论文标题
有限类型的二维集群品种的压缩的代数和符号观点
Algebraic and symplectic viewpoint on compactifications of two-dimensional cluster varieties of finite type
论文作者
论文摘要
在本文中，我们探讨了在第二个综合维度中有限类型的群集品种的压缩。群集品种可以看作是由theta函数产生的环的规格，并且可以通过该环上的分级给出这种品种的压缩，可以通过阳性多面体描述[17]。在我们利用的示例中，群集品种可以解释为Del Pezzo表面某些除数的补充。在符号观点中，可以通过$ \ r^2 $（完成后）上的几乎要折叠纤维来描述它们。一旦将它们识别为几乎要折叠的歧管，就可以在其他del Pezzo表面中符合地查看它们。因此，我们可以识别同一聚类品种的其他符号压缩，我们期望这也应该对应于不同的代数压缩。这两个观点都在这里介绍，几种压缩都具有相应的多面体。探索了簇突变的有限性，以在图表中提供循环，描述了通过几乎通过复曲面突变连接的del pezzo表面中的单调拉格朗日摩tori [34]。
In this article we explore compactifications of cluster varieties of finite type in complex dimension two. Cluster varieties can be viewed as the spec of a ring generated by theta functions and a compactification of such varieties can be given by a grading on that ring, which can be described by positive polytopes [17]. In the examples we exploit, the cluster variety can be interpreted as the complement of certain divisors in del Pezzo surfaces. In the symplectic viewpoint, they can be described via almost toric fibrations over $\R^2$ (after completion). Once identifying them as almost toric manifolds, one can symplectically view them inside other del Pezzo surfaces. So we can identify other symplectic compactifications of the same cluster variety, which we expect should also correspond to different algebraic compactifications. Both viewpoints are presented here and several compactifications have their corresponding polytopes compared. The finiteness of the cluster mutations are explored to provide cycles in the graph describing monotone Lagrangian tori in del Pezzo surfaces connected via almost toric mutation [34].
