加权射影线上的丛倾斜理论

本项目主要讨论加权射影线上的丛范畴的等价判别问题及加权射影线上的丛倾斜代数的模结构。具体研究内容如下:(1)拟证明特征为0的代数闭域上的一个代数的2-CY三角范畴与某加权射影线上的丛范畴等价当且仅当二者各自存在丛倾斜对象其自同态代数是同构的；讨论一些已有的2-CY三角范畴（如预投射代数的模范畴的某些特定子范畴的稳定范畴）何时和加权射影线上的丛范畴等价。(2)拟精确刻画加权射影线特别是tubular型加权射影线上的丛倾斜代数的rigid模，Schurian模以及函子有限的brick；拟证明权重不超过3的加权射影线上的丛倾斜代数的不可分解tau-rigid模由维数向量唯一决定；刻画tubular型丛代数的丛变量的分母向量与相应的tubular型丛倾斜代数的不可分解tau-rigid模维数向量之间的精确关系；拟证明tubular型丛代数不同的丛变量有不同的分母向量。

This is a proposal on cluster-tilting theory of weighted projective lines. It concerns a characterization for the cluster category of a weighted projective line and the module structure over a cluster-tilted algebra arising from a weighted projective line. More precisely, we are interested in the followings:.(1) We will prove that an algebraic 2-CY triangulated category over an algebraically closed field with characteristic zero is a cluster category of a weighted projective line if and only if it contains a cluster-tilting object whose endomorphism algebra is isomorphic to a cluster-tilted algebra arising from some weighted projective line. Moreover, we will also discuss the relations between cluster categories of weighted projective lines and some well-known 2-CY triangulated categories, such as certain subcategoies of module categories of preprojective algebras.. (2) For a cluster-tilted algebra arising from a weighted projective line, especially for a cluster-tilted algebra of tubular type, we will characterize its indecomposable rigid modules, Schurian modules and functorially finite bricks, and study the relationship among them. We also investigate the dimension vecotors of indecomposable tau-rigid modules. We will prove that the indecomposable tau-rigid modules are determined by their dimension vectors under the assumption that the associated weighted projective line has at most three weights. Moreover, for a cluster algebra of tubular type, we will give representation-theoretic formulas for the denominators of cluster variables and prove that different cluster variables have different denominators.