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.

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 $S$.

Equations

$$\mathrm{GKdim}A:=\inf \{\alpha \in \mathbb{R}\mid {\dim }_k(\sum _{i=0}^n{A}_i)\le n^{\alpha }\text{ for all }n\gg 0\}$$

$$k⟨x,y,z⟩/(f_1,f_2,f_3)\text{ (quadratic case) },\text{ or }k⟨x,y⟩/(g_1,g_2)\text{ (cubic case) }$$

$${\theta }_n(a{\theta }_l(b))={\theta }_n(a){\theta }_{n+l}(b)$$

$$a\ast b:=a{\theta }_l(b)\text{for all }a\in A_l,b\in A_m.$$

$$\mathcal{D}(w):=T(V)/({\partial }_{x_1}w,{\partial }_{x_2}w,{\partial }_{x_3}w).$$

$$(\theta \otimes \mathrm{id}\otimes \mathrm{id})\phi (w)=w,$$

$$w^{\theta }:=({\theta }^2\otimes \theta \otimes \mathrm{id})(w)$$

$$\mathrm{Aut}(w):=\{\theta \in \mathrm{GL}(V)\mid ({\theta }^{\otimes 3})(w)=\lambda w,\exists \lambda \in k\setminus \{0\}\}.$$

$$({\theta }^{\prime }\otimes \mathrm{id}\otimes \mathrm{id})(\phi (w^{\theta }))$$

$$={\lambda }^{-1}({\theta }^3\otimes {\theta }^2\otimes \theta )(w)={\lambda }^{-1}({\theta }^2\otimes \theta \otimes \mathrm{id})(\lambda w)=w^{\theta },$$

$$\mathcal{D}(w)$$

$$=k⟨x,y,z⟩/(yz-zy,zx-xz,xy-yx)=k[x,y,z].$$

$$w^{\theta }$$

$$=({\alpha }^2\beta xyz+{\beta }^2\gamma yzx+\alpha {\gamma }^{2}zxy)-(\beta {\gamma }^{2}zyx+\alpha {\beta }^{2}yxz+{\alpha }^2\gamma xzy).$$

$$\mathcal{D}(w^{\theta })$$

$$=k⟨x,y,z⟩/({\alpha }^2\beta yz-{\alpha }^2\gamma zy,{\beta }^2\gamma zx-\alpha {\beta }^{2}xz,\alpha {\gamma }^{2}xy-\beta {\gamma }^{2}yx)$$

$$=k⟨x,y,z⟩/(\beta yz-\gamma zy,\gamma zx-\alpha xz,\alpha xy-\beta yx).$$

$$\mathcal{V}(R):=\{(p,q)\in \mathbb{P}(V^{\ast })\times \mathbb{P}(V^{\ast })\mid f(p,q)=0\text{ for all }f\in R\}.$$

$$\mu :{\mathrm{H}}^0(E,\mathcal{L})\otimes {\mathrm{H}}^0(E,\mathcal{L})\to {\mathrm{H}}^0(E,\mathcal{L})\otimes {\mathrm{H}}^0(E,{\mathcal{L}}^{\sigma })\to {\mathrm{H}}^0(E,\mathcal{L}{\otimes }_{{\mathcal{O}}_E}{\mathcal{L}}^{\sigma })$$

$${\mathrm{H}}^0(E,\mathcal{L})={\mathrm{H}}^0(\mathbb{P}(V^{\ast }),{\mathcal{O}}_{\mathbb{P}(V^{\ast })}(1))=V\text{ as k-vector spaces.}$$

$$({\sigma }^2{)}^{\ast }\mathcal{L}{\otimes }_{{\mathcal{O}}_E}\mathcal{L}\cong {\sigma }^{\ast }\mathcal{L}{\otimes }_{{\mathcal{O}}_E}{\sigma }^{\ast }\mathcal{L}.$$

$$\{\begin{array}{ll}{f}_1& =x^2-y^2,\\ f_2& =xz-zy-\beta xy+(\beta +\gamma )y^2,\\ f_3& =yz-zx-\alpha yx+(\alpha +\gamma )x^2,\end{array}$$

$$\begin{array}[]{l}w=(1-\beta-\gamma)x^{3}-(\alpha+2\gamma)yx^{2}+zx^{2}-xy^{2}+\gamma y^{3}-zy^{2}\\ \quad\quad-x^{2}z+xzy+\beta x^{2}y-y^{2}z+yzx+\alpha y^{2}x\end{array}$$

$$\{\begin{array}{ll}{\partial }_{x}w& =(1-\beta -\gamma )x^2-y^2-xz+zy+\beta xy,\\ {\partial }_{y}w& =-(\alpha +2\gamma )x^2+\gamma y^2-yz+zx+\alpha yx,\\ {\partial }_{z}w& =x^2-y^2.\end{array}$$

$$\{\begin{array}{ll}& xy+yx,\\ & xz-zx-x^2,\\ & zy-yz+xy+x^2.\end{array}$$

$$\mathrm{Div}E:=\left\{\sum _{p\in E}{n}_p\cdot p|n_p\in \mathbb{Z}\right\}$$

$$\tilde{\sigma }\left(\sum _{p\in E}{n}_p\cdot p\right)=\sum _{p\in E}{n}_p\cdot {\sigma }^{-1}(p).$$

$${\sigma }^{\ast }({\mathcal{O}}_E(D))\cong {\mathcal{O}}_E(\tilde{\sigma }D).$$

$$|{\mathrm{Aut}}_k(E,p)|=\{\begin{array}{ll}2& \text{if }j(E)\ne 0,12^3,\\ 6& \text{if }j(E)=0,\\ 4& \text{if }j(E)=12^3.\end{array}$$

$$j(E)=\frac{27{\lambda }^3({\lambda }^3+8{)}^3}{({\lambda }^3-1{)}^3}.$$

$$[{\mathcal{O}}_E([n]p-o_E)]=[{\mathcal{O}}_E(n(p-o_E))].$$

$$\{\begin{array}{l}ayz+bzy+c{x}^2,\\ azx+bxz+c{y}^2,\\ axy+byx+c{z}^2.\end{array}$$

$$({\sigma }_p{\tau }^i{)}^{\ast }\mathcal{L}\ncong \mathcal{L}.$$

Full text

1. Introduction

In noncommutative algebraic geometry, an Artin-Schelter regular (AS-regular) algebra introduced by Artin-Schelter [AS] is one of the main interests. Artin-Tate-Van den Bergh [ATV] proved that there exists a one-to-one correspondence between $3$-dimensional AS-regular algebras and regular geometric pairs. This work convinced us that algebraic geometry is very useful to study even noncommutative algebras. Dubois-Violette [D] and Bocklandt-Schedler-Wemyss [BSW] showed that every $3$-dimensional quadratic AS-regular algebra $A$ is isomorphic to a derivation-quotient algebra $\mathcal{D}(w)$ of a twisted superpotential $w$, and Mori-Smith [MS1] showed that such $w$ is unique up to non-zero scalar multiples. So it is interesting to study AS-regular algebras using both algebraic geometry and twisted superpotentials. In fact, Mori-Smith [MS2] classified $3$-dimensional quadratic Calabi-Yau AS-regular algebras by using superpotentials.

