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arXiv · 2309.11112

Defining relations of 3-dimensional cubic AS-regular algebras of Type P, S and T

Masaki Matsuno·Yu Saito
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Abstract

Classification of AS-regular algebras is one of the most important projects in noncommutative algebraic geometry. Recently, Itaba and the first author gave a complete list of defining relations of $3$-dimensional quadratic AS-regular algebras by using the notion of geometric algebra and twisted superpotential. In this paper, we extend the notion of geometric algebra to cubic algebras, and give a geometric condition for isomorphism and graded Morita equivalence. One of the main results is a complete list of defining relations of $3$-dimensional cubic AS-regular algebras corresponding to $\mathbb{P}^1 \times \mathbb{P}^1$ or a union of irreducible divisors of bidegree $(1,1)$ in $\mathbb{P}^1 \times \mathbb{P}^1$. Moreover, we classify them up to isomorphism and up to graded Morita equivalence in terms of their defining relations.

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Masaki Matsuno, Yu Saito. 2023-09-20. Defining relations of 3-dimensional cubic AS-regular algebras of Type P, S and T. https://arxiv.org/abs/2309.11112

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