On Cox rings of K3 surfaces

Abstract

We study Cox rings of K3 surfaces. A first result is that a K3 surface has a finitely generated Cox ring if and only if its effective cone is rational polyhedral. Moreover, we investigate degrees of generators and relations for Cox rings of K3 surfaces of Picard number two, and explicitly compute the Cox rings of generic K3 surfaces with a nonsymplectic involution that have Picard number 2 to 5 or occur as double covers of del Pezzo surfaces.

Cox rings of K3 surfaces with Picard number 2

“…(i) follows from [14, 2.7] and ii) from the main result of [11]. Then (iii) and (iv) follow from [14, Proposition 2.6], while (v) follows from [14] and [1,Proposition 3.4].…”

“…It follows that F is the pullback of the quartic polynomial defining X under the embedding given by H. In particular, the elimination ideal (xy 1 …”

“…This problem was considered in [1] for many classes of K3 surfaces, including double covers of Del Pezzo surfaces. The aim of this paper is to extend some of these results and also present some new examples.…”

“…Cox rings of K3 surfaces were first studied by Artebani, Hausen and Laface in the recent paper [1]. In that paper it was shown that a K3 surface has a finitely generated Cox ring if and only if its effective cone is rational polyhedral.…”

Good quotients of Mori dream spaces

“…Since K is of finite index in K the algebra Γ (X, S) inherits finite generation from Γ (X, S ) by [1,Proposition 4.4]. Now, let U X be an arbitrary open subset.…”