Classifications of 3-dimensional cubic AS-regular algebras whose point schemes are not integral
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917367599071232 |
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| author | Itaba, Ayako Matsuno, Masaki Saito, Yu |
| author_facet | Itaba, Ayako Matsuno, Masaki Saito, Yu |
| contents | By the result of Artin--Tate--Van den Bergh, every $3$-dimensional cubic AS-regular algebra A can be expressed as a geometric algebra $A=\mathcal{A}(E,σ)$, 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 $σ\in \mathrm{Aut}_{k}E$. 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 $\mathcal{A}(E,σ)$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_02361 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Classifications of 3-dimensional cubic AS-regular algebras whose point schemes are not integral Itaba, Ayako Matsuno, Masaki Saito, Yu Rings and Algebras By the result of Artin--Tate--Van den Bergh, every $3$-dimensional cubic AS-regular algebra A can be expressed as a geometric algebra $A=\mathcal{A}(E,σ)$, 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 $σ\in \mathrm{Aut}_{k}E$. 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 $\mathcal{A}(E,σ)$, 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. |
| title | Classifications of 3-dimensional cubic AS-regular algebras whose point schemes are not integral |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2511.02361 |