In the preceding paper [IM] by the authors, in terms of geometric algebras defined by Mori [M], we determined all possible defining relations for geometric algebras whose point schemes are either $\mathbb{P}^{2}$ or cubic curves in $\mathbb{P}^{2}$ and classified them up to graded algebra isomorphism and up to graded Morita equivalence. However, in [IM], the authors did not check the AS-regularity of these classified geometric algebras. So, one of the aims of this paper is to check the AS-regularity of them. Note that, Iyudu-Shkarin [IS] recently gave a list of defining relations of $3$-dimensional AS-regular algebras by using twisted superpotentials, but no proof of AS-regularity of these algebras was given in their paper. For geometric algebras listed in [IM, Theorem 3.1], we give a list of candidates of twisted superpotentials to serve our purposes (see Proposition 3.1). By using this list, we give a complete list of superpotentials whose derivation-quotient algebras are $3$-dimensional quadratic Calabi-Yau AS-regular algebras whose point schemes are not elliptic curves (see Theorem 3.3). By using a twist of a superpotential (in the sense of [MS1]), we showed that potentials listed in Proposition 3.1 are in fact twisted superpotentials and derivation-quotient algebras of them are $3$-dimensional quadratic AS-regular algebras (see Theorems 3.4 and 3.5). For a geometric algebra $A$ whose point scheme is an elliptic curve in $\mathbb{P}^{2}$, we give a simple condition that $A$ is AS-regular (see Theorem 4.3). As an application of Corollary 3.7 and Theorem 4.3, we prove the following theorem (see Theorem 4.4).

Theorem 1.1**.**

For every $3$-dimensional quadratic AS-regular algebra $A$, there exists a Calabi-Yau AS-regular algebra $S$ such that $A$ and $S$ are graded Morita equivalent.

Theorem 1.1 tells us that, for a $3$-dimensional quadratic AS-regular algebra $A$, to study the noncommutative projective scheme ${\rm Proj}_{\rm nc}A$ of $A$ in the sense of Artin-Zhang [AZ] is reduced to study ${\rm Proj}_{\rm nc}S$ for the Calabi-Yau AS-regular algebra $S$. Note that [U, Example 14] gave one example of a $3$-dimensional cubic AS-regular algebra which is not graded Morita equivalent to any Calabi-Yau AS-regular algebra.

This paper is organized as follows: In Section 2, we recall the definition of an AS-regular algebra defined by Artin-Schelter [AS], a Calabi-Yau algebra by Ginzburg [G], a twisted superpotential and a twist of a superpotential in the sense of [MS1]. Also, we recall Zhang’s twist and twisted algebras from [Z] and some lemmas which are needed to show our Theorem 1.1. Moreover, we recall the definitions of a geometric algebra for quadratic algebras introduced by Mori [M], and the result of our preceding paper [IM]. In Section 3, we prove Theorem 1.1 for geometric algebras whose point schemes are not elliptic curves. Finally, in Section 4, we prove Theorem 1.1 for geometric algebras whose point schemes are elliptic curves in $\mathbb{P}^{2}$.

2. Preliminaries

Throughout this paper, let $k$ be an algebraically closed field of characteristic [math]. A graded $k$-algebra means an $\mathbb{N}$-graded algebra $A=\bigoplus_{i\in\mathbb{N}}A_{i}$. A connected graded $k$-algebra $A$ is a graded $k$-algebra such that $A_{0}=k$. We denote by ${\rm GrMod}\,A$ the category of graded right $A$-modules and graded right $A$-module homomorphisms and we say that two graded $k$-algebras $A$ and $B$ are graded Morita equivalent if two categories ${\rm GrMod}\,A$ and ${\rm GrMod}\,B$ are equivalent.

2.1. AS-regular algebras and Calabi-Yau algebras

Let $A$ be a connected graded $k$-algebra finitely generated by elements of positive degree. We recall that

[TABLE]

is called the Gelfand-Kirillov dimension of $A$.

Definition 2.1** ([AS, page 171]).**

A connected graded $k$-algebra $A$ is called a $d$-dimensional Artin-Schelter regular (AS-regular) algebra if $A$ satisfies the following conditions:

1. (i)

${\rm gldim}\,A=d<\infty$, 2. (ii)

${\rm GKdim}\,A<\infty$, 3. (iii)

(Gorenstein condition) {\rm Ext}_{A}^{i}(k,A)\cong\left\{\begin{array}[]{ll}k&\quad(i=d),\\ 0&\quad(i\neq d).\end{array}\right.

Any $3$-dimensional AS-regular algebra $A$ finitely generated in degree $1$ is a graded algebra isomorphic to an algebra of the form

[TABLE]

where $f_{i}$ are homogeneous polynomials of degree $2$ and $g_{i}$ are homogeneous polynomials of degree $3$ ([AS, Theorem 1.5 (i)]). In this paper, we focus on $3$-dimensional quadratic AS-regular algebras.

Let $V$ be a $3$-dimensional $k$-vector space and $T(V)$ the tensor algebra of $V$. We choose a basis $\{x_{1},x_{2},x_{3}\}$ of $V$. Also, for an algebra $T(V)/(R)$, we choose a basis $\{f_{1},f_{2},f_{3}\}$ of $R\subset V^{\otimes 2}$. We set $\bm{x}:=(x_{1},x_{2},x_{3})^{t}$ and $\bm{f}:=(f_{1},f_{2},f_{3})^{t}$, where, for a matrix $N$, $N^{t}$ means the transpose of $N$. There is a unique $3\times 3$ matrix $M$ with entries in $V$ such that $\bm{f}=M\bm{x}$ (see [AS, page 177]). From [ATV, page 34], $T(V)/(R)$ is called standard if there are bases for $V$ and $R$ such that the entries in $\bm{x}^{t}M$ are also a basis for $R$.

Theorem 2.2** ([ATV, Theorem 1]).**

Let $V$ be a $3$-dimensional $k$-vector space and $R$ a $3$-dimensional subspace of $V^{\otimes 2}$. Then $T(V)/(R)$ is a $3$-dimensional AS-regular algebra if and only if $T(V)/(R)$ is standard and the common zero locus in $\mathbb{P}^{2}$ of the $2\times 2$ minors of the matrix $M$ in the above is empty.

Here, we recall the definition of a Calabi-Yau algebra introduced by [G].

Definition 2.3** ([G, Definition 3.2.3]).**

A $k$-algebra $S$ is called $d$-dimensional Calabi-Yau if $S$ satisfies the following conditions:

1. (i)

$\mathrm{pd}_{S^{\rm e}}S=d<\infty$, 2. (ii)

$\mathrm{Ext}_{S^{\rm e}}^{i}(S,S^{\rm{e}})\cong\begin{cases}S&\text{ if }i=d,\\ 0&\text{ if }i\neq d\end{cases}$  (as right $S^{\rm e}$-modules)

where $S^{\rm e}=S^{{\rm op}}\otimes_{k}S$ is the enveloping algebra of $S$.

