Neron-Severi Group Research Articles

On the Hodge cycles of Prym varieties

We show that the Neron–Severi group of the Prym variety for a degree three unramified Galois covering of a hyperelliptic Riemann surface has a distinguished subgroup of rank three. For the general hyperelliptic curve, the algebra of Hodge cycles on the Prym variety is generated by this group of rank three. Mathematics Subject Classification (2000): 14H40 (primary); 14C30 (secondary). 1. – Introduction Let X be a hyperelliptic Riemann surface. Let f : Y −→ X be a unramified Galois covering with Galois group Z/3Z with Y connected. The Neron–Severi group of the Prym variety Prym( f ) has a certain distinguished subgroup of rank three. This subgroup is constructed as follows. We first prove that Prym( f ) is identified with Pic0(Z) × Pic0(Z), where Z is a Riemann surface constructed from the covering datum (Lemma 3.1). The two projections and the addition law of Pic0(Z) together give three maps from Pic0(Z)×Pic0(Z) to Pic0(Z). The pull backs, by these three maps, to Pic0(Z)×Pic0(Z) = Prym( f ) of the natural polarization on Pic0(Z) give the distinguished subgroup of the Neron–Severi group of Prym( f ). For the general hyperelliptic curve X , the algebra of Hodge cycles on Prym( f ) is generated by this subgroup of rank three of NS(Prym( f )) (Theorem 4.1). In particular, for general hyperelliptic X , the Neron–Severi group NS(Prym( f )) is of rank three, and all the Hodge cycles on Prym( f ) are algebraic. We also consider degree three ramified coverings of an arbitrary compact Riemann surface. Under an assumption on the covering, the algebra of Hodge cycles of the Prym variety for the general Riemann surface is again generated by the Neron–Severi group (Lemma 5.1). Pervenuto alla Redazione il 9 luglio 2003 e in forma definitiva il 25 giugno 2004.

Small codimension subvarieties in homogeneous spaces

We prove Bertini type theorems for the inverse image, under a proper morphism, of any Schubert variety in an homogeneous space. Using generalisations of Deligne's trick, we deduce connectedness results for the inverse image of the diagonal in X 2 where X is any isotropic grassmannian. We also deduce simple connectedness properties for subvarieties of X. Finally we prove transplanting theorems à la Barth-Larsen for the Picard group of any isotropic grassmannian of lines and for the Neron-Severi group of some adjoint and coadjoint homogeneous spaces.

K3 surfaces with Picard number one and infinitely many rational points

In general, not much is known about the arithmetic of K3 surfaces. Once the geometric Picard number, which is the rank of the Neron–Severi group over an algebraic closure of the base field, is high enough, more structure is known and more can be said. However, until recently not a single explicit K3 surface was known to have geometric Picard number one. We give explicit examples of such surfaces over the rational numbers. This solves an old problem that has been attributed to Mumford. The examples we give also contain infinitely many rational points, thereby answering a question of Swinnerton-Dyer and Poonen.

Polyhedral groups and pencils of K3-surfaces with maximal Picard number

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 pencil has Picard number 19 and each pencil contains four surfaces with Picard number 20. These surfaces are obtained as the minimal resolution of quotients X/G, where $G\subset SO(4,\R)$ is some finite subgroup and $X\subset \P_3(\C)$ denotes a G-invariant surface. The singularities of X/G come from fix points of G on X or from singularities of X. In any case the singularities on X/G are A-D-E surface singularities. The rational curves which resolve them and some extra 2-divisible sets, resp. 3-divisible sets of rational curves generate the Neron-Severi group of the minimal resolution.

Effective divisor classes on a ruled surface

The Neron-Severi group of divisor classes modulo algebraic equivalence on a smooth algebraic surface is often not difficult to calculate, and has classically been studied as one of the fundamental invariants of the surface. A more difficult problem is the determination of those divisor classes which can be represented by effective divisors; these divisor classes form a monoid contained in the Neron-Severi group. Despite the finite generation of the whole Neron-Severi group, the monoid of effective divisor classes may or may not be finitely generated, and the methods used to explicitly calculate this monoid seem to vary widely as one proceeds from one type of surface to another in the standard classification scheme (see Rosoff, 1980, 1981). In this paper we shall use concrete vector bundle techniques to describe the monoid of effective divisor classes modulo algebraic equivalence on a complex ruled surface over a given base curve. We will find that, over a base curve of genus 0, the monoid of effective divisor classes is very simple, having two generators (which is perhaps to be expected), while for a ruled surface over a curve of genus 1, the monoid is more complicated, having either two or three generators. Over a base curve of genus 2 or greater, we will give necessary and sufficient conditions for a ruled surface to have its monoid of effective divisor classes finitely generated; these conditions point to the existence of many ruled surfaces over curves of higher genus for which finite generation fails.

The Hodge conjecture for general Prym varieties

The space of Hodge cycles of the general Prym variety is proved to be generated by its Neron-Severi group.

Parabolic bundles and representations¶of the fundamental group

Let X be a smooth complex projective variety with Neron–Severi group isomorphic to ℤ, and D an irreducible divisor with normal crossing singularities. Assume 1<r≤ 3. We prove that if π1(X) doesn't have irreducible PU(r) representations, then π1(X- D) doesn't have irreducible U(r) representations. The proof uses the non-existence of certain stable parabolic bundles. We also obtain a similar result for GL(2) when D is smooth.

On non-projective normal surfaces

In this note we construct examples of non-projective normal proper algebraic surfaces and discuss the somewhat pathological behaviour of their Neron-Severi group. Our surfaces are birational to the product of a projective line and a curve of higher genus.

