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

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