Section rings of $\mathbb{Q}$-divisors on minimal rational surfaces

TL;DR

This paper establishes bounds on the degrees of generators and relations for section rings of arbitrary rational divisors on projective spaces and Hirzebruch surfaces, optimizing known bounds for effective divisors.

Contribution

It provides the first comprehensive bounds for section rings of $Q$-divisors on these surfaces, including optimal bounds for effective divisors on projective spaces.

Findings

Bounds on degrees of generators and relations for section rings.

Optimal bounds for effective $Q$-divisors on projective spaces.

General bounds applicable to all dimensions and surfaces.

Abstract

We give bounds on the degree of generation and relations of section rings associated to arbitrary $\mathbb{Q}$-divisors on projective spaces of all dimensions and Hirzebruch surfaces. For section rings of effective $\mathbb{Q}$-divisors on projective spaces, we find the best possible bound on the degrees of generators and relations.