For example, it is known that an $n$-th polynomial ring $k[x_{1},x_{2},\ldots,x_{n}]$ is $n$-dimensional Calabi-Yau.

Note that, for a $3$-dimensional quadratic AS-regular algebra $A$, the quadratic dual $A^{!}$ of $A$ is a Frobenius algebra by [S, Proposition 5.10]. Hence, we can consider the Nakayama automorphism $\nu_{A^{!}}$ of $A^{!}$. By using the following consequence proved by Reyes-Rogalski-Zhang [RRZ], we can determine whether these algebras $A$ are Calabi-Yau algebras or not.

Lemma 2.4** ([RRZ, comments after the proof of Example 1.4]).**

*Let $A$ be a $3$-dimensional quadratic AS-regular algebra. Then $A$ is Calabi-Yau if and only if the Nakayama automorphism $\nu_{A^{!}}$ of $A^{!}$ is the identity (that is, $A^{!}$ is symmetric). *

2.2. Twisted algebras

In this subsection, we recall the notion of twisting system and twisted algebra introduced by Zhang [Z].

A set of graded $k$-linear automorphisms of $A$, say $\theta=\{\theta_{i}\mid i\in\mathbb{N}\}$, is called a twisting system of $A$ if

[TABLE]

for all $l,m,n\in\mathbb{N}$ and all $a\in A_{l},b\in A_{m}$ ([Z, Definition 2.1]). Let $\theta=\{\theta_{i}\mid i\in\mathbb{N}\}$ be a twisting system of $A$. Then, a new graded and associative multiplication $\ast$ on the underlying graded $k$-vector space $\bigoplus_{i\in\mathbb{N}}A_{i}$ is defined by

[TABLE]

We denote by $1_{\theta}$ the identity with respect to $\ast$. The graded $k$-algebra $(\bigoplus_{i\in\mathbb{N}}A_{i},\ast,1_{\theta})$ is called the twisted algebra of $A$ by $\theta$ and is denoted by $A^{\theta}$ ([Z, Definition 2.3]). Any graded algebra automorphism $\theta\in{\rm Aut}\,A$ defines a twisting system of $A$ by $\{\theta^{i}\}_{i\in\mathbb{N}}$. The twist of $A$ by this twisting system is denoted by $A^{\theta}$ instead of $A^{\{\theta^{i}\}_{i\in\mathbb{N}}}$ which is called the twist of $A$ by $\theta$.

Lemma 2.5** ([Z, Theorem 3.5]).**

Let $A$ and $A^{\prime}$ be two connected graded $k$-algebras with $A_{1}\neq 0$. Then $A^{\prime}$ is isomorphic to a twisted algebra of $A$ if and only if ${\rm GrMod}\,A$ and ${\rm GrMod}\,A^{\prime}$ are equivalent.

Lemma 2.6** ([Z, Theorem 5.11 (b)]).**

Let $A$ be a connected graded $k$-algebra and $\theta$ be a twisting system of $A$. Then $A$ is a $3$-dimensional quadratic AS-regular algebra if and only if the twist $A^{\theta}$ of $A$ by $\theta$ is also a $3$-dimensional quadratic AS-regular algebra.

2.3. Derivation-quotient algebras

Now, we recall the definitions of superpotentials, twisted superpotentials and derivation-quotient algebras from [BSW] and [MS1]. Also, we recall the definition of a twist of a superpotential due to [MS1] (see Definition 2.9).

Fix a basis $\{x_{1},x_{2},x_{3}\}$ for $V$. For $w\in V^{\otimes 3}$, there exist unique $w_{i}\in V^{\otimes 2}$ such that $w=\sum_{i=1}^{3}x_{i}\otimes w_{i}$. Then the partial derivative of $w$ with respect to $x_{i}$ ($i=1,2,3$) is $\partial_{x_{i}}(w):=w_{i},$ and the derivation-quotient algebra of $w$ is

[TABLE]

Note that we call an element $w\in V^{\otimes 3}$ a potential in this paper. We define the $k$-linear map $\varphi$: $V^{\otimes 3}\longrightarrow V^{\otimes 3}$ by $\varphi(v_{1}\otimes v_{2}\otimes v_{3}):=v_{3}\otimes v_{1}\otimes v_{2}$. We write ${\rm GL}(V)$ for the general linear group of $V$.

Definition 2.7** ([BSW, Introduction], [MS1, Definition 2.5]).**

Let $w\in V^{\otimes 3}$ be a potential.

1. (1)

If $\varphi(w)=w$,then $w$ is called a superpotential. 2. (2)

If there exists $\theta\in{\rm GL}(V)$ such that

[TABLE]

then $w$ is called a twisted superpotential.

Remark 2.8**.**

By Dubois-Violette [D] and Bocklandt-Schedler-Wemyss [BSW], every $3$-dimensional quadratic AS-regular algebra $A$ is isomorphic to a derivation-quotient algebra $\mathcal{D}(w)$ of a twisted superpotential $w$ (see [D, Theorem 5] and [BSW, Theorem 6.8]), and by Mori-Smith [MS1], such $w$ is unique up to non-zero scalar multiples (see [MS1, Proposition 2.12]).

Definition 2.9** ([MS1, page 390]).**

For a superpotential $w\in V^{\otimes 3}$ and $\theta\in{\rm GL}(V)$,

[TABLE]

is called a Mori-Smith twist (MS twist) of $w$ by $\theta$.

For a potential $w\in V^{\otimes 3}$, we set

[TABLE]

For any $w\in V^{\otimes 3}$, it follows from [MS1, Lemma 3.1] that ${\rm Aut}\,(w)\subset{\rm Aut}\,\mathcal{D}(w)$.

Lemma 2.10** ([MS1, Proposition 5.2]).**

For a superpotential $w\in V^{\otimes 3}$ and $\theta\in{\rm Aut}\,(w)$, we have that $\mathcal{D}(w^{\theta})\cong\mathcal{D}(w)^{\theta}$.

Lemma 2.11**.**

If $w\in V^{\otimes 3}$ is a superpotential and $\theta\in{\rm Aut}\,(w)$, then the MS twist $w^{\theta}$ of a superpotential $w$ by $\theta$ is a twisted superpotential.

Proof.

