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

204 Pages·1999·9.88 MB·English

by  Gerard van der Geer

Carel Faber
Eduard Looijenga (Eds.)
Moduli of Curves and
Abelian Varieties
Aspects of Mathematics
Edited by Klas Diederich
Vol. E 3: G. Hector/ U. Hirsch: Introduction to the Geometry of Foliations, Part B
Vol. E 6: G. Faltings/G. Wustholz et al.: Rational Points*
Vol. E 9: A. Howard/P.-M. Wong (Eds.): Contribution to Several Complex Variables
Vol. E 15: J .-P Serre: Lectures on the Mordell-Weil Theorem
Vol. E 16: K. Iwasaki/H. Kimura/S. Shimomura/M. Yoshida: From Gauss to Painleve
Vol. E 17: K. Diederich (Ed.): Complex Analysis
Vol. E 18: W. W. J . Hulsbergen: Conjectures in Arithmetic Algebraic Geometry
Vol. E 19: R. Racke: Lectures on Nonlinear Evolution Equations
Vol. E 20: F. Hirzebruch/Th. Berger /R. Jung: Manifolds and Modular Forms*
Vol. E 21: H. Fujimoto: Value Distribution Theory of the Gauss Map of Minimal
Surfaces in Rm
Vol. E 22: D. V. Anosov / A. A. Bolibruch: The Riemann-Hilbert Problem
Vol. E 24: D. S. Alexander: A History of Complex Dynamics
Vol. E 25: A. Tikhomirov / A. Tyurin (Eds.): Algebraic Geometry and its Applications
Vol. E 27: D. N. Akhiezer: Lie Group Actions in Complex Analysis
Vol. E 28: R. Gerard/H. Tahara: Singular Nonlinear Partial Differential Equations
Vol. E29: R.-P. Holzapfel: Ball and Surface Arithmetics
Vol. E 30: R. Huber: Etale Cohomology of Rigid Analytic Varieties and Adic Spaces
Vol. E 31: D. Huybrechts/M. Lehn: The Geometry of Moduli Spaces of Sheaves*
Vol. E 32: M. Yoshida: Hypergeometric Functions, My Love
Vol. E 33: C. Faber/E. Looijenga (Eds.): Moduli of Curves and Abelian Varieties
* A Publication of the Max-Planck-Institut fur Mathematik, Bonn
Carel Faber
Eduard Looijenga
(Eds. )
Moduli of Curves and
Abelian Varieties
The Dutch Intercity Seminar on Moduli
~
vleweg
Prof. Dr. Eduard Looijenga
Dept. uf Mathematics
University of Utrecht
PO Box 80. 010
NL-3508 TA Utrecht
Dr. Carel Faber
Dept. of Mathematics
Oklahoma State University
Stillwater, OK 74078-1058
USA
and
Dept. of Mathematics
Royal Institute of Technology
5-100 44 Stockholm
Prof. Dr. Klas Diederich (Series Editor)
Dept. of Mathematics
University of Wuppertal
0-42119 Wuppertal
AI I rig h ts rese rved
© Fricdr. Vieweg & Sohn Verlagsgesel\schaft mbH, Braunschweig/ Wiesbaden, 1999
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ISBN 978-3-322-90174-3 ISBN 978-3-322-90172-9 (eBook)
DOI 10.1007/978-3-322-90172-9
v
Preface
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.
Bonn/Utrecht, May 1999
Carel Faber
Eduard Looijenga
Contents
Preface v
Moduli of Abelian Varieties: A Short Introduction and Survey 1
Gerard van der Geer and Frans Dart
1 Introduction. . 1
2 Elliptic Curves . . . . 2
3 Abelian Varieties ... 4
4 The Torelli Morphism 9
5 Cycles on the Moduli Space of Abelian Varieties 12
6 The Moduli Space Ag in Positive Characteristic 14
7 The Moduli Space in Mixed Characteristic 17
Remarks on Moduli of Curves 23
Carel Faber and Eduard Laaijenga
1 Introduction ..... 23
2 Mapping Class Groups. 24
3 The Torelli Group . . . 25
4 Moduli Spaces of Curves. 26
5 Deligne-Mumford-Knudsen Completion 27
6 Covers of Moduli Stacks 28
7 Tautological Classes . . . . . . . . . . . 29
8 Stability. . . . . . . . . . . . . . . . . . 30
9 A Proarithmetic Hull of the Mapping Class Group 31
10 The Witten Conjecture ... . . . . . . 32
11 Complete Subvarieties of Moduli Spaces . 33
12 Intersection Theory ........... . 37
13 Stable Maps and the Virasoro Conjecture 39
A Stratification of a Moduli Space of Polarized Abelian Varieties in
Positive Characteristic 47
Frans Dart
1 Introduction 47
2 Elementary Sequences and Final Filtrations . 48
3 Explicit Description of Some of the Strata . . 52
Contents Vll
4 Standard Types 5:3
5 Moving in a Stratum. 54
6 The Raynaud Trick 56
7 Results ..... 59
8 Some Questions 60
Cycles on the Moduli Space of Abelian Varieties 65
Gerard van der Geer
1 Introduction ........... 65
2 The Tautological Subring of Ag . 66
Ag ..
3 The Tautological Ring of 70
4 Cycle Relations from Grothendieck-Riemann-Roch 72
5 On the Torsion of the Class).,g . . . . . . . . . . . 76
6 The Ekedahl-Oort Stratification in Positive Characteristic 76
7 The Ekedahl-Oort Strata as Degeneracy Loci 78
8 The Degeneracy Locus U0 and fo(p) 82
9 The Cycle Classes of the Strata. . 83
10 Effectivity of Tautological Classes 87
11 Some Additional Results. . . . . . 88
Locally Symmetric Families of Curves and Jacobians 91
Richard Hain
1 Introduction ........ . . . . . . . 91
2 Background and Definitions . . . . . . . 94
3 Homomorphisms to Mapping Class Groups 96
4 Maps of Lattices to Mapping Class Groups 97
5 Locally Symmetric Hypersurfaces in Locally Symmetric Varieties 100
