Cluster Structures on Algebraic Varieties

在代数几何和表示理论的前沿，簇代数及其相关簇变在镜像对称和射影变的环退化构造中发挥着越来越重要的作用。然而，fmin - zelevinsky（2002）和Fock-Goncharov（2009）对它们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 .人们普遍认为簇簇变体是环面变体的概括，但除了代数环面本身之外，不知道如何为给定的环面变体产生簇簇结构。该项目的第一个目标就是设计这样一个结构聚类品种的构造很容易导致非诺瑟型品种，并且没有一般的标准来确定聚类品种是诺瑟型还是有限型。该项目的另一个目标是提供这种标准在上述两个目标方向上的结果将为集群品种的构建提供新的思路。该项目的第三个目标是利用这些新信息来描述更多的代数-几何方法来构建簇品种。这些目标的出发点是Gross-Hacking-Keel通过初等变换将二维簇变描述为仿射平面的胶合。初等变换在高维中也存在，用初等变换来推广Groos-Hacking-Keel构造是可行的。此外，使用这种结构来描述所有的二维环面品种作为簇品种是实现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.