Canonical rings of Q-divisors on P^1

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.