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

2025/11/04 by Itaba, Ayako, Matsuno, Masaki, Saito, Yu
Mathematics · #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Homotopy and Cohomology in Algebraic Topology #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2511.02361

openalex publication_date 2025/11/04 · openalex created_date 2025/11/06 · openalex updated_date 2026/07/28

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=A(E,σ), where E is either ℙ¹× ℙ¹ or a curve of bidegree (2,2) in ℙ¹× ℙ¹ and σ∈ Aut_kE. 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 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.

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