G.B.M. van der Geer

On the geometry of a Siegel modular threefold

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 1. Compactifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 2. Humbert Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 3. The Chern Numbers of S~(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 4. Symmetric Quartic Threefolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 5. The Modular Threefold of Level 2 . . . . . . . . . . . . . . . . . . . . . . . . . 335 6. Kummer Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 7. Prym Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 8. Humbert Surfaces Again . . . . . . . . . . . . . . . . : . . . . . . . . . . . . 345

Effectivity of Arakelov divisors and the theta divisor of a number field

We introduce the notion of an effective Arakelov divisor for a number field and the arithmetical analogue of the dimension of the space of sections of a line bundle. We study the analogue of the theta divisor for a number field.

On generalized Hamming weights of BCH codes

Introduces methods from algebraic geometry to determine generalized Hamming weights of BCH codes. As an application of these methods the authors determine for primitive 2- (resp., 3-) error correcting BCH codes the third (resp., the second) generalized Hamming weight. >

Generalized Hamming weights of BCH(3) revisited

Determines the first five generalized Hamming weights of 3-error-correcting primitive binary BCH-codes. >

Formal Brauer Groups and Moduli of Abelian Surfaces

LetXbe an algebraic surface over an algebraically closed fieldkof characteristicp >0. We denote by Фx the formal Brauer group ofXand byh = h(Ф x )the height of Фx. In a previous paper, [6], we examined the structure of the stratification given by the heighthin the moduli space of K3 surfaces, and we determined the cycle class of each stratum. We also showed that the final stratum is non-reduced. In this paper, we use the methods of [6] to treat the case of abelian surfaces. In this case, the situation is more concrete, and so we can more easily determine the structure of the stratification given by the height h(Ф.A) in the moduli of abelian surfaces. For the local structure we refer to [20].

Families of abelian surfaces with real multiplication over Hilbert modular surfaces

Around the beginning of this century G. Humbert ([9]) made a detailed study of the properties of compact complex surfaces which can be parametrized by singular abelian functions. A surface parametrized by singular abelian functions is the image under a holomorphic map of a singular abelian surface (i.e. an abelian surface whose endomorphism ring is larger than the ring of rational integers). Humbert showed that the periods of a singular abelian surface satisfy a quadratic relation with integral coefficients and he constructed an invariant D of such a relation with respect to the action of the integral symplectic group on the periods.