本项目主要利用几何模型研究代数表示理论，通过构造skewed-gentle的几何模型，几何实现其有界导出范畴中的不可分解对象，并进一步研究skewed-gentle代数的导出表示型与导出等价的相关问题，体现利用几何模型实现代数对象及通过几何模型发掘代数性质两方面的重要意义。具体地说，我们将：(1)利用分次曲面构造skewed-gentled的几何模型，并给出分次曲线与其有界导出范畴中不可分解对象之间的一一对应；(2)在几何模型中描述skewed-gentle代数导出范畴中不可分解对象的AR平移函子及对象间态射空间的维数；(3)利用曲线与不可分解对象的对应，证明skewed-gentle代数有界导出范畴的Bongartz-Ringel定理，即不可分解对象的上同调长度是连续的；(4)研究skewed-gentle代数在几何模型上的导出等价判定，并刻画其中导出唯一的代数。

This project is to study representation theory of algebras via geometric model, which is mainly to construct the geometric model of skewed-gentle algebras and realize geometrically the indecomposables in their derived categories, then investigate the derived representation type and derived equivalences on the geometric model and so on. To be more precise, we will consider the questions as follows: (1) realize derived categories of skewed-gentle algebras by surface with punctured points, establish a bijection between the indecomposables in the derived category and certain curves with degree; (2) describe the AR-translation functor of derived category and the dimension of morphisms between indecomposalbes via geometric items; (3) prove the Bongartz-Ringel's theorem for derived categories of skewed-gentle algebras, i.e., there is no gaps in the sequences of the cohomological lengths of indecomposables in derived categories; (4) consider the derived equivalences in the geometry model and then describe the derived unique skewed-gentle algebras.