Let $w\in V^{\otimes 3}$ be a superpotential and $\theta\in{\rm Aut}\,(w)$. By definition, there exists $\lambda\in k\setminus\{0\}$ such that $(\theta^{\otimes 3})(w)=\lambda w$. We set $\theta^{\prime}:=\lambda^{-1}\theta^{3}\in{\rm GL}(V)$. Since $w$ is a superpotential,

[TABLE]

so the MS twist $w^{\theta}$ is a twisted superpotential. ∎

Remark 2.12**.**

If $\theta\in{\rm GL}(V)\setminus{\rm Aut}\,(w)$, then the MS twist $w^{\theta}$ of a superpotential $w\in V^{\otimes 3}$ by $\theta$ may not be a twisted superpotential. Indeed, let $w:=x^{3}\in V^{\otimes 3}$. Since $\varphi(w)=w$, we see that $w$ is a superpotential. Take \theta:=\left(\begin{array}[]{ccc}0&1&0\\ 1&0&0\\ 0&0&1\end{array}\right)\in{\rm GL}_{3}(k). Then $w^{\theta}=(\theta^{2}\otimes\theta\otimes{\rm id})(w)=xyx.$ Since, for any $\theta^{\prime}\in{\rm GL}(V)$, $(\theta^{\prime}\otimes{\rm id}\otimes{\rm id})(\varphi(w^{\theta}))=\theta^{\prime}(x)xy\neq w^{\theta}$, the MS twist $w^{\theta}$ is not a twisted superpotential. Note that we have $\theta\not\in{\rm Aut}\,(w)$ by $(\theta^{\otimes 3})(w)=y^{3}\neq w$.

Definition 2.13**.**

Let $w\in V^{\otimes 3}$ be a potential.

1. (1)

A potential $w$ is called regular if the derivation-quotient algebra $\mathcal{D}(w)$ is a $3$-dimensional quadratic AS-regular algebra. 2. (2)

A potential $w$ is called Calabi-Yau if the derivation-quotient algebra $\mathcal{D}(w)$ is a $3$-dimensional Calabi-Yau AS-regular algebra.

Remark 2.14**.**

By Bockdlandt [B], every $3$-dimensional quadratic Calabi-Yau AS-regular algebra is isomorphic to a derivation-quotient algebra $\mathcal{D}(w)$ of a superpotential $w$ ([B, Theorem 3.1]).

Lemma 2.15** ([MS1, Corollary 4.5]).**

Let $w\in V^{\otimes 3}$ be regular. Then $w$ is Calabi-Yau if and only if it is a superpotential.

Lemma 2.16**.**

If $w$ is a Calabi-Yau superpotential and $\theta\in{\rm Aut}\,(w)$, then the MS twist $w^{\theta}$ of a superpotential $w$ by $\theta$ is a regular twisted superpotential.

Proof.

Let $w\in V^{\otimes 3}$ be a Calabi-Yau superpotential and $\theta\in{\rm Aut}\,(w)$. By Lemma 2.11, the MS twist $w^{\theta}$ of $w$ by $\theta$ is a twisted superpotential, so it is sufficient to show that $w^{\theta}$ is regular. Since $w$ is Calabi-Yau, $\mathcal{D}(w)$ is Calabi-Yau AS-regular, and since $\theta\in{\rm Aut}\,(w)$, by Lemma 2.10, we have that $\mathcal{D}(w^{\theta})\cong\mathcal{D}(w)^{\theta}$. Since it holds from Lemma 2.6 that AS-regularity is presented by twisting, $\mathcal{D}(w^{\theta})$ is AS-regular, that is, the MS twist $w^{\theta}$ is regular. ∎

Example 2.17**.**

We set $w:=(xyz+yzx+zxy)-(zyx+yxz+xzy)\in V^{\otimes 3}$. Then, we see that $\varphi(w)=w$. So, $w$ is a superpotential. The derivation-quotient algebra of $w$ is

[TABLE]

It is known that a polynomial ring $k[x,y,z]$ is Calabi-Yau AS-regular. So, by Definition 2.13 (2), $w$ is a Calabi-Yau superpotential. Take \theta:=\left(\begin{array}[]{ccc}\alpha&0&0\\ 0&\beta&0\\ 0&0&\gamma\end{array}\right)\in{\rm GL}_{3}(k). Calculating the MS twist $w^{\theta}$ of the superpotential $w$ by $\theta$,

[TABLE]

Therefore, the derivation-quotient algebra of $w^{\theta}$ is as follows:

[TABLE]

Since ${\rm Aut}\,(w)={\rm GL}\,(V)$, we see that $\theta\in{\rm Aut}\,(w)$. By Lemma 2.16, $w^{\theta}$ is a regular twisted superpotential, so, $\mathcal{D}(w^{\theta})$ is an AS-regular algebra.

2.4. Geometric algebras

Let $(E,\mathcal{O}_{E})$ be a scheme where $\mathcal{O}_{E}$ is the structure sheaf on $E$. An invertible sheaf on $E$ is defined to be a locally free $\mathcal{O}_{E}$-module of rank $1$. For a quadratic algebra $A=T(V)/(R)$, we set

[TABLE]

Let $E\subset\mathbb{P}(V^{\ast})$ be a closed $k$-subscheme and $\sigma$ an automorphism of $E$. For the rest of the paper, we fix

1. (a)

$\pi:E\rightarrow\mathbb{P}(V^{\ast})$ is the embedding, 2. (b)

$\mathcal{L}:=\pi^{\ast}(\mathcal{O}_{\mathbb{P}(V^{\ast})}(1))$.

In this case, $\mathcal{L}$ becomes an invertible sheaf on $E$. The map

[TABLE]

of $k$-vector spaces is defined by $v\otimes w\mapsto v\otimes w^{\sigma}$ where $\mathcal{L}^{\sigma}=\sigma^{\ast}\mathcal{L}$ and $w^{\sigma}=w\circ\sigma$.

For a quadratic algebra, a geometric algebra was introduced by Mori [M].

Definition 2.18** ([M, Definition 4.3]).**

A quadratic algebra $A=T(V)/(R)$ is called geometric if there is a pair $(E,\sigma)$ where $E\subset\mathbb{P}(V^{\ast})$ is a closed $k$-subscheme, and $\sigma$ is a $k$-automorphism of $E$ such that

• •

(G1): $\mathcal{V}(R)=\{(p,\sigma(p))\in\mathbb{P}(V^{\ast})\times\mathbb{P}(V^{\ast})\mid p\in E\}$, and

• •

(G2): $R={\rm ker}\mu$ with the identification

[TABLE]

When $A$ satisfies the condition (G2), we write $A=\mathcal{A}(E,\sigma)$.

Let $A=T(V)/(R)$ be a quadratic algebra. If $A=\mathcal{A}(E,\sigma)$ is a geometric algebra, then $E$ is called the point scheme of $A$. If $E$ is reduced, then the condition (G2) is equivalent to the condition (G2’): $R=\{f\in V\otimes V\mid f|_{\mathcal{V}(R)}=0\}$ (see [M]).

