The canonical ring of a stacky curve

TL;DR

This paper generalizes classical theorems to describe the generators and relations of the canonical ring of a stacky curve, including explicit Gr"obner bases, and applies these results to modular forms from Fuchsian groups.

Contribution

It provides a comprehensive description of the canonical ring of a stacky curve, extending classical results and including explicit computational tools.

Findings

Explicit generators and relations for the canonical ring of a stacky curve.

An explicit Gr"obner basis for the canonical ring.

Presentation of graded rings of modular forms from Fuchsian groups.

Abstract

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 Gr\"obner 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.