Moduli of Curves and Abelian Varieties: The Dutch Intercity Seminar on Moduli

by Carel Faber (Editor), Eduard Looijenga (Editor)

Overview

The present volume, with contributions of R. Dijkgraaf, C. Faber, G. van der Geer, R. Rain, E. Looijenga, and F. Oort, originates from the Dutch Intercity Seminar on Moduli (year 1995-96). Some of the articles here were discussed, in preliminary form, in the seminar; others are completely new. Two introductory papers, on moduli of abelian varieties and on moduli of curves, accompany the articles. Topics include a stratification of a moduli space of abelian varieties in positive characteristic, and the calculation of the classes of the strata, tautological classes for moduli of abelian varieties as well as for moduli of curves, correspondences between moduli spaces of curves, locally symmetric families of curves and jaco­ bians, and the role of symmetric product spaces in quantum field theory, string theory and matrix theory. This Intercity Seminar is part of the long term project "Algebraic curves and Riemann surfaces: geometry, arithmetic and applications" , sponsored hy the Netherlands Organization for Scientific Research (NWO), that has been running since 1994. Its ancestry can be traced back to joint activities in the seventies (if not earlier), which as of 1980 had evolved into active biweekly research seminars. These have been a focal point of Dutch algebraic geometry and singularity theory since. We are grateful to NWO for its support for the project. C.F. thanks the Max-Planck-Institut fur Mathematik, Bonn, for support during the final stages of the preparation of this volume.

Product Details

ISBN-13:	9783322901743
Publisher:	Vieweg+Teubner Verlag
Publication date:	11/06/2012
Series:	Aspects of Mathematics , #33
Edition description:	Softcover reprint of the original 1st ed. 1999
Pages:	200
Product dimensions:	0.00(w) x 0.00(h) x 0.02(d)

About the Author

Prof. Dr. Eduard Looijenga ist Mathematiker an der Universität Utrecht, Niederlande.

Dr. Carel Faber ist jetzt am Max-Planck-Institut für Mathematik in Bonn tätig.

Table of Contents

Moduli of Abelian Varieties: A Short Introduction and Survey.- 1 Introduction.- 2 Elliptic Curves.- 3 Abelian Varieties.- 4 The Torelli Morphism.- 5 Cycles on the Moduli Space of Abelian Varieties.- 6 The Moduli Space Ag in Positive Characteristic.- 7 The Moduli Space in Mixed Characteristic.- Remarks on Moduli of Curves.- 1 Introduction.- 2 Mapping Class Groups.- 3 The Torelli Group.- 4 Moduli Spaces of Curves.- 5 Deligne-Mumford-Knudsen Completion.- 6 Covers of Moduli Stacks.- 7 Tautological Classes.- 8 Stability.- 9 A Proarithmetic Hull of the Mapping Class Group.- 10 The Witten Conjecture.- 11 Complete Subvarieties of Moduli Spaces.- 12 Intersection Theory.- 13 Stable Maps and the Virasoro Conjecture.- A Stratification of a Moduli Space of Polarized Abelian Varieties in Positive Characteristic.- 1 Introduction.- 2 Elementary Sequences and Final Filtrations.- 3 Explicit Description of Some of the Strata.- 4 Standard Types.- 5 Moving in a Stratum.- 6 The Raynaud Trick.- 7 Results.- 8 Some Questions.- Cycles on the Moduli Space of Abelian Varieties.- 1 Introduction.- 2 The Tautological Subring of Ag.- 3 The Tautological Ring of A? g.- 4 Cycle Relations from Grothendieck-Riemann-Roch.- 5 On the Torsion of the Class— g.- 6 The Ekedahl-Oort Stratification in Positive Characteristic.- 7 The Ekedahl-Oort Strata as Degeneracy Loci.- 8 The Degeneracy Locus Uø and—0(p).- 9 The Cycle Classes of the Strata.- 10 Effectivity of Tautological Classes.- 11 Some Additional Results.- Locally Symmetric Families of Curves and Jacobians.- 1 Introduction.- 2 Background and Definitions.- 3 Homomorphisms to Mapping Class Groups.- 4 Maps of Lattices to Mapping Class Groups.- 5 Locally Symmetric Hypersurfaces in Locally Symmetric Varieties.- 6 Geometry of the Jacobian Locus.- 7 LocallySymmetric Families of Curves.- 8 Locally Symmetric Families of Jacobians.- 9 Appendix: An Example.- A Conjectural Description of the Tautological Ring of the Moduli Space of Curves.- 1 Introduction.- 2 Known Results.- 3 The Conjectures and the Evidence.- 4 Calculations.- 5 Other Relations.- Correspondences between Moduli Spaces of Curves.- 1 Introduction.- 2 Correspondences for Mapping Class Groups: Closed Surfaces.- 3 Action of the Correspondences on the Tautological Classes.- 4 Hecke Operators Attached to Finite Abelian Covers.- 5 Correspondences via Ramified Covers.- 6 Correspondences Acting on a Lie Algebra.- Fields, Strings, Matrices and Symmetric Products.- 1 Introduction.- 2 Particles, Symmetric Products and Fields.- 3 Second-quantized Strings.- 4 Light-Cone Quantization of Quantum Field Theories.- 5 Light-Cone Quantization of String Theories.- 6 Matrix Strings and Interactions.- 7 String Theories in Six Dimensions.- Addresses.