We introduce a new class of algebraic varieties which we call frieze varieties. Each frieze variety is determined by an acyclic quiver. The frieze variety is defined in an elementary recursive way by constructing a set of points in affine space. From a more conceptual viewpoint, the coordinates of these points are specializations of cluster variables in the cluster algebra associated to the quiver. We give a new characterization of the finite--tame--wild trichotomy for acyclic quivers in terms of their frieze varieties. We show that an acyclic quiver is representation finite, tame, or wild, respectively, if and only if the dimension of its frieze variety is $0,1$, or $\ge2$, respectively.

中文翻译：

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带状变体：非循环箭袋的有限驯服-野生三分法的特征

我们引入了一类新的代数簇，我们称之为 frieze 簇。每个饰带品种都由一个非循环的箭袋决定。通过在仿射空间中构造一组点，以基本递归方式定义了饰带变体。从更概念化的角度来看，这些点的坐标是与箭袋相关联的簇代数中簇变量的特化。我们根据楣变体对非循环箭袋的有限-驯服-野生三分法进行了新的表征。我们表明，当且仅当其楣变体的维度分别为 $0,1$ 或 $\ge2$ 时，非循环箭袋分别是有限的、驯服的或狂野的表示。