交换扭曲通过 Bridgeland-Chen 触发器函子与双有理几何相关联。
Given a quasi-projective 3-fold X with only Gorenstein terminal singularities, we prove that the flop functors beginning at X satisfy higher degree braid relations, with the combinatorics controlled by a real hyperplane arrangement H. This leads to a general theory, incorporating known special cases with degree 3 braid relations, in which we show that higher degree relations can occur even for two smooth rational curves meeting at a point. This theory yields an action of the fundamental group of the complexified complement of H on the derived category of X, for any such 3-fold that admits individually floppable curves. We also construct such an action in the more general case where individual curves may flop analytically, but not algebraically, and furthermore we lift the action to a form of affine pure braid group under the additional assumption that X is Q-factorial. Along the way, we produce two new types of derived autoequivalences. One uses commutative deformations of the scheme-theoretic fibre of a flopping contraction, and the other uses noncommutative deformations of the fibre with reduced scheme structure, generalising constructions of Toda and the authors which considered only the case when the flopping locus is irreducible. For type A flops of irreducible curves, we show that the two autoequivalences are related, but that in other cases they are very different, with the noncommutative twist being linked to birational geometry via the Bridgeland-Chen flop-flop functor.