Cluster Structures on Algebraic Varieties

在代数几何与表示论的边缘，簇代数及其相关的簇簇在射影簇的镜像对称性和环形退化的构造中起着越来越重要的作用。然而，Fomin-Zlevinsky在2002年和Fock-Goncharov在2009年给出的定义在代数几何学者看来仍然很奇怪，甚至很难回答关于这些结构的基本代数几何问题：1人们普遍认为簇簇是环面簇的推广，但除了代数环面本身之外，还不知道如何产生给定环面簇的簇结构。该项目的第一个目标是设计这样一种结构2.簇变种的构建很容易导致非以太品种，并且没有一般标准来确定一个簇品种何时是以太品种或有限类型品种。该项目的另一个目标是提供这样的标准。3上述两个目标方向的结果将为集群品种的建设带来新的曙光。该项目的第三个目标是利用这些新的信息来描述更多的代数几何方法来构造簇簇。这些目标的起点是Gross-hking-Keel通过初等变换将二维簇簇描述为仿射平面的粘合。初等变换也存在于更高的维度中，尝试用它们来推广Groos-hking-Keel的构造是合理的。此外，使用这种结构将所有的二维环变元描述为簇变元是迈向第一个目标的重要的第一步。

At the border of Algebraic Geometry and Representation Theory, cluster algebras and their associated cluster varieties play a more and more important role in mirror symmetry and the construction of toric degenerations of projective varieties. However, their definition by Fomin-Zelevinsky in 2002 and Fock-Goncharov in 2009 still looks bizarre to Algebraic Geometers and makes it difficult to answer even basic algebraic-geometric questions on these structures:1 It is widely assumed that cluster varieties are generalizations of toric varieties, but it is not known how to produce a cluster structure for a given toric varieties besides the algebraic torus itself. The first objective of the project is to devise such a construction.2 The construction of a cluster variety can easily lead to non-Noetherian varieties, and there are no general criteria to decide when a cluster variety is Noetherian or of finite type. Another objective of the project is to provide such criteria.3 Results in the directions of both objectives above will shed new light on the construction of a cluster variety. A third objective of the project is to use this new information to descibe more algebraic-geometric ways to construct cluster varieties.A starting point for these objectives is given by Gross-Hacking-Keel's description of 2-dimensional cluster varieties as gluings of affine planes by elementary transformations. Elementary transformations also exist in higher dimension, and it is plausible to try to use them to generalize Groos-Hacking-Keel's construction. Also, using this construction to describe all 2-dimensional toric varieties as cluster varieties is an important first step towards the first objective.