# AS-regularity of geometric algebras of plane cubic curves

## TL;DR

This paper classifies 3-dimensional quadratic AS-regular algebras related to plane cubic curves, providing explicit superpotentials and criteria, and shows their Morita equivalence to Calabi-Yau AS-regular algebras, simplifying their geometric study.

## Contribution

It offers a complete classification of superpotentials for these algebras and establishes Morita equivalence to Calabi-Yau AS-regular algebras, advancing understanding of their structure.

## Key findings

- Complete list of superpotentials for Type EC algebras
- Criterion for AS-regularity in Type EC algebras
- Existence of Morita equivalent Calabi-Yau AS-regular algebras

## Abstract

Let $k$ be an algebraically closed field of characteristic $0$ and $A$ a graded $k$-algebra finitely generated in degree $1$. In this paper, for $3$-dimensional quadratic AS-regular algebras except for Type EC, we give a complete list of twisted superpotentials and a complete list of superpotentials such that derivation-quotient algebras are $3$-dimensional quadratic Calabi-Yau AS-regular algebras. For an algebra $A$ of Type EC, we give a criterion when $A$ is AS-regular. As an application, for an algebra $A$ of any type, we show that there exists a Calabi-Yau AS-regular algebra $S$ such that $A$ and $S$ are graded Morita equivalent. This result tells us that, for a $3$-dimensional quadratic AS-regular algebra $A$, to study the noncommutative projective scheme for $A$ defined by Artin-Zhang is reduced to study the noncommutative projective scheme for $S$ for the Calabi-Yau AS-regular algebra $S$.