Theorem 2.19** ([ATV, Theorem 3]).**

Let $A$ be a quadratic algebra. Then $A$ is a $3$-dimensional AS-regular algebra if and only if $A$ is isomorphic to a geometric algebra $\mathcal{A}(E,\sigma)$ which satisfies one of the following conditions:**

1. (1)

$E=\mathbb{P}^{2}$* and $\sigma\in{\rm Aut}_{k}\,\mathbb{P}^{2}$.* 2. (2)

$E$* is a cubic curve in $\mathbb{P}^{2}$ and $\sigma\in{\rm Aut}_{k}\,E$ such that $\sigma^{\ast}\mathcal{L}\not\cong\mathcal{L}$ and*

[TABLE]

The types of $(E,\sigma)$ of $3$-dimensional quadratic AS-regular algebras are defined in [MU] which are slightly modified from the original types defined in [AS] and [ATV]. We extend the types defined in [MU] as follows (see [IM, Subsection 2.3]):

(1) Type P:

$E$ is $\mathbb{P}^{2}$, and $\sigma\in\mathrm{Aut}_{k}\mathbb{P}^{2}=\mathrm{PGL}_{3}(k)$ (Type $\mathbb{P}^{2}$ is divided into Type Pi ($i=1,2,3$) in terms of the Jordan canonical form of $\sigma$).

(2-1) Type S1:

$E$ is a triangle, and $\sigma$ stabilizes each component.

(2-2) Type S2:

$E$ is a triangle, and $\sigma$ interchanges two of its components.

(2-3) Type S3:

$E$ is a triangle, and $\sigma$ circulates three components.

(3-1) Type S’1:

$E$ is a union of a line and a conic meeting at two points, and $\sigma$ stabilizes each component and two intersection points.

(3-2) Type S’2:

$E$ is a union of a line and a conic meeting at two points, and $\sigma$ stabilizes each component and interchanges two intersection points.

(4-1) Type T1:

$E$ is a union of three lines meeting at one point, and $\sigma$ stabilizes each component.

(4-2) Type T2:

$E$ is a union of three lines meeting at one point, and $\sigma$ interchanges two of its components.

(4-3) Type T3:

$E$ is a union of three lines meeting at one point, and $\sigma$ circulates three components.

(5) Type T’:

$E$ is a union of a line and a conic meeting at one point, and $\sigma$ stabilizes each component.

(6) Type CC:

$E$ is a cuspidal cubic curve.

(7) Type NC:

$E$ is a nodal cubic curve (Type NC is divided into Type NCi $(i=1,2)$).

(8) Type WL:

$E$ is a union of a double line and a line (Type WL is divided into Type WLi $(i=1,2,3)$).

(9) Type TL:

$E$ is a triple line (Type TL is divided into Type TLi $(i=1,2,3,4)$).

(10) Type EC:

$E$ is an elliptic curve.

Remark 2.20**.**

All possible defining relations of $3$-dimensional quadratic AS-regular algebras are listed in each type up to isomorphism from (1) through (9) in [IM, Theorem 3.1], and (10) in [IM, Theorem 4.9].

3. Classifications of twisted superpotentials

In this section, we will give complete lists of superpotentials and twisted superpotentials whose derivation-quotient algebras are $3$-dimensional quadratic AS-regular algebras except for Type EC, by using the following three steps:

**Step I: **

(Proposition 3.1) Find the candidates of regular twisted superpotentials corresponding to defining relations listed in [IM, Theorem 3.1].

**Step II: **

(Theorem 3.3) Find all superpotentials among the above candidates and show that they are Calabi-Yau superpotentials.

**Step III: **

(Theorem 3.4, Theorem 3.5) Show that all above candidates can be written as MS twists of Calabi-Yau superpotentials and that they are in fact regular twisted superpotentials.

As a byproduct, we will prove that, for any $3$-dimensional quadratic AS-regular algebra $A$ except for Type EC, there exist a Calabi-Yau AS-regular algebra $S$ and $\theta\in{\rm Aut}\,S$ such that $A$ is isomorphic to $S^{\theta}$ as graded $k$-algebras. This result is needed to prove our main result Theorem 1.1.

Proposition 3.1**.**

Every $3$-dimensional quadratic AS-regular algebra except for Type EC is isomorphic to $\mathcal{D}(w)$ of a potential $w$ in Table $1$.

[TABLE]

[TABLE]

[TABLE]

Proof.

All possible defining relations $f_{1},f_{2},f_{3}$ of $3$-dimensional quadratic AS-regular algebras except for Type EC were given in [IM, Theorem 3.1]. In each type, it is enough to find $w$ such that $(\partial_{x}{w},\partial_{y}{w},\partial_{z}{w})=(f_{1},f_{2},f_{3})$.

We will give a proof for Type T2 algebras. For the other types, the proofs are similar. From [IM, Theorem 3.1], Type T2 algebras are given as $A=k\langle x,y,z\rangle/(f_{1},f_{2},f_{3})$:

[TABLE]

where $\alpha+\beta+\gamma\neq 0$.

Taking a potential

[TABLE]

as in Table 1, we have

[TABLE]

Since $\partial_{z}w=x^{2}-y^{2}=f_{1}$, $\partial_{x}w=(1-\beta-\gamma)f_{1}-f_{2}$ and $\partial_{y}w=-\gamma f_{1}-f_{3}$, it follows that $\mathcal{D}(w)=k\langle x,y,z\rangle/(\partial_{x}w,\partial_{y}w,\partial_{z}w)=k\langle x,y,z\rangle/(f_{1},f_{2},f_{3})=A$. ∎

Remark 3.2**.**

(1) For an algebra $A=k\langle x,y,z\rangle/(f_{1},f_{2},f_{3})$ of any type, we can take a potential $w=xf_{1}+yf_{2}+zf_{3}$ such that $A=\mathcal{D}(w)$. But it is difficult to check that this potential $w$ is a regular twisted superpotential and, in many cases, it is not so. The potentials $w$ listed in Proposition 3.1 were chosen so that they are candidates of regular twisted superpotentials. By [MS1, Theorem 4.4], every regular twisted superpotential $w$ satisfies $(\nu^{-1}\otimes\mathrm{id}\otimes\mathrm{id})(\varphi(w))=w$ where $\nu$ is the Nakayama automorphism of $A^{!}$, so in the above proposition, we take a potential $w$ such that $((\nu_{A^{!}})^{-1}\otimes\mathrm{id}\otimes\mathrm{id})(\varphi(w))=w$ where $\nu_{A^{!}}$ is the Nakayama automorphism of $A^{!}$ listed in Table $1$.

(2) For Type TL4 in [IM, Theorem 3.1], we gave the defining relations

[TABLE]

The relation $xy+yx$ is a typo, and the relation $xy-yx$ is correct, which is given in Table 1 as above.

Next, we give a complete list of Calabi-Yau superpotentials as follows:

Theorem 3.3**.**

For every type except for Type EC, the following table is a complete list of Calabi-Yau superpotentials $w_{0}$.

[TABLE]

Proof.

