Cox rings of K3 surfaces of Picard number three

Let X be a projective K3 surface over C. We prove that its Cox ring has a generating set whose degrees are either classes of smooth rational curves, sums of at most three elements of the Hilbert basis of the nef cone, or of the form 2(f + f'), where f, f' are classes of smooth elliptic curves with f . f' = 2. This result and techniques using Koszul's type exact sequences are then applied to determine a generating set for the Cox ring of all Mori dream K3 surfaces of Picard number three which given in some special cases with few generators. (C) 2020 Elsevier Inc. All rights reserved.