Spin canonical rings of log stacky curves

TL;DR

This paper establishes tight bounds on the degrees of generators and relations for log spin canonical rings of stacky curves, leading to improved understanding of the algebraic structure of modular forms of arbitrary integral weight.

Contribution

It provides the first precise bounds on the degrees of minimal generators and relations for log spin canonical rings, extending to rings of modular forms of any integral weight.

Findings

Bounded degrees of minimal generators and relations for log spin canonical rings.

Extended bounds to rings of modular forms of all integral weights.

Enhanced understanding of the algebraic structure of modular forms.

Abstract

Consider modular forms arising from a finite-area quotient of the upper-half plane by a Fuchsian group. By the classical results of Kodaira-Spencer, this ring of modular forms may be viewed as the log spin canonical ring of a stacky curve. In this paper, we tightly bound the degrees of minimal generators and relations of log spin canonical rings. As a consequence, we obtain a tight bound on the degrees of minimal generators and relations for rings of modular forms of arbitrary integral weight.