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Classifications of 3-dimensional cubic AS-regular algebras whose point schemes are not integral

Ayako Itaba, Masaki Matsuno, Yu Saito

TL;DR

This paper classifies 3-dimensional cubic AS-regular algebras whose point schemes are not integral by leveraging the Artin–Tate–Van den Bergh geometric correspondence. It focuses on geometric algebras ${\mathcal A}(E,\sigma)$ where $E$ is a non-integral curve of Type WL/TWL (and related S$'$, T$'$, FL configurations) inside $\mathbb{P}^1\times\mathbb{P}^1$, derives explicit defining relations via a six-step program, and verifies AS-regularity through twisted superpotentials. The authors provide complete lists of defining relations for the non-integral cases and classify these algebras up to isomorphism and graded Morita equivalence, relying on the geometric data $(E,\sigma)$ and MS-twists. By integrating previous results (MaS) with their analysis, the work delivers a full classification in this non-integral setting, clarifying how geometric degeneracies influence algebraic structure and equivalence classes. The findings advance the program of compiling comprehensive catalogs of noncommutative projective surfaces in dimension three and illuminate the role of point schemes in shaping graded equivalences of AS-regular algebras.

Abstract

In noncommutative algebraic geometry, Artin--Tate--Van den Bergh showed that a $3$-dimensional cubic AS-regular algebra $A$ is {\it geometric}. So, we can write $A=\mathcal{A}(E,σ)$ where $E$ is $\mathbb{P}^{1}\times \mathbb{P}^{1}$ or curves of bidegree (2,2) in $\mathbb{P}^{1}\times \mathbb{P}^{1}$, and $σ\in \mathrm{Aut}_{k}E$. In this paper, for each case that $E$ is either (i) a conic and two lines in a triangle, (ii) a conic and two lines intersecting in one point, or (iii) quadrangle, we give the complete list of defining relations of $A$ and classify them up to graded algebra isomorphisms and graded Morita equivalences in terms of their defining relations. By the results of the second and third authors and our result in this paper, we give classifications of $3$-dimensional cubic AS-regular algebra whose point schemes are not integral.

Classifications of 3-dimensional cubic AS-regular algebras whose point schemes are not integral

TL;DR

This paper classifies 3-dimensional cubic AS-regular algebras whose point schemes are not integral by leveraging the Artin–Tate–Van den Bergh geometric correspondence. It focuses on geometric algebras where is a non-integral curve of Type WL/TWL (and related S, T, FL configurations) inside , derives explicit defining relations via a six-step program, and verifies AS-regularity through twisted superpotentials. The authors provide complete lists of defining relations for the non-integral cases and classify these algebras up to isomorphism and graded Morita equivalence, relying on the geometric data and MS-twists. By integrating previous results (MaS) with their analysis, the work delivers a full classification in this non-integral setting, clarifying how geometric degeneracies influence algebraic structure and equivalence classes. The findings advance the program of compiling comprehensive catalogs of noncommutative projective surfaces in dimension three and illuminate the role of point schemes in shaping graded equivalences of AS-regular algebras.

Abstract

In noncommutative algebraic geometry, Artin--Tate--Van den Bergh showed that a -dimensional cubic AS-regular algebra is {\it geometric}. So, we can write where is or curves of bidegree (2,2) in , and . In this paper, for each case that is either (i) a conic and two lines in a triangle, (ii) a conic and two lines intersecting in one point, or (iii) quadrangle, we give the complete list of defining relations of and classify them up to graded algebra isomorphisms and graded Morita equivalences in terms of their defining relations. By the results of the second and third authors and our result in this paper, we give classifications of -dimensional cubic AS-regular algebra whose point schemes are not integral.

Paper Structure

This paper contains 14 sections, 19 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. AS-regular algebras and twisted superpotentials
  4. Geomertic algebras for cubic algebras
  5. Type WL and TWL
  6. Defining relations of Type S$'$, T$'$ and FL
  7. Step 1: Classify $E$ up to equivalence and $2$-equivalence
  8. Step 2: Find all automorphisms $\sigma \in \operatorname{Aut}_k^G E$
  9. Step 3: Find the defining relations of ${\mathcal{A}}(E,\sigma)$ for each $\sigma \in \operatorname{Aut}_k^G E$
  10. Step 4: Check AS-regularity of ${\mathcal{A}}(E,\sigma)$ via finding twisted superpotentials.
  11. Classifications of $3$-dimensional cubic AS-regular algebras whose point schemes are not integral
  12. Step 5: Classify them up to isomorphisms of graded algebras in terms of their defining relations
  13. Step 6: Classify them up to graded Morita equivalences in terms of their defining relations
  14. Summary

Key Result

Lemma 2.3

Let $V$ be a $2$-dimensional vector space with a basis $\{x_1, x_2\}$ and $\omega \in V^{\otimes 4} \setminus \{0\}$. Then $\omega$ is a twisted superpotential if and only if $(\partial_{x_1} \omega, \partial_{x_2} \omega)=(\omega \partial_{x_1}, \omega \partial_{x_2})$ as homogeneous two-sided idea

Theorems & Definitions (40)

  • Definition 2.1: AS
  • Definition 2.2: BSW, MoS1
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Definition 2.5: MoS1, cf. IM2
  • Lemma 2.6: MaS
  • Theorem 2.7: MoS1
  • Theorem 2.8: ATV1
  • Proposition 2.9
  • ...and 30 more