Central curves on noncommutative surfaces

TL;DR

This paper extends the correspondence between hereditary orders and stacky curves to include non-hereditary orders and singular curves, providing new classifications and geometric insights into noncommutative conics and skew cubics.

Contribution

It generalizes the existing dictionary to non-hereditary orders and singular stacky curves, and applies this to classify noncommutative conics and skew cubics.

Findings

Classification of noncommutative conics in graded Clifford algebras.

Complete understanding of skew cubics, extending Kanazawa's work.

Geometric proof of existing results on noncommutative conics.

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.