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The canonical ring of a stacky curve

About this Title

John Voight and David Zureick-Brown

Publication: Memoirs of the American Mathematical Society
Publication Year: 2022; Volume 277, Number 1362
ISBNs: 978-1-4704-5228-5 (print); 978-1-4704-7094-4 (online)
Published electronically: March 28, 2022
Keywords: Canonical rings, canonical embeddings, stacks, algebraic curves, modular forms, automorphic forms, generic initial ideals, Gröbner bases

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Table of Contents


  • 1. Introduction
  • 2. Canonical rings of curves
  • 3. A generalized Max Noether’s theorem for curves
  • 4. Canonical rings of classical log curves
  • 5. Stacky curves
  • 6. Rings of modular forms
  • 7. Canonical rings of log stacky curves: genus zero
  • 8. Inductive presentation of the canonical ring
  • 9. Log stacky base cases in genus 0
  • 10. Spin canonical rings
  • 11. Relative canonical algebras
  • Appendix: Tables of canonical rings


Generalizing the classical theorems of Max Noether and Petri, we describe generators and relations for the canonical ring of a stacky curve, including an explicit Gröbner basis. We work in a general algebro-geometric context and treat log canonical and spin canonical rings as well. As an application, we give an explicit presentation for graded rings of modular forms arising from finite-area quotients of the upper half-plane by Fuchsian groups.

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