In noncommutative algebraic geometry an Artin–Schelter regular (AS-regular) algebra is one of the main interests, and every three-dimensional quadratic AS-regular algebra is a geometric algebra, introduced by Mori, whose point scheme is either $\mathbb {P}^{2}$ or a cubic curve in $\mathbb {P}^{2}$ by Artin et al. [‘Some algebras associated to automorphisms of elliptic curves’, in: The Grothendieck Festschrift, Vol. 1, Progress in Mathematics, 86 (Birkhäuser, Basel, 1990), 33–85]. In the preceding paper by the authors Itaba and Matsuno [‘Defining relations of 3-dimensional quadratic AS-regular algebras’, Math. J. Okayama Univ. 63 (2021), 61–86], we determined all possible defining relations for these geometric algebras. However, we did not check their AS-regularity. In this paper, by using twisted superpotentials and twists of superpotentials in the Mori–Smith sense, we check the AS-regularity of geometric algebras whose point schemes are not elliptic curves. For geometric algebras whose point schemes are elliptic curves, we give a simple condition for three-dimensional quadratic AS-regular algebras. As an application, we show that every three-dimensional quadratic AS-regular algebra is graded Morita equivalent to a Calabi–Yau AS-regular algebra.

在非交换代数几何中，Artin-Schelter 正则 (AS-regular) 代数是主要兴趣之一，每个三维二次 AS-正则代数都是由 Mori 引入的几何代数，其点格式为 $\mathbb { P}^{2}$ 或Artin等人在 $\mathbb {P}^{2}$ 中的三次曲线。['一些与椭圆曲线自同构相关的代数'，见：格洛腾迪克音乐节，卷。1, 数学进展, 86 (Birkhäuser, Basel, 1990), 33–85]。在作者 Itaba 和 Matsuno 的前一篇论文中 ['Defining Relations of 3-dimensional quadratic AS-regular algebras', Math. J.冈山大学 63 (2021), 61-86]，我们确定了这些几何代数的所有可能定义关系。但是，我们没有检查他们的 AS-regularity。在本文中，通过使用 Mori-Smith 意义上的扭曲超电势和扭曲超电势，我们检查了点方案不是椭圆曲线的几何代数的 AS 正则性。对于点格式为椭圆曲线的几何代数，我们给出了三维二次AS正则代数的简单条件。作为一个应用，我们展示了每个 3 维二次 AS 正则代数的 Morita 等价于一个 Calabi-Yau AS 正则代数。