相关论文: Determinantal quartics and the computation of the …
We test R. van Luijk's method for computing the Picard group of a $K3$ surface. The examples considered are the resolutions of Kummer quartics in $\bP^3$. Using the theory of abelian varieties, in this case, the Picard group may be computed…
We present a method to compute the geometric Picard rank of a $K3$ surface over $\bbQ$. Contrary to a widely held belief, we show it is possible to verify Picard rank $1$ using reduction only at a single prime. Our method is based on…
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…
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 $32$…
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…
In this paper we describe six pencils of K3-surfaces which have large Picard-Number and contain precisely five singular fibers: four have A-D-E singularities and one is non-reduced. In particular we describe these surfaces as cyclic…
We solve the Picard number problem for complex quintic surfaces by proving that every number between 1 and 45 occurs as Picard number of a quintic surface over the rationals. Our main technique consists in arithmetic deformations of…
We study the fixed singularities imposed on members of a linear system of surfaces in P^3_C by its base locus Z. For a 1-dimensional subscheme Z \subset P^3 with finitely many points p_i of embedding dimension three and d >> 0, we determine…
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,…
Using the invariant algebra of the reflection group denoted by $G\_{32}$ in Shephard-Todd classification, we construct three irreducible surfaces in $P^3$ with many singularities: one of them has degree $24$ and contains $1440$ quotient…
We show, in this second part, that the maximal number of singular points of a quartic surface $X \subset \mathbb{P}^3_K$ defined over an algebraically closed field $K$ of characteristic 2 is at most 14, and that, if we have 14…
We study the behavior of geometric Picard ranks of K3 surfaces over the rationals under reduction modulo primes. We compute these ranks for reductions of smooth quartic surfaces modulo all primes $p<2^{16}$ in several representative…
We discuss some aspects of the behavior of specialization at a finite place of N\'eron-Severi groups of K3 surfaces over number fields. We give optimal lower bounds for the Picard number of such specializations, thus answering a question of…
Based on high precision computation of periods and lattice reduction techniques, we compute the Picard group of smooth surfaces. We also study the lattice reduction technique that is employed in order to quantify the possibility of…
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…
Motivated by an application to LDPC (low density parity check) algebraic geometry codes described by Voloch and Zarzar, we describe a computational procedure for establishing an upper bound on the arithmetic or geometric Picard number of a…
The article covers the unique complex K3 surface with maximal Picard rank and discriminant four. We discuss smooth, rational curves and identify generators of its automorphism group with certain Cremona transformations of $\mathbb{P}^2$.…
We compute the integral Picard group of the moduli stack of polarized K3 surfaces of fixed degree whose singularities are at most rational double points. We also compute the integral Picard group of the stack of quasi-polarized K3 surfaces,…
We study finite abelian groups acting on three-dimensional rationally connected varieties. We concentrate on the groups of K3 type, that is, abelian extensions by a cyclic group of groups that faithfully act on a K3 surface. In particular,…
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…