Central curves on noncommutative surfaces

Thilo Baumann; Pieter Belmans; Okke van Garderen

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arXiv · 2410.07620

Central curves on noncommutative surfaces

Thilo Baumann·Pieter Belmans·Okke van Garderen

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Abstract

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.

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Thilo Baumann, Pieter Belmans, Okke van Garderen. 2024-10-10. Central curves on noncommutative surfaces. https://arxiv.org/abs/2410.07620

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