Classification of AS-regular algebras is one of the main interests in noncommutative algebraic geometry. We say that a $3$-dimensional quadratic AS-regular algebra is of Type EC if its point scheme is an elliptic curve in $\mathbb{P}^{2}$. In this paper, we give a complete list of geometric pairs and a complete list of twisted superpotentials corresponding to such algebras. As an application, we show that there are only two exceptions up to isomorphism among all $3$-dimensional quadratic AS-regular algebras which cannot be written as a twist of a Calabi-Yau AS-regular algebra by a graded algebra automorphism.

中文翻译：

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类型 EC 的 3 维二次正则代数的完整分类

AS-正则代数的分类是非交换代数几何的主要兴趣之一。如果一个$3$维二次AS-正则代数的点格式是$\mathbb{P}^{2}$中的椭圆曲线，我们就说它是EC类型的。在本文中，我们给出了完整的几何对列表和与这些代数对应的扭曲超势的完整列表。作为一个应用，我们证明了在所有 $3$-维二次 AS-正则代数中只有两个同构例外，它们不能被分级代数自同构写成 Calabi-Yau AS-正则代数的扭曲。