Quantum projective planes and Beilinson algebras of 33-dimensional quantum polynomial algebras for Type S'

Let $A=\mathcal{A}(E,\sigma)$ be a $3$-dimensional quantum polynomial algebra where $E$ is $\mathbb{P}^{2}$ or a cubic divisor in $\mathbb{P}^{2}$, and $\sigma\in \mathrm{Aut}_{k}E$. Artin-Tate-Van den Bergh proved that $A$ is finite over its center if and only if the order $|\sigma|$ of $\sigma$ is finite. As a categorical analogy of their result, the author and Mori showed that the following conditions are equivalent; (1) $|\nu^{\ast}\sigma^{3}|<\infty$, where $\nu$ is the Nakayama automorphism of $A$. (2) The norm $\|\sigma\|$ of $\sigma$ is finite. (3) The quantum projective plane $\mathsf{Proj}_{{\rm nc}}A$ is finite over its center. In this paper, we will prove for Type S' algebra $A$ that the following conditions are equivalent; (1) $\mathsf{Proj}_{{\rm nc}}A$ is finite over its center. (2) The Beilinson algebra $\nabla A$ of $A$ is $2$-representation tame. (3) The isomorphism classes of simple $2$-regular modules over $\nabla A$ are parametrized by $\mathbb{P}^{2}$.Comment: 18 pages. arXiv admin note: substantial text overlap with arXiv:2010.1309

Defining relations of 3-dimensional quadratic AS-regular algebras

Classification of AS-regular algebras is one of the main interests in non-commutative algebraic geometry. Recently, a complete list of superpotentials (defining relations) of all 3-dimensional AS-regular algebras which are Calabi-Yau was given by Mori-Smith (the quadratic case) and Mori-Ueyama (the cubic case), however, no complete list of defining relations of all 3-dimensional AS-regular algebras has not appeared in the literature. In this paper, we give all possible defining relations of 3-dimensional quadratic AS-regular algebras. Moreover, we classify them up to isomorphism and up to graded Morita equivalence in terms of their defining relations in the case that their point schemes are not elliptic curves. In the case that their point schemes are elliptic curves, we give conditions for isomorphism and graded Morita equivalence in terms of geometric data