Related papers: K3 Surfaces, Picard Numbers and Siegel Disks

Using elliptic structures, we show that any supersingular K3 surface of Artin invariant $1$ in characteristic $p \not= 5$, $7$, $13$ has an automorphism the entropy of which is the natural logarithm of a Salem number of degree $22$.

We determine all posible orders of automorphisms of finite order of complex K3 surfaces or of K3 surfaces in characteristic $p>3$. E.g., a positive integer $N$ is the order of an automorphism of a complex K3 surface if and only if…

We derive a characterization of the complex projective K3 surfaces which have automorphisms of positive entropy in term of their N\'eron-Severi lattices. Along the way, we classify the projective K3 surfaces of zero entropy with infinite…

Using toric geometry, lattice theory, and elliptic surface techniques, we compute the Picard Lattice of certain K3 surfaces. In particular, we examine the generic member of each of M. Reid's list of 95 families of Gorenstein K3 surfaces…

In each characteristic $p\neq 2, 3$, it was shown in a previous work that the order of an automorphism of a K3 surface is bounded by 66, if finite. Here, it is shown that in each characteristic $p\neq 2, 3$ a K3 surface with a cyclic action…

We classify K3 surfaces with a non-symplectic finite automorphism of high order. It is shown that such an automorphism cannot be of order 60, and for each of the orders 38, 44, 48, 50, 54 and 66, there exists a unique K3 surface with such…

We provide a criterion for when Hilbert squares of complex projective K3 surfaces with Picard number one are strongly ambiguous. This criterion is the same as [DM, Proposition 3.14], but is obtained by a different method. In particular,…

We explain how to use the computer algebra system OSCAR to find all elliptic fibrations (up to automorphism) on a given surface and compute their Weierstrass models. This is illustrated for Vinberg's most algebraic K3 surface, the unique K3…

We show that a characteristic $0$ model $X_R\to \Spec R$, with Picard number $1$ over a geometric generic point, of a K3 surface in characteristic $p\ge 3$, essentially kills all automorphisms (Theorem 5.1). We show that there is an…

A K3-surface is a (smooth) surface which is simply connected and has trivial canonical bundle. In these notes we investigate three particular pencils of K3-surfaces with maximal Picard number. More precisely the general member in each…

The present paper provides several results on automorphisms of hyperk\"ahler (or irreducible holomorphic symplectic) manifolds. In particular it focuses on the symplectic case and contains a classification of prime order symplectic…

Let S be a K3 surface that admits a non-symplectic automorphism $\rho$ of order 3. We divide $S\times \mathbb{P}^1$ by $\rho\times\psi$ where $\psi$ is an automorphism of order 3 of $\mathbb{P}^1$. There exists a threefold ramified cover of…

We complete the classification of order $5$ nonsymplectic automorphisms on hyper-K\"ahler fourfolds deformation equivalent to the Hilbert square of a K3 surface. We then compute the topological Lefschetz number of natural automorphisms of…

In this paper we classify complex K3 surfaces with non-symplectic automorphism of order 8 that leaves invariant a smooth elliptic curve. We show that the rank of the Picard group is either 10, 14 or 18 and the fixed locus is the disjoint…

We construct a K3 surface over an algebraically closed field of characteristic 2 which contains two sets of 21 disjoint smooth rational curves such that each curve from one set intersects exactly 5 curves from the other set. This…

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…

We give a short proof that every supersingular K3 surface (except possibly in characteristic $2$ with Artin invariant $\sigma=10$) has an automorphism of Salem degree 22. In particular an infinite subgroup of the automorphism group does not…

We construct examples of $K3$ surfaces of geometric Picard rank $1$. Our method is a refinement of that of R. van Luijk. It is based on an analysis of the Galois module structure on \'etale cohomology. This allows to abandon the original…

We study the maximal Salem degree of automorphisms of K3 surfaces via elliptic fibrations. By generalizing \cite{EOY14}, we establish a characterization of such maximum in terms of elliptic fibrations with infinite automorphism groups. As…

We present a method to generate many automorphisms of a supersingular K3 surface in odd characteristic. As an application, we show that, if p is an odd prime less than or equal to 7919, then every supersingular K3 surface in characteristic…