Let $A$ be a $3$-dimensional quadratic AS-regular algebra except for Type EC. By Lemma 2.4, $A$ is Calabi-Yau if and only if the Nakayama automorphism $\nu_{A^{!}}$ is the identity, where $A^{!}$ is the quadratic dual of $A$. Considering the condition that $\nu_{A^{!}}$ in Table 1 is the identity, we have a superpotential, so it is sufficient to show that $w_{0}$ is regular, that is, $\mathcal{D}(w_{0})=k\langle x,y,z\rangle/(\partial_{x}w_{0},\partial_{y}w_{0},\partial_{z}w_{0})$ is a $3$-dimensional quadratic AS-regular algebra. In fact, if $w$ is regular, then by Lemma 2.15, it is Calabi-Yau. In order to prove AS-regularity of $\mathcal{D}(w_{0})$, we will check that $\mathcal{D}(w_{0})$ satisfies the conditions of Theorem 2.2. Note that if $w$ is a superpotential, then the derivation-quotient algebra $\mathcal{D}(w)$ is standard if and only if the partial derivatives $\partial_{x}x$, $\partial_{y}w$, $\partial_{z}w$ are linearly independent (for example, see [MS2, Proposition 2.6]).

We will give a proof for Type T1 algebras. For the other types, the proofs are similar. Let $w_{0}=xyz+yzx+zxy-(xzy+zyx+yxz)+(x^{2}y+xyx+yx^{2})-(y^{2}x+yxy+xy^{2})$. It is easy to check that $\partial_{x}w_{0},\partial_{y}w_{0},\partial_{z}w_{0}$ are linearly independent. For the potential $w_{0}$, we have the unique $3\times 3$ matrix

M:=\left(\begin{array}[]{ccc}y&x-y-z&y\\ x-y+z&-x&-x\\ -y&x&0\end{array}\right) such that \left(\begin{array}[]{c}\partial_{x}w_{0}\\ \partial_{y}w_{0}\\ \partial_{z}w_{0}\end{array}\right)=M\left(\begin{array}[]{c}x\\ y\\ z\end{array}\right).

By calculation, we have \left(\begin{array}[]{ccc}x&y&z\end{array}\right)M=\left(\begin{array}[]{ccc}\partial_{x}w_{0}&\partial_{y}w_{0}&\partial_{z}w_{0}\end{array}\right). Hence, $\mathcal{D}(w_{0})$ is standard. We denote by $\Delta_{ij}$ the $(i,j)$-th $2\times 2$ minors of the matrix $M$ ($1\leq i,j\leq 3$). Since $\Delta_{11}=-x^{2}$, $\Delta_{22}=y^{2}$ and $\Delta_{33}=-x^{2}+xy-y^{2}+z^{2}$, we have that $\mathcal{V}(\{\Delta_{ij}\mid 1\leq i,j\leq 3\})=\emptyset$.

Therefore, by Theorem 2.2, $\mathcal{D}(w_{0})$ is a $3$-dimensional quadratic AS-regular algebra, that is, $w_{0}$ is a Calabi-Yau superpotential. ∎

Theorem 3.4**.**

For a potential $w$ in Table $1$, there exist a Calabi-Yau superpotential $w_{0}$ in Table $2$ and $\theta\in{\rm Aut}\,(w_{0})$ such that $\mathcal{D}(w)\cong\mathcal{D}((w_{0})^{\theta})$ as in Table $3$.

[TABLE]

[TABLE]

[TABLE]

Proof.

By direct computation, for a potential $w$ in Table 1, we find a Calabi-Yau superpotential $w_{0}$ in Table 2 and $\theta\in{\rm Aut}\,(w_{0})$ such that $\mathcal{D}(w)\cong\mathcal{D}((w_{0})^{\theta})$ as in Table 3.

We will give a proof for Type T1 algebra. For the other types, the proofs are similar. Let $w$ be a potential of Type T1 in Table $1$. We take a superpotential $w_{0}=xyz+yzx+zxy-(xzy+yxz+zyx)+(x^{2}y+xyx+yx^{2})-(xy^{2}+yxy+y^{2}x)$ and \theta=\left(\begin{array}[]{ccc}1&0&0\\ 0&1&0\\ \lambda\nu^{-1}&\mu\nu^{-1}&1\end{array}\right) where $\lambda:=\frac{1}{3}(-\alpha+2\beta-\gamma)$, $\mu:=\frac{1}{3}(2\alpha-\beta-\gamma)$ and $\nu:=\frac{1}{3}(\alpha+\beta+\gamma)$. Since $(\theta^{\otimes 3})(w_{0})=w_{0}$, $\theta\in{\rm Aut}\,(w_{0})$. By calculation, we have that $(w_{0})^{\theta}=(xyz+yzx+zxy)-(xzy+yxz+zyx)+\nu^{-1}(\beta x^{2}y+(\alpha-\beta+\gamma)xyx+\beta yx^{2})-\nu^{-1}(\alpha y^{2}x+(-\alpha+\beta+\gamma)yxy+\alpha xy^{2})$. By taking \theta^{\prime}=\left(\begin{array}[]{ccc}1&0&0\\ 0&1&0\\ 0&0&\nu^{-1}\end{array}\right)\in{\rm GL}_{3}(k), it follows that $(\theta^{\prime\otimes^{3}})((w_{0})^{\theta})=\nu^{-1}w$, so $\mathcal{D}((w_{0})^{\theta})\cong\mathcal{D}(w)$. ∎

We remark how we find $\theta$ in Table $3$. We take the third root of the matrix $(\nu_{A^{!}})^{-1}$, where $\nu_{A^{!}}$ is in Table $1$. Comparering the third root of the matrix $(\nu_{A^{!}})^{-1}$, we decide $\theta$ in Table $3$ (see Lemma 2.11 and Remark 3.2 (1)).

By Lemma 2.11, Lemma 2.16, Proposition 3.1, Theorem 3.3 and Theorem 3.4, the following theorem immediately holds.

Theorem 3.5**.**

*Any potential $w$ in Table $1$ is a regular twisted superpotential. *

Remark 3.6**.**

It turns out from Theorem 3.5 that the defining relations listed in [IM, Theorem 3.1] are in fact those of $3$-dimensional quadratic AS-regular algebras (see also Remark 2.20).

By Theorem 3.4, for a $3$-dimensional AS-regular algebra $A$ except for Type EC, there exist a Calabi-Yau superpotential $w_{0}$ and $\theta\in{\rm Aut}(w_{0})$ in Table $3$ such that $A\cong\mathcal{D}((w_{0})^{\theta})$. Since $\theta\in{\rm Aut}(w_{0})$, we have that $\mathcal{D}((w_{0})^{\theta})\cong\mathcal{D}(w_{0})^{\theta}$ by Lemma 2.10. Since $\mathcal{D}(w_{0})$ is Calabi-Yau AS-regular, we have the following corollary:

Corollary 3.7**.**

For a $3$-dimensional quadratic AS-regular algebra $A$ except for Type EC, there exist a Calabi-Yau AS-regular algebra $S$ and $\theta\in{\rm Aut}\,S$ such that $A$ is isomorphic to $S^{\theta}$ as graded $k$-algebras.

4. Geometric algebras of Type EC

We say that a geometric algebra $A=\mathcal{A}(E,\sigma)$ is of Type EC if $E$ is an elliptic curve in $\mathbb{P}^{2}$. In this section, we give a criterion when a geometric algebra of Type EC is a $3$-dimensional quadratic AS-regular algebra.

