We develop a mixed-characteristic version of the MoriMukai technique for producing rational curves on K3 surfaces. We reduce modulo p, produce rational curves on the resulting K3 surface over a finite field, and lift to characteristic zero. As an application, we prove that all complex K3 surfaces with Picard group generated by a class of degree two have an infinite number of rational curves.

We compute all K3 surfaces with Picard rank 20 over Q. Our proof uses modularity, the Artin-Tate conjecture and class group theory. With different techniques, the result has been established by Elkies to show that Mordell-Weil rank 18 over Q is impossible for an elliptic K3 surface. We also apply our methods to general singular K3 surfaces, i.e. with geometric Picard rank 20, but not necessaril...

The aim of this paper is to describe algebraic K3 surfaces with an even set of rational curves or of nodes. Their minimal possible Picard number is nine. We completely classify these K3 surfaces and after a carefull analysis of the divisors contained in the Picard lattice we study their projective models, giving necessary and sufficient conditions to have an even set. Moreover we investigate th...

The aim of this paper is to describe algebraic K3 surfaces with an even set of rational curves or of nodes. Their minimal possible Picard number is nine. We completely classify these K3 surfaces and after a careful analysis of the divisors contained in the Picard lattice we study their projective models, giving necessary and sufficient conditions to have an even set. Moreover we investigate the...

We study Cox rings of K3-surfaces. A first result is that a K3surface 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 non-symplectic involution that have Picard number 2 to 5...

We present M-theory compactifications on K3 × K3 with membranes near the An or Dn singularities of the K3 spaces. By realizing each of these compactifications in two different ways as type I’ models with 2and 6-branes, we explain the three-dimensional duality between gauge theories recently found by Intriligator and Seiberg. We also find new pairs of dual gauge theories, which we briefly descri...

We determine all possible configurations of rational double points on complex normal algebraic K3 surfaces, and on normal supersingular K3 surfaces in characteristic p > 19.