arXiv subjects

Masaki Matsuno

Publications and source records attributed to Masaki Matsuno.

7 recordsLinked to original sources

arXiv2026

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

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.

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arXiv2023

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

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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arXiv2022

Twisted algebras of geometric algebras

A twisting system is one of the major tools to study graded algebras, however, it is often difficult to construct a (non-algebraic) twisting system if a graded algebra is given by generators and relations. In this paper, we show that a twisted algebra of a geometric algebra is determined by a certain automorphism of its point variety. As an application, we classify twisted algebras of $3$-dimensional geometric Artin-Schelter regular algebras up to graded algebra isomorphism.

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arXiv2022

Noncommutative conics in Calabi-Yau quantum projective planes

In noncommutative algebraic geometry, noncommutative quadric hypersurfaces are major objects of study. In this paper, we focus on studying noncommutative conics $\operatorname{Proj_{nc}} A$ embedded into Calabi-Yau quantum projective planes. In particular, we give complete classifications of homogeneous coordinate algebras $A$ of noncommutative conics up to isomorphism of graded algebras, and of noncommutive conics $\operatorname{Proj_{nc}} A$ up to isomorphism of noncommutative schemes.

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arXiv2021

AS-regularity of geometric algebras of plane cubic curves

Let $k$ be an algebraically closed field of characteristic $0$ and $A$ a graded $k$-algebra finitely generated in degree $1$. In this paper, for $3$-dimensional quadratic AS-regular algebras except for Type EC, we give a complete list of twisted superpotentials and a complete list of superpotentials such that derivation-quotient algebras are $3$-dimensional quadratic Calabi-Yau AS-regular algebras. For an algebra $A$ of Type EC, we give a criterion when $A$ is AS-regular. As an application, for an algebra $A$ of any type, we show that there exists a Calabi-Yau AS-regular algebra $S$ such that $A$ and $S$ are graded Morita equivalent. This result tells us that, for a $3$-dimensional quadratic AS-regular algebra $A$, to study the noncommutative projective scheme for $A$ defined by Artin-Zhang is reduced to study the noncommutative projective scheme for $S$ for the Calabi-Yau AS-regular algebra $S$.

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arXiv2019

A complete classification of $3$-dimensional quadratic AS-regular algebras of Type EC

Classification of AS-regular algebras is one of the main interests in noncommutative algebraic geometry. We say that a $3$-dimensional quadratic AS-regular algebra is of Type EC if its point scheme is an elliptic curve in $\mathbb{P}^{2}$. In this paper, we give a complete list of geometric pairs and a complete list of twisted superpotentials corresponding to such algebras. As an application, we show that there are only two exceptions up to isomorphism among all $3$-dimensional quadratic AS-regular algebras which cannot be written as a twist of a Calabi-Yau AS-regular algebra by a graded algebra automorphism.

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arXiv2019

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.

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