Related papers: Cox rings of K3 surfaces with Picard number two

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 polyhedral. Moreover, we investigate degrees of generators and relations for Cox rings of…

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…

We construct explicit examples of $K3$ surfaces over ${\mathbb Q}$ having real multiplication. Our examples are of geometric Picard rank 16. The standard method for the computation of the Picard rank provably fails for the surfaces…

We determine the Cox rings of the minimal resolutions of cubic surfaces with at most rational double points, of blow ups of the projective plane at non-general configurations of six points and of three dimensional smooth Fano varieties of…

In this paper we determine a minimal set of generators for the Cox rings of extremal rational elliptic surfaces. Moreover, we develop a technique for computing the ideal of relations between them which allows, in all but three cases, to…

We determine explicit generators for the ring of modular forms associated with the moduli spaces of K3 surfaces with automorphism group $(\mathbb{Z}/2\mathbb{Z})^2$ and of Picard rank 13 and higher. The K3 surfaces in question carry a…

We study the geometry of the K3 surfaces $X$ with a finite number automorphisms and Picard number $\geq 3$. We describe these surfaces classified by Nikulin and Vinberg as double covers of simpler surfaces or embedded in a projective space.…

We investigate Cox rings of minimal resolutions of surface quotient singularities and provide two descriptions of these rings. The first one is the equation for the spectrum of a Cox ring, which is a hypersurface in an affine space. The…

In this paper we investigate the relation between the finite generation of the Cox ring R(X) of a smooth projective surface X and its anticanonical Iitaka dimension k(-K_X).

We investigate the Cox ring of a normal complete variety X with algebraic torus action. Our first results relate the Cox ring of X to that of a maximal geometric quotient of X. As a consequence, we obtain a complete description of the Cox…

We review recent developments in the arithmetic of K3 surfaces. Our focus lies on aspects of modularity, Picard number and rational points. Throughout we emphasise connections to geometry.

In this paper we study the geometry of the $14$ families of K3 surfaces of Picard number four with finite automorphism group, whose N\'eron-Severi lattices have been classified by \`E.B. Vinberg. We provide projective models, we identify…

We bring examples of toric varieties blown up at a point in the torus that do not have finitely generated Cox rings. These examples are generalizations of previous work where toric surfaces of Picard number 1 were studied. In this article…

This paper concerns K3 surfaces with automorphisms of order 11 in arbitrary characteristic. Specifically we study the wild case and prove that a general such surface in characteristic 11 has Picard number 2. We also construct K3 surfaces…

In this paper we study the automorphisms group of some K3 surfaces which are double covers of the projective plane ramified over a smooth sextic plane curve. More precisely, we study some particlar case of a K3 surface of Picard rank two.

We study families of $K3$ surfaces obtained by double covering of the projective plane branching along curves of $(2,3)$-torus type. In the first part, we study the Picard lattices of the families, and a lattice duality of them. In the…

The Cox rings of del Pezzo surfaces are closely related to the Lie groups E_n. In this paper, we generalize the definition of Cox rings to G- surfaces defined by us earlier, where the Lie groups G=A_n, D_n or E_n. We show that the Cox ring…

We study Cox rings of normal threefolds on which SL2 acts with a dense orbit. Exploiting the method of U-invariants, we obtain combinatorial criteria for the total coordinate space and the base variety to have log terminal singularities.…

We give examples of K3 surfaces over $\mathbb{Q}$ of degree $10$ whose geometric Picard group has rank~$1$. These K3 surfaces are intersections in $\mathbb{P}^9$ of three hyperplanes, one quadric and the image of the Pl\"ucker embedding of…

The subject of this paper is the study of various families of quartic K3 surfaces which are invariant under a certain $(\mathbb{Z}/2\mathbb{Z})^{4}$ action. In particular, we describe families whose general member contains $8,16,24$ or $32$…