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 with non-integral point schemes, detailing their relations, isomorphisms, and superpotential realizations.

Contribution

It provides a comprehensive classification of these algebras for specific geometric configurations, completing the broader classification effort.

Findings

All defining relations for the considered algebras are listed.

Algebras are classified up to graded isomorphism and Morita equivalence.

Explicit superpotentials are constructed to realize these algebras.

Abstract

By the result of Artin--Tate--Van den Bergh, every $3$-dimensional cubic AS-regular algebra A can be expressed as a geometric algebra $A=\mathcal{A}(E,\sigma)$, where $E$ is either $\mathbb{P}^{1}\times \mathbb{P}^{1}$ or a curve of bidegree ($2$,$2$) in $\mathbb{P}^{1}\times \mathbb{P}^{1}$ and $\sigma\in \mathrm{Aut}_{k}E$. In particular, we treat the following three configurations: (1) a conic and two lines in a triangle, (2) a conic and two lines intersecting in one point, and (3) a quadrangle. For each of these cases, we (i) list all defining relations of the corresponding algebras $\mathcal{A}(E,\sigma)$, and (ii) classify them up to graded algebra isomorphism and graded Morita equivalence. Furthermore, we present explicit (twisted) superpotentials whose derivation-quotient algebras realize these algebras and verify that the resulting algebras are AS-regular. Combining our results with existing classifications for the remaining types (including Types P, S, T, WL, and TWL), we thereby complete the classification of 3-dimensional cubic AS-regular algebras whose point schemes are not integral.