Classification of noncommutative central conics
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866908808143437824 |
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| author | Hu, Haigang Mori, Izuru Wu, Wenchao |
| author_facet | Hu, Haigang Mori, Izuru Wu, Wenchao |
| contents | Classification of noncommutative quadric hypersurfaces is one of the major projects in noncommutative algebraic geometry. In recent years, we are dedicated to complete the classification of noncommutative central conics. To achieve this goal, we and other authors develop some theories to study and classify some classes of noncommutative quadric hypersurfaces in a series of papers. Finally, in this paper, we completely classify noncommutative central conics by developing the general theory of homogenization and dehomogenization for noncommutative algebras and by previous results. As a main result, we show that there are bijections among the following sets of objects (i) the set of isomorphism classes of $4$-dimensional Frobenius algebras, (ii) the set of isomorphism classes of noncommutative affine pencils of conics, and (iii) the set of isomorphism classes of noncommutative central conics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_03236 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Classification of noncommutative central conics Hu, Haigang Mori, Izuru Wu, Wenchao Rings and Algebras Algebraic Geometry 16E65, 16S38, 16W50 Classification of noncommutative quadric hypersurfaces is one of the major projects in noncommutative algebraic geometry. In recent years, we are dedicated to complete the classification of noncommutative central conics. To achieve this goal, we and other authors develop some theories to study and classify some classes of noncommutative quadric hypersurfaces in a series of papers. Finally, in this paper, we completely classify noncommutative central conics by developing the general theory of homogenization and dehomogenization for noncommutative algebras and by previous results. As a main result, we show that there are bijections among the following sets of objects (i) the set of isomorphism classes of $4$-dimensional Frobenius algebras, (ii) the set of isomorphism classes of noncommutative affine pencils of conics, and (iii) the set of isomorphism classes of noncommutative central conics. |
| title | Classification of noncommutative central conics |
| topic | Rings and Algebras Algebraic Geometry 16E65, 16S38, 16W50 |
| url | https://arxiv.org/abs/2602.03236 |