Picard and Lefschetz numbers of real algebraic surfaces

Two Picard numbers and two Lefschetz numbers are defined for a real algebraic surface. They are similar to the Picard number and the Lefschetz number of a complex algebraic surface. For these numbers, some estimates and relations in the form of inequalities are proved.Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 847–852, June, 1998.     

On the Rank of Picard Groups of Modular Varieties Attached to Orthogonal Groups

We derive lower bounds for the rank of Picard groups of modular varieties associated with natural congruence subgroups of the orthogonal group of an even lattice of signature (2, l). As an example we consider the Siegel modular group of genus 2. The analytic part of this paper also leads to certain class number identities.     

Transcendental obstructions to weak approximation on general K3 surfaces

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.     

Self-correspondences of K3 surfaces via moduli of sheaves and arithmetic hyperbolic reflection groups

In a series of our papers with Carlo Madonna (2002–2008), we described self-correspondences of a K3 surface over ℂ via moduli of sheaves with primitive isotropic Mukai vectors for the Picard number 1 or 2 of the K3 surfaces. Here we give a natural and functorial answer to the same problem for an arbitrary Picard number. As an application, we characterize, in terms of self-correspondences via moduli of sheaves, K3 surfaces with reflective Picard lattice, that is, when the automorphism group of the lattice is generated by reflections up to finite index. It is known since 1981 that the number of reflective hyperbolic lattices is finite. We also formulate some natural unsolved related problems.     

Wehler K3 surfaces with Picard number 3 and 4. Appendix to: “Orbits of points on certain K3 surfaces”, by Arthur Baragar

We show that Wehler K3 surfaces with Picard number three, which are the focus of the previous paper by Arthur Baragar, do indeed exist over the rational numbers.     

Canonical vector heights on K3 surfaces with Picard number three-- An argument for nonexistence

In this paper, we investigate a K3 surface with Picard number three and present evidence that strongly suggests a canonical vector height cannot exist on this surface.

     

Generalized Shioda–Inose structures on K3 surfaces

In this note, we study the action of finite groups of symplectic automorphisms on K3 surfaces which yield quotients birational to generalized Kummer surfaces. For each possible group, we determine the Picard number of the K3 surface admitting such an action and for singular K3 surfaces we show the uniqueness of the associated abelian surface. Received: 9 April 1998 / Revised version: 17 July 1998     

Equivalences of twisted K3 surfaces

We prove that two derived equivalent twisted K3 surfaces have isomorphic periods. The converse is shown for K3 surfaces with large Picard number. It is also shown that all possible twisted derived equivalences between arbitrary twisted K3 surfaces form a subgroup of the group of all orthogonal transformations of the cohomology of a K3 surface.The passage from twisted derived equivalences to an action on the cohomology is made possible by twisted Chern characters that will be introduced for arbitrary smooth projective varieties.     

Picard-Zahlen komplexer Tori

We give explicit equations for the calculation of Chern classes of holomorphic line bundles on a complex torus X. As easy applications we deduce properties of the Picard numbers ρ(X) of n-dimensional tori, when the complex structure changes. The tori with ρ(X)≥k form a countable union of analytic subsets in a moduli space M; furthermore the set of tori with ρ(X)=k is empty or dense in M. For n-dimensional tori one has O≤ρ(X)≤n², but for n≥3 not all numbers 0≤k≤n² occur as Picard numbers. We conclude our considerations with a list of examples and with some remarks about this gap phenomenon in the distribution of Picard numbers of complex tori.     

PICARD LATTICES OF FAMILIES OF K3 SURFACES

ABSTRACT

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 which occur as hypersurfaces in weighted projective 3-spaces. As an application, we are able to determine whether the mirror family (in the sense of mirror symmetry for K3 surfaces) for each one is also on Reid's list.     

K3 surfaces with Picard number three and canonical vector heights

In this paper we construct the first known explicit family of K3 surfaces defined over the rationals that are proved to have geometric Picard number . This family is dense in one of the components of the moduli space of all polarized K3 surfaces with Picard number at least . We also use an example from this family to fill a gap in an earlier paper by the first author. In that paper, an argument for the nonexistence of canonical vector heights on K3 surfaces of Picard number was given, based on an explicit surface that was not proved to have Picard number . We redo the computations for one of our surfaces and come to the same conclusion.

     

Transcendental lattices of some K 3‐surfaces

In a previous paper, [12], we described six families of K 3‐surfaces (over ?) with Picard‐number 19, and we identified surfaces with Picard‐number 20. In these notes we classify some of the surfaces by computing their transcendental lattices. Moreover, we show that the surfaces with Picard‐number 19 are birational to a Kummer surface which is the quotient of a non‐product type abelian surface by an involution. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some Remarks about the FM-partners of <Emphasis Type="Italic">K</Emphasis>3 Surfaces with Picard Numbers 1 and 2

In this paper we describe some results about K3 surfaces with Picard number 1 and 2. In particular, we give a new simple proof of a theorem due to Oguiso which shows that, given an integer N, there is a K3 surface with Picard number 2 and at least N non-isomorphic FM-partners. We describe also the Mukai vectors of the moduli spaces associated to the FM-partners of K3 surfaces with Picard number 1.     

Symplectic Automorphisms and the Picard Group of a K3 Surface

We consider the symplectic action of a finite group G on a K3 surface X. The Picard group of X has a primitive sublattice determined by G. We show how to compute the rank and discriminant of this sublattice. We then investigate the classification of symplectic actions by a fixed finite group, using moduli spaces of K3 surfaces with symplectic G-action.     

Irreducibility of the Igusa tower

We shall give a simple （basically） the Igusa tower over Shimura varieties of PEL purely in characteristic p proof of the irreducibility of type. Our result covers Shimura variety of type A and type C classical groups, in particular, the Siegel modular varieties, the Hilbert-Siegel modular varieties, Picard surfaces and Shimura varieties of inner forms of unitary and symplectic groups over totally real fields.     

Arithmetic Properties of Generalized Hypergeometric <Emphasis Type="Italic">F</Emphasis>-Series

A generalization of the Siegel–Shidlovskii method in the theory of transcendental numbers is used to prove the infinite algebraic independence of elements (generated by generalized hypergeometric series) of direct products of fields \(\mathbb{K}_v\), which are completions of an algebraic number field \(\mathbb{K}\) of finite degree over the field of rational numbers with respect to valuations v of \(\mathbb{K}\) extending p-adic valuations of the field ? over all primes p, except for a finite number of them.     

A note on a smooth projective surface with Picard number 2

We characterize the integral Zariski decomposition of a smooth projective surface with Picard number 2 to partially solve a problem of B. Harbourne, P. Pokora, and H. Tutaj‐Gasinska [Electron. Res. Announc. Math. Sci. 22 (2015), 103–108].     

Brill–Noether general curves on Knutsen K3 surfaces

This article classifies Knutsen K3 surfaces all of whose hyperplane sections are irreducible and reduced. As an application, this gives infinite families of K3 surfaces of Picard number two whose general hyperplane sections are Brill–Noether general curves.     

Cox rings of K3 surfaces with Picard number 2

We study presentations of Cox rings of K3 surfaces of Picard number 2. In particular we consider the Cox rings of classical examples of K3 surfaces, such as quartic surfaces containing a line and doubly elliptic K3 surfaces.