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A complete classification of 3-dimensional quadratic as-regular algebras of type EC. (English) Zbl 07333162
Summary: 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 that cannot be written as a twist of a Calabi-Yau AS-regular algebra by a graded algebra automorphism.
##### MSC:
 16E65 Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.) 16W50 Graded rings and modules (associative rings and algebras) 16S37 Quadratic and Koszul algebras 14A22 Noncommutative algebraic geometry 16S38 Rings arising from noncommutative algebraic geometry 14H52 Elliptic curves
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