Real algebraic -surfaces

We prove necessary and sufficient conditions for a real algebraic surface to be a -surface. We calculate the Neron-Severi group , the Brauer group and the algebraic cohomology group , where is a real projective surface. We also prove Nikulin's congruence for an arbitrary orientable -surface

Heights for line bundles on arithmetic varieties

Let X be an arithmetic variety and L be an element of the Neron-Severi group of its generic fiber X K . Then there are only finitely many line bundles on X, generically belonging to L, such that the degrees of on the irreducible components of the special fibers of X and the height of are bounded. The concept of a height used here is recalled. Several elementary properties of this height are proven.

Vector bundles with holomorphic connection over a projective manifold with tangent bundle of nonnegative degree

For a projective manifold whose tangent bundle is of nonnegative degree, a vector bundle on it with a holomorphic connection actually admits a compatible flat holomorphic connection, if the manifold satisfies certain conditions. The conditions in question are on the Harder-Narasimhan filtration of the tangent bundle, and on the Neron-Severi group.

On the Picard group of a compact flat projective variety

In this note, we describe the Picard group of the class of compact, smooth, flat, projective varieties. In view of Charlap’s work and Johnson’s characterization, we construct line bundles over such manifolds as the holonomy-invariant elements of the Neron-Severi group of a projective flat torus covering the manifold. We prove a generalized version of the Appell-Humbert theorem which shows that the nontrivial elements of the Picard group are precisely those coming from the above construction. Our calculations finally give an estimate for the set of positive line bundles for such varieties.

Neron-Severi group for nonalgebraic elliptic surfaces I Elliptic bundle case

Neron-Severi group for nonalgebraic elliptic surfaces I Elliptic bundle case

Divisors on symmetric products of curves

For a curve with general moduli, the Neron-Severi group of its symmetric products is generated by the classes of two divisors x and θ. In this paper we give bounds for the cones of effective and ample divisors in the xθ-plane

On the Néron-Severi group of abelian complex Lie groups

In this paper we want to discuss some phenomena concerning Chern classes of holomorphic line bundles on not necessarily compact quotients X of a complex vector space by a lattice. We give explicit equations describing the Neron-Severi group NS(X), from which we can deduce some facts on the behaviour of Picard numbers in complete families of abelian complex Lie groups. The equations enable us to calculate long lists of Picard numbers of abelian Lie groups of given dimension and rank. In contrary to the compact case the Chern classes of holomorphic line bundles on X are in general not of type (1, 1). We point out a representation of the (1, 1) -part of the Neron-Severi group, which is analoguous to the description of NS(X) as a subgroup of Hom(X, X), when X is an abelian variety. The projection of NS(X) to the space of cohomology classes of type (1, 1) is in general not infective; however if X is a toroidal group this is the case, but in this situation the image of NS(X) in-H1,1(X) is in general not a discrete subgroup.

Elliptic Curves Over Function Fields and the Picard Number

Introduction. The purpose of this paper is to study the rank of the group of rational points on an elliptic curve defined over a function field and simultaneously the Picard number of the corresponding elliptic surface. The results utilize the general theory of elliptic surfaces due to Kodaira [5] and [6], and properties of the Gauss-Manin connexion associated with an elliptic surface (Stiller [15]). Let E be an elliptic surface having a global section over its base curve X. We assume throughout that the functional invariant J is non-constant and that E has no exceptional curves of the first kind in the fibres. We denote by NS(E) the Neron-Severi group of the surface E and by p the rank of NS(E) which is called the Picard number of E. Additionally, letting K(X) be the function field of X, we denote by Egen (K(X)) the group of K(X)-rational points on the generic fibre of E over X, whose rank we denote by r. We refer the reader to Shioda [14] for a description of NS(E) and its relation to the group Egen (K(X)). In particular recall

Local contributions to global deformations of surfaces

In [23], Schiffer and Spencer prove that all small deformations of complex structure on a compact Riemann surface may be realized by altering the complex structure only within an arbitrarily small neighborhood of a point on the surface. It seems interesting in general to consider whether it is possible to construct deformations of an algebraic variety or complex manifold from deformations of neighborhoods of certain subvarieties. Further motivation for trying to understand the role of subvarieties in deformations is suggested by the instability under deformation of the Neron-Severi group, i.e., the group of divisors modulo numerical equivalence. As an example, one may consider the family of affine surfaces Vt: x 2 + y 2 + z 2 = t z (t is a parameter; V o has a nodal singularity at the origin). This family admits a resolution {Xt} --, {V~}, with {X,} a smooth family of non-singular surfaces, and each X~ is a minimal resolution of V t ([4] or [5]). The exceptional curve E in X o is a IP 1 with self-intersection 2 which does not appear in any Xt, for t#:0. One may ask whether every smooth surface X with such a curve in it admits a one-parameter family of deformations arising from this local model. Furthermore, if X contains several disjoint such curves, does each one independently contribute one dimension to the moduli of X? The Hartogs ' theorem of [14] says that one cannot simply plumb in the local deformation, leaving the structure of X unchanged outside a small neighborhood of E. Moreover, old examples of Segre [26] show that the nodes on certain hypersurfaces V in IP 3 are not "independent", i.e., there aren't enough deformations of the resolution X of V to allow for a one-dimensional contribution from each node. Theorem (3.7) of this paper says that the regularity of the Kuranishi variety of X is sufficient for the deformations of X to realize independently