Inertia groups and abelian surfaces

This paper classifies the finite groups that occur as inertia groups associated to abelian surfaces. These groups can be viewed as Galois groups for the smallest totally ramified extension over which an abelian surface over a local field acquires semistable reduction. The results extend earlier elliptic curves results of Serre and Kraus.     

Sheaves on abelian surfaces and strange duality

We formulate three versions of a strange duality conjecture for sections of the Theta bundles on the moduli spaces of sheaves on abelian surfaces. As supporting evidence, we check the equality of dimensions on dual moduli spaces, answering a question raised by Göttsche et al. (K-theoretic Donaldson invariants via instanton counting. arXiv:math/0611945).     

Singular abelian covers of algebraic surfaces

In this paper we generalise F. Cataneses work on singular (/2)²-covers to arbitrary finite abelian covers of algebraic surfaces. We determine the contribution of singularities to the invariants , K ², q, p _g , P _n and the canonical sheaf. We use these computations to construct a surface of general type with birational canonical map ₁, p _g =4 and K ² =31. Mathematics Subject Classification (2000):14J29, 14J17, 14J25, 14E20     

Nef cone volumes and discriminants of abelian surfaces

The nef cone volume appeared first in work of Peyre in a number-theoretic context on Fano varieties, and was then studied by Derenthal and co-authors in a series of papers on del Pezzo surfaces. The idea was subsequently extended to also measure the Zariski chambers of del Pezzo surfaces. We start in this paper to explore the possibility to use this attractive concept to effectively measure the size of the nef cone on algebraic surfaces in general. This provides an interesting way of measuring in how big a space an ample line bundle can be moved without destroying its positivity. We give here complete results for simple abelian surfaces that admit a principal polarization and for products of elliptic curves.     

Primitive higher order embeddings of abelian surfaces

In recent years several concepts of higher order embeddings have been studied: -spannedness, -very ampleness and -jet ampleness. In the present note we consider primitive line bundles on abelian surfaces and give numerical criteria which allow to check whether a given ample line bundle satisfies these properties.

     

Embeddings of general blowups of abelian surfaces

Blowups of algebraic surfaces polarized by tensor powers of ample line bundles were studied by Coppens [8]. In this note we complete the picture in the case of abelian surfaces studying blowups of surfaces polarized by primitive line bundles. Received: 16 November 1999 / Revised version: 13 June 2000     

A note on general blow-ups of abelian surfaces

In the paper we give numerical conditions for a line bundle on a general blow-up of abelian surface to be k-very ample.     

Seshadri constants in finite subgroups of abelian surfaces

Given an étale quotient q : X → Y of smooth projective varieties we relate the simple Seshadri constant of a line bundle M on Y with the multiple Seshadri constant of q ^* M in the points of the fiber. We apply this method to compute the Seshadri constant of polarized abelian surfaces in the points of a finite subgroup.      

Torsion points on some abelian surfaces of CM-type

Let A be a two-dimensional abelian variety of CM-type defined over Q, which is not simple over C. Let p be a prime number. We show that torsion points of A(Q) of prime order p are possible only for p≦7.     

Primes of superspecial reduction for QM abelian surfaces

We show that any abelian surface with multiplication by thequaternion -algebra of discriminant 6, with field of moduli and which is a Jacobian in characteristic 2 and 3, has infinitelymany primes of superspecial reduction. This is done by examiningcomplex multiplication points in characteristic 0 and p andthe values of a certain j-function on the associated modulispace at these points.     

Deformations of equivelar Stanley–Reisner abelian surfaces

The versal deformation of Stanley–Reisner schemes associated to equivelar triangulations of the torus is studied. The deformation space is defined by binomials and there is a toric smoothing component which I describe in terms of cones and lattices. Connections to moduli of abelian surfaces are considered. The case of the Möbius torus is especially nice and leads to a projective Calabi–Yau 3-fold with Euler number 6.     

Moduli spaces of stable sheaves on abelian surfaces

In this paper, we consider basic problems on moduli spaces of stable sheaves on abelian surfaces. Our main assumption is the primitivity of the associated Mukai vector. We determine the deformation types, albanese maps, Bogomolov factors and their weight 2 Hodge structures. We also discuss the deformation types of moduli spaces of stable sheaves on K3 surfaces. Received: 28 February 2000 / Revised version: 15 September 2000 / Published online: 24 September 2001     

Modular curves and Poncelet polygons

Parts of the results and the essential techniques of this note are taken from the Erlangen thesis (1991) of the second author. They were circulated as Nr. 122 of Schriftenreihe Komplexe Mannigfaltigkeiten. Our research was supported by DFG grant Ba 423/3-3 and the European Science Project Geometry of Algebraic Varieties SCI-0398-C(A)