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A Complete Classification of 3-dimensional Quadratic AS-regular Algebras of Type EC

Part of: Rings and algebras with additional structure Rings and algebras arising under various constructions Curves Homological methods Algebraic geometry: Foundations

Published online by Cambridge University Press:  27 April 2020

Masaki Matsuno
Masaki Matsuno*
Affiliation:
Graduate School of Science and Technology Educational Division, Shizuoka University Ohya 836, Shizuoka422-8529, Japan
*
e-mail: matsuno.masaki.14@shizuoka.ac.jp
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Abstract

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.

Keywords

AS-regular algebrasgeometric algebrasCalabi-Yau algebrassuperpotentialselliptic curves

MSC classification

Primary: 14A22: Noncommutative algebraic geometry
Secondary: 14H52: Elliptic curves 16W50: Graded rings and modules 16S37: Quadratic and Koszul algebras 16E65: Homological conditions on rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 1 , March 2021 , pp. 123 - 141
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Copyright
© Canadian Mathematical Society 2020

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