Canonical rings of Q-divisors on P^1

TL;DR

This paper investigates the structure of canonical rings associated with Q-divisors on the projective line, providing explicit descriptions for simple cases and bounds for more complex divisors, with results depending only on divisor coefficients.

Contribution

It offers a complete description of the canonical ring for divisors with up to two points and bounds for generators and relations in general, highlighting dependence only on divisor coefficients.

Findings

Complete description of S_D for divisors with up to two points

Bounds on generators and relations for general divisors

Generators and relations depend only on divisor coefficients

Abstract

The canonical ring $S_D = \bigoplus_{d \geq 0} H^0(X, \lfloor dD \rfloor)$ of a divisor D on a curve X is a natural object of study; when D is a Q-divisor, it has connections to projective embeddings of stacky curves and rings of modular forms. We study the generators and relations of S_D for the simplest curve X = P^1. When D contains at most two points, we give a complete description of S_D; for general D, we give bounds on the generators and relations. We also show that the generators (for at most five points) and a Groebner basis of relations between them (for at most four points) depend only on the coefficients in the divisor D, not its points or the characteristic of the ground field; we conjecture that the minimal system of relations varies in a similar way. Although stated in terms of algebraic geometry, our results are proved by translating to the combinatorics of lattice points in simplices and cones.