Torsten Ekedahl

Hurwitz numbers and intersections on moduli spaces of curves

This article is an extended version of preprint math.AG/9902104. We find an explicit formula for the number of topologically different ramified coverings of a sphere by a genus g surface with only one complicated branching point in terms of Hodge integrals over the moduli space of genus g curves with marked points.

Canonical models of surfaces of general type in positive characteristic

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On The Adic Formalism

This article is concerned with developing a formalism for complexes of l-adic sheaves instead of just for the sheaves themselves as was done in [SGA5:Exp.V]. The need for such a generalisation has become apparent from the theory of perverse sheaves, which by their definition are complexes of l-adic sheaves. When trying to carry through such an extension one is immediately faced with two problems. On the one hand it is clear already from the case of l-adic sheaves that — contrary to the case of torsion sheaves — one is not dealing with actual sheaves but rather inverse systems of sheaves. On the other hand one wants to pretend that one is dealing with sheaves and not some more abstract objects.

On hurwitz numbers and Hodge integrals

Abstract In this paper we find an explicit formula for the number of topologically different ramified coverings C → CP1 (C is a compact Riemann surface of genus g) with only one complicated branching point in terms of Hodge integrals over the moduli space of genus g curves with marked points.

Cycle classes of the E-O stratification on the moduli of abelian varieties

We introduce a stratification on the space of symplectic flags on the de Rham bundle of the universal principally polarized abelian variety in positive characteristic. We study its geometric properties, such as irreducibility of the strata, and we calculate the cycle classes. When the characteristic p is treated as a formal variable these classes can be seen as a deformation of the classes of the Schubert varieties for the corresponding classical flag variety (the classical case is recovered by putting p equal to 0). We relate our stratification with the E-O stratification on the moduli space of principally polarized abelian varieties of a fixed dimension and derive properties of the latter. Our results are strongly linked with the combinatorics of the Weyl group of the symplectic group.

SPLITTING ALGEBRAS, SYMMETRIC FUNCTIONS AND GALOIS THEORY

We present a theory for splitting algebras of monic polynomials over rings, and apply the results to symmetric functions, and Galois theory.

Cycles representing the top Chern class of the Hodge bundle on the moduli space of abelian varieties

We give upper and lower bounds for the order of the top Chern class of the Hodge bundle on the moduli space of principally polarized abelian varieties. We also give a generalization to higher genera of the famous formula

Isolated rational curves on K3-fibered Calabi–Yau threefolds

12 \lambda_1=\delta

Boundary behaviour of Hurwitz schemes

for genus 1.

The order of the top Chern class of the Hodge bundle on the moduli space of abelian varieties

Abstract:In this paper we study 16 complete intersection K3-fibered Calabi--Yau variety types in biprojective space ℙn1}×ℙ1. These are all the CICY-types that are K3 fibered by the projection on the second factor. We prove existence of isolated rational curves of bidegree (d,0) for every positive integer d on a general Calabi–Yau variety of these types. The proof depends heavily on existence theorems for curves on K3-surfaces proved by S. Mori and K. Oguiso. Some of these varieties are related to Calabi–Yau varieties in projective space by a determinantal contraction, and we use this to prove existence of rational curves of every degree for a general Calabi–Yau variety in projective space.