Canonical Rings of ${\mathbb{Q}}$-Divisors on ${\mathbb{P}^1}$

Abstract

The canonical ring ${S_{D} = \oplus_{d\geq0}H^{0}(X, \lfloor dD\rfloor)}$ of a divisor D on a curve X is a natural object of study; when D is a ${\mathbb{Q}}$-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 ${X = \mathbb{P}^1}$. 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 Gröbner 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.