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Central curves on noncommutative surfaces

Thilo Baumann, Pieter Belmans, Okke van Garderen

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.

Central curves on noncommutative surfaces

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.

Paper Structure

This paper contains 36 sections, 40 theorems, 137 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Restricting the dictionary to central curves
  3. Application: noncommutative plane curves
  4. Noncommutative conics in graded Clifford algebras
  5. Noncommutative (Fermat) cubics in skew polynomial algebras
  6. Notation and conventions
  7. Acknowledgements
  8. Dictionaries between orders and stacks
  9. Generalized Rees algebras
  10. Hereditary orders and stacky curves
  11. Using generalized Rees algebras
  12. Tame orders of global dimension 2 and Azumaya algebras on stacky surfaces
  13. The root stack
  14. The canonical stack
  15. Restricting to central curves
  16. ...and 21 more sections

Key Result

lemma 3

Let $\Lambda$ be a tame order over a noetherian integrally closed domain $R$, ramified over primes $\mathfrak{p}_1,\ldots,\mathfrak{p}_n$. If each $\mathfrak{p}_i$ is principal, then $\mathop{\mathrm{Z}}\nolimits(\widetilde{\Lambda}) = \widetilde{R}(\mathfrak{p}_1,\ldots,\mathfrak{p}_n; e_1,\ldots,

Figures (1)

  • Figure 1: The 6 possible intersections $L\cap\Delta$

Theorems & Definitions (91)

  • definition 1
  • definition 2
  • lemma 3
  • proof
  • corollary 4
  • definition 5
  • theorem 6: Chan--Ingalls
  • lemma 7
  • proof
  • proposition 8
  • ...and 81 more