4.1. Divisors on curves and Hesse forms

Let $E$ be a projective smooth curve over $k$. The Picard group of $E$, denoted by $({\rm Pic}\,E,[\mathcal{O}_{E}],\otimes)$, is the group of isomorphism classes of invertible sheaves on $E$ under the operation $\otimes$ (see [H, page 143]).

A divisor on $E$ is an element of the free abelian group

[TABLE]

where only finitely many $n_{p}$ are different from zero. We write the group of divisors $({\rm Div}\,E,0,+)$ where [math] is the zero divisor, that is, $n_{p}=0$ for any $p\in E$. For any divisor $D\in{\rm Div}\,E$, there exists an invertible sheaf on $E$, denoted by $\mathcal{O}_{E}(D)$, and the map $D\mapsto\mathcal{O}_{E}(D)$ gives a surjective homomorphism from $({\rm Div}\,E,0,+)$ to $({\rm Pic}\,E,[\mathcal{O}_{E}],\otimes)$ (see [H, Proposition II 6.13] and [H, Corollary II 6.16]), that is, for any $[\mathcal{M}]\in{\rm Pic}\,E$ there exists a divisor $D\in{\rm Div}\,E$ such that $[\mathcal{M}]=[\mathcal{O}_{E}(D)]$. The zero divisor [math] maps to the isomorphism class $[\mathcal{O}_{E}]\in{\rm Pic}\,E$.

For $\sigma\in{\rm Aut}_{k}\,E$, we define a map $\tilde{\sigma}:\operatorname{Div}E\rightarrow\operatorname{Div}E$ by

[TABLE]

This map $\tilde{\sigma}$ is a group automorphism of $(\operatorname{Div}E,0,+)$. On the other hand, for $\sigma\in{\rm Aut}_{k}\,E$, the rule $\mathcal{M}\mapsto\sigma^{\ast}\mathcal{M}$ where $\mathcal{M}$ is an invertible sheaf on $E$ induces a group automorphism of the Picard group $\sigma^{\ast}:\operatorname{Pic}E\rightarrow\operatorname{Pic}E$. It follows from [H, II Ex. 6.8] that, if $D\in\operatorname{Div}E$, then

[TABLE]

Let $E$ be an elliptic curve in $\mathbb{P}^{2}$. It is well-known that the $j$-invariant $j(E)$ classifies elliptic curves up to isomorphism, that is, two elliptic curves $E$ and $E^{\prime}$ in $\mathbb{P}^{2}$ are isomorphic if and only if $j(E)=j(E^{\prime})$ (see [H, Theorem IV 4.1(b)]). For $p\in E$, we define ${\rm Aut}_{k}\,(E,p):=\{\sigma\in{\rm Aut}_{k}\,E\mid\sigma(p)=p\}$. It follows from [H, Corollary IV 4.7] that, for every point $p\in E$, ${\rm Aut}_{k}\,(E,p)$ becomes a cyclic group of order

[TABLE]

For each point $o\in E$, we can define an addition $\oplus$ on $E$ so that $(E,o,\oplus)$ is an abelian group with the zero element $o$ and, for $p\in E$, the map $\sigma_{p}$ defined by $\sigma_{p}(q):=p\oplus q$ is a scheme automorphism of $E$, called the translation by a point $p$.

4.2. Type EC

Throughout this subsection, for an elliptic curve $E$ in $\mathbb{P}^{2}$, we use a Hesse form $E=\mathcal{V}(x^{3}+y^{3}+z^{3}-3\lambda xyz)$ where $\lambda\in k$ with $\lambda^{3}\neq 1$. The $j$-invariant of a Hesse form is given by the following formula (see [F, Proposition 2.16]):

[TABLE]

We fix the group structure on $E$ with the zero element $o_{E}:=(1:-1:0)\in E$. Every automorphism $\sigma\in{\rm Aut}_{k}\,E$ can be written as $\sigma=\sigma_{p}\tau^{i}$ where $p\in E$, $\tau$ is a generator of ${\rm Aut}_{k}\,(E,o_{E})$ and $i\in\mathbb{Z}_{|\tau|}$ ([IM, Proposition 4.5 and Theorem 4.6]).

We call a point $p\in E$ $3$-torsion if $p\oplus p\oplus p=o_{E}$. We set $E[3]:=\{p\in E\mid p\oplus p\oplus p=o_{E}\}$. For $p\in E$ and $i\in\mathbb{Z}$, $A=\mathcal{A}(E,\sigma_{p}\tau^{i})$ is of Type EC if and only if $p\in E\setminus E[3]$ ([IM, Lemma 4.14]).

The map $p\mapsto[\mathcal{O}_{E}(p-o_{E})]$ is an injective homomorphism from $(E,o_{E},\oplus)$ to $(\operatorname{Pic}E,[\mathcal{O}_{E}],\otimes)$ (see [H, Example IV 1.3.7]). For $p\in E$ and $n\in\mathbb{Z}$, we use a notation $[n]p:=\underbrace{p\oplus\cdots\oplus p}_{n}$. It is easy to check the following lemma.

Lemma 4.1**.**

Let $(E,o_{E},\oplus)$ be an elliptic curve in $\mathbb{P}^{2}$, $p\in E$ and $n\in\mathbb{Z}$. Then

[TABLE]

Since the zero element $o_{E}=(1:-1:0)$ is an inflection point of $E$, it follows that $\mathcal{L}\cong\mathcal{O}_{E}(3o_{E})$ where $\mathcal{L}=\pi^{\ast}(\mathcal{O}_{\mathbb{P}^{2}}(1))$.

Lemma 4.2** (cf. [M, Lemma 4.5]).**

Let $\pi:E\rightarrow\mathbb{P}^{2}$ be the embedding and $\mathcal{L}=\pi^{\ast}\mathcal{O}_{\mathbb{P}^{2}}(1)$. Then an automorphism $\sigma\in{\rm Aut}_{k}E$ can be extended to an automorphism of $\mathbb{P}^{2}$ if and only if $\sigma^{\ast}\mathcal{L}\cong\mathcal{L}$.

A $3$-dimensional Sklyanin algebra is defined by $\mathcal{A}(E,\sigma_{p})$ where $p=(a:b:c)\in E\setminus E[3]$ and the defining relations are given as follows:

[TABLE]

A $3$-dimensional Sklyanin algebra $\mathcal{A}(E,\sigma_{p})$ is a $3$-dimensional quadratic AS-regular algebra by [ATV, Section 1]. It follows from [IM, Theorem 4.12 (1)] that $\sigma_{p}\tau^{i}$ is not extended to an automorphism of $\mathbb{P}^{2}$, so, by Lemma 4.2,

[TABLE]

For a geometric algebra $A=\mathcal{A}(E,\sigma)$ of Type EC, we give a criterion when $A$ is AS-regular.

