Cox rings of K3 surfaces of Picard number three

2019/09/03 by Artebani, Michela, Correa, Claudia, Laface, Antonio
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1909.01267

Abstract

Let X be a projective K3 surface over \mathbb C. We prove that its Cox ring R(X) 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 elliptic fibrations with f⋅ f'=2. This result and techniques using Koszul's type exact sequences allow to determine a generating set for the Cox ring of all Mori dream K3 surfaces of Picard number three which is minimal in most cases. A presentation for the Cox ring is given in some special cases with few generators.

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