2024/10/10 by Thilo Baumann, Baumann, Thilo, Pieter Belmans +3
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Commutative Algebra and Its Applications
paper · pdf · doi:10.48550/arxiv.2410.07620
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.