We study generators and relations for Cox rings of K3 surfaces of Picard number 2. The main results are explicit descriptions of the Cox rings when X is a double cover of P ramified over a sextic with a tritangent, a quartic surface in P with a line or a K3 surface with intersection matrix

In this article, we study the discriminant of those K3 surfaces with invo-lution which were introduced and investigated by Matsumoto, Sasaki, and Yoshida. We extend several classical results on the discriminant of elliptic curves to the dis-criminant of Matsumoto-Sasaki-Yoshida's K3 surfaces.

We show that the Mathieu groups M22 and M11 can act on the supersingular K3 surface with Artin invariant 1 in characteristic 11 as symplectic automorphisms. More generally we show that all maximal subgroups of the Mathieu group M23 with three orbits on 24 letters act on a supersingular K3 surface with Artin invariant 1 in a suitable characteristic.

We investigate configurations of rational double points with the total Milnor number 21 on supersingular K3 surfaces. The complete list of possible configurations is given. As an application, we also give the complete list of extremal (quasi-)elliptic fibrations on supersingular K3 surfaces.

A few facts concerning the phrase ”the automorphism groups become larger at special points of the moduli of K3 surfaces” are presented. It is also shown that the automorphism groups are of infinite order over a dense subset in any onedimensional non-trivial family of projective K3 surfaces.

In a Type III degeneration of K3-surfaces the dual graph of the central fibre is a triangulation of S2. We realise the tetrahedral, octahedral and especially the icosahedral triangulation in families of K3-surfaces, preferably with the associated symmetry groups acting.

We construct an explicit K3 surface over the field of rational numbers that has geometric Picard rank one, and for which there is a transcendental Brauer-Manin obstruction to weak approximation. To do so, we exploit the relationship between polarized K3 surfaces endowed with particular kinds of Brauer classes and cubic fourfolds.

The Matrix theory description of Type IIA string theory on a compact K3 surface as the theory of Neveu-Schwarz five-branes oñ K3 × S 1 is analyzed. The full multiplet of space-time BPS states is identified in the five-brane world-volume as fluxes. October 1997

In this paper, the chromaticity of K3-gluings of two wheels is studied. For each even integer n ≥ 6 and each odd integer 3 ≤ q ≤ [n/2] all K3-gluings of wheels Wq+2 and Wn−q+2 create an χ-equivalent class.