A complete classification of $3$-dimensional quadratic AS-regular algebras of Type EC

TL;DR

This paper provides a comprehensive classification of 3-dimensional quadratic AS-regular algebras of Type EC, characterized by elliptic point schemes, and identifies unique algebraic structures within this class.

Contribution

It offers a complete list of geometric pairs and twisted superpotentials for these algebras, advancing the understanding of their structure and classification.

Findings

Only two exceptions exist among these algebras that are not twists of Calabi-Yau AS-regular algebras.

Complete classification of geometric pairs associated with Type EC algebras.

Explicit list of twisted superpotentials for all such algebras.

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 which cannot be written as a twist of a Calabi-Yau AS-regular algebra by a graded algebra automorphism.