Theorem 4.3**.**

Let $A=\mathcal{A}(E,\sigma)$ be a geometric algebra of Type EC where $\sigma=\sigma_{p}\tau^{i}$, $p\in E\backslash E[3]$ and $i\in\mathbb{Z}_{|\tau|}$. Then the following are equivalent:**

1. (1)

$A$* is a $3$-dimensional quadratic AS-regular algebra.* 2. (2)

$p\ominus\tau^{i}(p)\in E[3]$. 3. (3)

$A$* is graded Morita equivalent to a $3$-dimensional Sklyanin algebra $\mathcal{A}(E,\sigma_{p})$.*

Proof.

(1) $\Rightarrow$ (2): Assume that $A=\mathcal{A}(E,\sigma)$ is a $3$-dimensional quadratic AS-regular algebra. By Theorem 2.19, it holds that

[TABLE]

Since $\mathcal{L}\cong\mathcal{O}_{E}(3o_{E})$,

[TABLE]

where $q:=\sigma^{-1}(o_{E})=\ominus\tau^{-i}(p)$ and $r:=\sigma^{-2}(o_{E})=\sigma^{-1}(q)=q\oplus\tau^{-i}(q)$, so

[TABLE]

Therefore,

[TABLE]

Since $q=\ominus\tau^{-i}(p)$, $q\ominus\tau^{-i}(q)=\ominus\tau^{-i}(p)\oplus\tau^{-2i}(p)=\tau^{-2i}(p\ominus\tau^{i}(p))$. Hence we have $p\ominus\tau^{i}(p)\in E[3]$.

(2) $\Rightarrow$ (3): Assume that $p\ominus\tau^{i}(p)\in E[3]$. By [IM, Theorem 4.20], $A=\mathcal{A}(E,\sigma_{p}\tau^{i})$ and $\mathcal{A}(E,\sigma_{p})$ are graded Morita equivalent.

(3) $\Rightarrow$ (1): Assume that $A=\mathcal{A}(E,\sigma_{p}\tau^{i})$ is graded Morita equivalent to a $3$-dimensional Sklyanin algebra $\mathcal{A}(E,\sigma_{p})$. By Lemma 2.5, $A$ is isomorphic to a twisted algebra of $\mathcal{A}(E,\sigma_{p})$. Since being AS-regular is invariant under twisting system by Lemma 2.6, a twisted algebra of $\mathcal{A}(E,\sigma_{p})$ is a $3$-dimensional quadratic AS-regular algebra. Therefore, $A=\mathcal{A}(E,\sigma_{p}\tau^{i})$ is a $3$-dimensional quadratic AS-regular algebra. ∎

Now, we are ready to prove Theorem 1.1 in the Introduction.

Theorem 4.4**.**

For every $3$-dimensional quadratic AS-regular algebra $A$, there exists a Calabi-Yau AS-regular algebra $S$ such that $A$ and $S$ are graded Morita equivalent.

Proof.

Except for Type EC: By Lemma 2.5 and Corollary 3.7, the statement holds.

For Type EC: Let $A=\mathcal{A}(E,\sigma_{p})=\mathcal{D}(w)$ be a $3$-dimensional Sklyanin algebra where $p=(a:b:c)\in E\setminus E[3]$ and $w=a(xyz+yzx+zxy)+b(xzy+zyx+yxz)+c(x^{3}+y^{3}+z^{3})$. Since $w$ is a superpotential, by 2.15, $\mathcal{D}(w)$ is Calabi-Yau AS-regular. By Theorem 4.3, the statement holds. ∎

In Section 3, Corollary 3.7 tells us that, for a $3$-dimensional AS-regular algebra except for Type EC, there exist a Calabi-Yau AS-regular algebra $S$ and $\theta\in{\rm Aut}\,S$ such that $A$ is isomorphic to $S^{\theta}$ as graded $k$-algebras. We prove this by using Theorem 3.4, that is, for a potential $w$ in Theorem 3.1, there exist a superpotential $w_{0}$ in Theorem 3.3 and $\theta\in{\rm Aut}\,(w_{0})$ such that $\mathcal{D}(w)\cong\mathcal{D}((w_{0})^{\theta})$. On the other hand, it follows from [IM, Theorem 4.9] that, for a potential $w$ of a geometric algebra of Type EC, there exist a Calabi-Yau superpotential $w_{0}$ and $\theta\in{\rm GL}\,(V)$ induced by $\tau^{i}\in{\rm Aut}_{k}\,E$ such that $w=(w_{0})^{\theta}$. But this $\theta$ is not necessary in ${\rm Aut}\,(w_{0})$, so $w=(w_{0})^{\theta}$ may not be a twisted superpotential nor regular (see Example 4.6). This means that we do not know whether Theorem 3.4 holds or not for Type EC. So, we need to divide the proof of Theorem 1.1 into two cases of non Type EC and Type EC.

Example 4.5**.**

Let $E=\mathcal{V}(x^{3}+y^{3}+z^{3}-3\lambda xyz)$ be an elliptic curve in $\mathbb{P}^{2}$ with $j(E)\neq 0,12^{3}$ and $A=\mathcal{A}(E,\sigma_{p}\tau)$ where $p\in E\setminus E[3]$. In this case, we have that $\tau(p)=\ominus p$. By Theorem 4.3, $A$ is a $3$-dimensional quadratic AS-regular algebra if and only if $[2]p\in E[3]$, that is, $p\in E[6]$ where $E[6]:=\{q\in E\mid[6]q=o_{E}\}$.

Example 4.6**.**

In general, it is not true that if $w$ is a regular superpotential and $\theta\in{\rm GL}(V)$, then the MS twist $w^{\theta}$ is regular. Let $E=\mathcal{V}(x^{3}+y^{3}+z^{3}-3\lambda xyz)$ be an elliptic curve with $j(E)\neq 0,12^{3}$ and $A=\mathcal{A}(E,\sigma_{p}\tau)$ where $p=(a:b:c)\in E\setminus E[6]$. By Theorem 4.3, $A$ is not AS-regular. By [IM, Theorem 4.6] and [IM, Theorem 4.9], we have that $A=\mathcal{A}(E,\sigma_{p}\tau)=\mathcal{D}(w^{\theta})$ where $w=a(xyz+yzx+zxy)+b(xzy+yxz+zyx)+c(x^{3}+y^{3}+z^{3})$ and \theta=\left(\begin{array}[]{ccc}0&1&0\\ 1&0&0\\ 0&0&1\end{array}\right)\in{\rm GL}_{3}(k). Note that if $a\neq b$, then $\theta\notin{\rm Aut}\,(w)$. Since $A$ is not a $3$-dimensional quadratic AS-regular algebra, $w^{\theta}$ is not regular.

Acknowledgements.

The authors thank the referee for helpful comments in improving the paper. They are also grateful to Professor Izuru Mori for his support and helpful discussions. Moreover, they appreciate Shinichi Hasegawa and Kosuke Shima for their helping to build and to check Table 1 in Proposition 3.1. For Remark 3.2 (2), the authors thank Professor Andrew Conner and Professor Peter Goetz for telling them a typo for the relation of Type TL4. The first author was supported by Grants-in-Aid for Young Scientific Research 18K13397 Japan Society for the Promotion of Science.