Mori Dream Spaces extremal contractions of K3 surfaces

We will give a criterion to assure that an extremal contraction of a K3 surface which is not a Mori Dream Space produces a singular surface which is a Mori Dream Spaces. We list the possible N\'eron--Severi groups of K3 surfaces with this property and an extra geometric condition such that the Picard number is greater then or equal to 10. We give a detailed description of two geometric examples for which the Picard number of the K3 surface is 3, i.e. the minimal possible in order to have the required property. Moreover we observe that there are infinitely many examples of K3 surfaces with the required property and Picard number equal to 3.Comment: V2: 25 pages. Lemma 2.6 of the old version contains a mistake pointed out by an anonimous referee. It is now substituted by Lemma 2.5, which contains an extra hypothesis, added also to Proposition 2.7. The other results are essentially unchanged (it is now checked that the examples satisfy the extra hypothesis). To appear in OJ

The dihedral group \Dh_5 as group of symplectic automorphisms on K3 surfaces

We prove that if a K3 surface $X$ admits $\Z/5\Z$ as group of symplectic automorphisms, then it actually admits \Dh_5 as group of symplectic automorphisms. The orthogonal complement to the \Dh_5-invariants in the second cohomology group of $X$ is a rank 16 lattice, $L$. It is known that $L$ does not depend on $X$: we prove that it is isometric to a lattice recently described by R. L. Griess Jr. and C. H. Lam. We also give an elementary construction of $L$.Comment: 11 pages. Arguments revised, results unchanged. Final version, to appear in Proc. Amer. Math. So

K3 surfaces with a non-symplectic automorphism and product-quotient surfaces with cyclic groups

We classify all the K3 surfaces which are minimal models of the quotient of the product of two curves $C_1\times C_2$ by the diagonal action of either the group $\Z/p\Z$ or the group $\Z/2p\Z$. These K3 surfaces admit a non-symplectic automorphism of order $p$ induced by an automorphism of one of the curves $C_1$ or $C_2$. We prove that most of the K3 surfaces admitting a non-symplectic automorphism of order $p$ (and in fact a maximal irreducible component of the moduli space of K3 surfaces with a non-symplectic automorphism of order $p$) are obtained in this way.\\ In addition, we show that one can obtain the same set of K3 surfaces under more restrictive assumptions namely one of the two curves, say $C_2$, is isomorphic to a rigid hyperelliptic curve with an automorphism $\delta_p$ of order $p$ and the automorphism of the K3 surface is induced by $\delta_p$.\\ Finally, we describe the variation of the Hodge structures of the surfaces constructed and we give an equation for some of them.Comment: 30 pages, 2 figure

On certain isogenies between K3 surfaces

The aim of this paper is to construct "special" isogenies between K3 surfaces, which are not Galois covers between K3 surfaces, but are obtained by composing cyclic Galois covers, induced by quotients by symplectic automorphisms. We determine the families of K3 surfaces for which this construction is possible. To this purpose we will prove that there are infinitely many big families of K3 surfaces which both admit a finite symplectic automorphism and are (desingularizations of) quotients of other K3 surfaces by a symplectic automorphism. In the case of involutions, for any $n\in\mathbb{N}_{>0}$ we determine the transcendental lattices of the K3 surfaces which are $2^n:1$ isogenous (by a non Galois cover) to other K3 surfaces. We also study the Galois closure of the $2^2:1$ isogenies and we describe the explicit geometry on an example.Comment: 28 page

Symplectic automorphisms of prime order on K3 surfaces

The aim of this paper is to study algebraic K3 surfaces (defined over the complex number field) with a symplectic automorphism of prime order. In particular we consider the action of the automorphism on the second cohomology with integer coefficients. We determine the invariant sublattice and its perpendicular complement, and show that the latter coincides with the Coxeter-Todd lattice in the case of automorphism of order three. We also compute many explicit examples, with particular attention to elliptic fibrations.Comment: 24 pages, final versio