arXiv subjects

Bert van Geemen

Publications and source records attributed to Bert van Geemen.

50 records · Page 3Linked to original sources

arXiv2006

Nikulin involutions on K3 surfaces

We study the maps induced on cohomology by a Nikulin (i.e. a symplectic) involution on a K3 surface. We parametrize the eleven dimensional irreducible components of the moduli space of algebraic K3 surfaces with a Nikulin involution and we give examples of the general K3 surface in various components. We conclude with some remarks on Morrison-Nikulin involutions, these are Nikulin involutions which interchange two copies of E_8(-1) in the Neron Severi group.

math.AG↗

arXiv2005

A family of marked cubic surfaces and the root system D_5

We define and study a family of cubic surfaces in the projectivized tangent bundle over a four dimensional projective space associated to the root system D_5. The 27 lines are rational over the base and we determine the classifying map to the moduli space of marked cubic surfaces. This map has degree two and we use it to get short proofs for some results on the Chow group of the moduli space of marked cubic surfaces.

math.AG↗

arXiv2004

Some remarks on Brauer groups of K3 surfaces

We discuss the geometry of the genus one fibrations associated to an elliptic fibration on a K3 surface. We show that the two-torsion subgroup of the Brauer group of a general elliptic fibration is naturally isomorphic to the two-torsion of the Jacobian of a curve associated to the fibration. We remark that this is related to Recillas' trigonal construction. Finally we discuss the two-torsion in the Brauer group of a general K3 surface with a polarisation of degree two.

math.AG↗

arXiv2004

Hessians and the moduli space of cubic surfaces

The Hessian of a general cubic surface is a nodal quartic surface, hence its desingularisation is a K3 surface. We determine the transcendental lattice of the Hessian K3 surface for various cubic surfaces (with nodes and/or Eckardt points for example). Classical invariant theory shows that the moduli space of cubic surfaces is a weighted projective space. We describe the singular locus and some other subvarieties of the moduli space.

math.AG↗

arXiv2003

An isogeny of K3 surfaces

In a recent paper Ahlgren, Ono and Penniston described the L-series of K3 surfaces from a certain one parameter family in terms of those of a particular family of elliptic curves. The Tate conjecture predicts the existence of a correspondence between these K3 surfaces and certain Kummer surfaces related to these elliptic curves. A geometric construction of this correspondence is given here, using results of D. Morrison on Nikulin involutions.

math.AG↗

arXiv2002

The Chow group of the moduli space of marked cubic surfaces

Naruki gave an explicit construction of the moduli space of marked cubic surfaces, starting from a toric variety and proceeding with blow ups and contractions. Using his result, we compute the Chow groups and the Chern classes of this moduli space.As an application we relate a recent result of Freitag on the Hilbert polynomial of a certain ring of modular forms to the Riemann-Roch theorem for the moduli space.

math.AG↗

arXiv2001

A linear system on Naruki's moduli space of marked cubic surfaces

Allcock and Freitag recently showed that the moduli space of marked cubic surfaces is a subvariety of a nine dimensional projective space which is defined by cubic equations. They used the theory of automorphic forms on ball quotients to obtain these results. Here we describe the same embedding using Naruki's toric model of the moduli space. We also give an explicit parametrization of the tritangent divisors, we discuss another way to find equations for the image and we show that the moduli space maps, with degree at least ten, onto the unique quintic hypersurface in a five dimensional projective space which is invariant under the action of the Weyl group of the root system E_6.

math.AG↗

arXiv2000

Half twists and the cohomology of hypersurfaces

A Hodge structure V of weight k on which a CM field acts defines, under certain conditions, a Hodge structure of weight k-1, its half twist. In this paper we consider hypersurfaces in projective space with a cyclic automorphism which defines an action of a cyclotomic field on a Hodge substructure in the cohomology. We determine when the half twist exists and relate it to the geometry and moduli of the hypersurfaces. We use our results to prove the existence of a Kuga-Satake correspondance for certain cubic 4-folds.

math.AG↗

arXiv2000

Half twists of Hodge structures of CM-type

To a Hodge structure V of weight k with CM by a field K we associate Hodge structures V_{-n/2} of weight k+n for n positive and, under certain circumstances, also for n negative. We show that these `half twists' come up naturally in the Kuga-Satake varieties of weight two Hodge structures with CM by an imaginary quadratic field.

math.AG↗

arXiv1999

Kuga-Satake varieties and the Hodge conjecture

This paper gives an introduction to Kuga-Satake varieties and discusses some aspects of the Hodge conjecture related to them. Kuga-Satake varieties are abelian varieties associated to certain weight two Hodge structures, for example the second cohomology group of a K3 surface. We give a detailed account of the construction of Kuga-Satake varieties and of their decomposition in simple subvarieties. We recall the Hodge conjecture and we point out a connection between the Hodge conjecture for abelian fourfolds and Kuga-Satake varieties. We discuss the implications of the Hodge conjecture on the geometry of surfaces whose second cohomology group has a Kuga-Satake variety. We conclude with some recent results on Kuga-Satake varieties of Hodge structures on which an imaginary quadratic field acts.

math.AG↗

arXiv1997

On Hitchin's Connection

The aim of this paper is to give an explicit expression for Hitchin's connection in the case of rank 2 bundles with trivial determinant over curves of genus 2. We recall the definition of this connection (which arose in Quantum Field Theory) and characterize it for families of genus two curves. We then consider a particular family (over a configuration space), the corresponding vector bundles are (almost) trivial. The Heat equations which give the connection these bundles are related to the Lie algebra so(6). We compute some local monodromy representations. As a byproduct we obtain, for any representation space V of so(2g+2), a flat connection on the trivial bundle with fiber V over a configuration space and thus a monodromy representation of a pure braid group.

alg-geom↗

arXiv1996

Modular forms on GL(3) and Galois representations

This paper gives an expository account of our experiments concerning relations between modular forms for congruence subgroups of SL(3,Z) and three dimensional Galois representations. The main new result presented here is a calculation of the variations of the Hodge structure corresponding to the motives we consider in realizing the Galois representations. It turns out that the period spaces for the Hodge structures are four dimensional, while the geometric realizations of such Hodge structures can appear in subspaces of dimension at most one.

math.NT↗

arXiv1994

On the Hitchin System

The Hitchin system is a completely integrable hamiltonian system (CIHS) on the cotangent space to the moduli space of semi-stable vector bundles over a curve. We consider the case of rank-two vector bundles with trivial determinant. Such a bundle $E$ defines a divisor $D_E$ in the Jacobian of the curve and for any smooth point of $D_E$ we define a cotangent vector (a Higgs field). The Hitchin map on these Higgs fields is then determined in terms of the Gauss map on the divisor $D_E$. We apply the results to the $g=2$ case and show how Hitchin's system is related to classical line geometry in $\PP^3$.

alg-geom↗

arXiv1992

Note on curves in a Jacobian

For a curve C, viewed as a cycle in its Jacobian, we study its image n_*C under multiplication by n on JC. We prove that the subgroup generated by these cycles, in the Chow group modulo algebraic equivalence, has rank at most d-1, where d is the gonality of C. We also discuss some general facts on the action of n_* on the Chow groups.

alg-geom↗