Central curves on noncommutative surfaces
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909344199606272 |
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| author | Baumann, Thilo Belmans, Pieter van Garderen, Okke |
| author_facet | Baumann, Thilo Belmans, Pieter van Garderen, Okke |
| contents | There exists a dictionary between hereditary orders and smooth stacky curves, resp. tame orders of global dimension 2 and Azumaya algebras on smooth stacky surfaces. We extend this dictionary by explaining how the restriction of a tame order to a curve on the underlying surface corresponds to the fiber product of the curve with the stacky surface. By considering "bad" intersections we can start extending the dictionary in the 1-dimensional case to include non-hereditary orders and singular stacky curves. Two applications of these results are a novel description and classification of noncommutative conics in graded Clifford algebras, giving a geometric proof of results of Hu-Matsuno-Mori, and a complete understanding and classification of skew cubics, generalizing the work of Kanazawa for Fermat skew cubics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_07620 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Central curves on noncommutative surfaces Baumann, Thilo Belmans, Pieter van Garderen, Okke Algebraic Geometry Rings and Algebras Representation Theory There exists a dictionary between hereditary orders and smooth stacky curves, resp. tame orders of global dimension 2 and Azumaya algebras on smooth stacky surfaces. We extend this dictionary by explaining how the restriction of a tame order to a curve on the underlying surface corresponds to the fiber product of the curve with the stacky surface. By considering "bad" intersections we can start extending the dictionary in the 1-dimensional case to include non-hereditary orders and singular stacky curves. Two applications of these results are a novel description and classification of noncommutative conics in graded Clifford algebras, giving a geometric proof of results of Hu-Matsuno-Mori, and a complete understanding and classification of skew cubics, generalizing the work of Kanazawa for Fermat skew cubics. |
| title | Central curves on noncommutative surfaces |
| topic | Algebraic Geometry Rings and Algebras Representation Theory |
| url | https://arxiv.org/abs/2410.07620 |