6 Geometry of the Jacobian Locus . . . . . 100
7 Locally Symmetric Families of Curves . . 103
8 Locally Symmetric Families of Jacobians 104
9 Appendix: An Example . . . . . . . . . . 106
A Conjectural Description of the Tautological Ring of the
Moduli Space of Curves 109
Carel Faber
1 Introd uction 109
2 Known Results .......... . 110
3 The Conjectures and the Evidence 114
4 Calculations .. 119
5 Other Relations ......... . 125
Correspondences between Moduli Spaces of Curves 131
Eduard Looijenga
1 Introduction ...... . . . . . . . . . . . . . . . . . . . .. 131
2 Correspondences for Mapping Class Groups: Closed Surfaces 133
V III Contents
3 Action of the Correspondences on the Tautological Classes 1:37
4 Hecke Operators Attached to Finite Abelian Covers 139
5 Correspondences via Ramified Covers . . 141
6 Correspondences Acting on a Lie Algebra 148
Fields, Strings, Matrices and Symmetric Products 151
Robbert DijkgraaJ
1 Introduction .............. 151
2 Particles, Symmetric Products and Fields 155
3 Second-quantized Strings . . . . . . . . . 163
4 Light-Cone Quantization of Quantum Field Theories . 173
5 Light-Cone Quantization of String Theories 179
6 Matrix Strings and Interactions. . 188
7 String Theories in Six Dimensions . . . . . 194
Addresses 200
Moduli of Abelian Varieties: A Short
Introduction and Survey
Gerard van der Geer and Frans Dart
Abstract
We give a short introduction to the moduli of abelian varieties as well as a survey of
recent research and open questions.
1 Introduction
The term moduli was introduced in 1857 by Riemann to describe the essential
parameters of Riemann surfaces of given genus. The use of the term has since
then been extended to a name for the essential parameters of algebraic varieties
(of a given type). Moduli spaces are playing an increasingly important role in
mathematics and since a number of years also in mathematical physics. Not only
can we learn about families of algebraic varieties of a certain type by studying
their moduli, but these moduli spaces are themselves most interesting mathe
matical objects with intriguing arithmetic properties. Often moduli spaces can
be understood much better than arbitrary varieties because their invariants and
properties have an intrinsic meaning, which makes them much more amenable to
treatment; certain deep conjectures about algebraic varieties can be proved for
certain moduli spaces, but still defy verification in general.
The moduli spaces most studied so far are the moduli of elliptic curves.
That they possess fascinating arithmetic properties was recognized in the late
19th century and by now they have become a central object of study in number
theory.
An elliptic curve is a (regular, complete, absolutely irreducible) curve of
genus 1 together with a marked point. It can be given as a regular cubic curve in
the projective plane. The crucial point is that these curves carry the structure of
C. Faber et al. (eds.), Moduli of Curves and Abelian Varieties
© Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden 1999
2 Gerard van der Geer and Frans Oort
a group variety, obtained by identifying the points of the curve with the group of
divisor classes of degree 0 on the curve.
The concept 'elliptic curve' admits two natural generalizations, depending on
whether we stick to dimension 1 or want to retain the notion of a group variety:
• a curve of genus g,
• an abelian variety of dimension g.
These generalizations lead to two types of moduli spaces: the moduli spaces Mg of
curves of genus 9 and the (various) moduli spaces of (polarized) abelian varieties.
These spaces have been the focus of intense study in the last decades. As a result
we learned a lot about them, though it seems fair to say that our understanding
of them is still very limited and that many basic and important questions are
still open.
Although Riemann introduced the concept of moduli already in the middle
of the 19th century and although the moduli of curves were studied by Ital
ian geometers around the turn of the century, the concept was first put on a
firm foundation around 1960 by Grothendieck, who introduced the notion of rep
resentability of a functor. This made it possible to give a precise meaning to
'moduli'. The existence of moduli spaces in this sense for curves and polarized
abelian varieties was shown in the sixties by Mumford in his book [45].
2 Elliptic Curves
Let us illustrate these concepts for elliptic curves. An elliptic curve E over a field
is a complete, geometrically irreducible regular curve of genus 1 with a marked
point. If P is the marked point then the linear system defined by 3P defines an
embedding of the curve E into the projective plane and the image is given as
a smooth plane cubic curve; with the image of P equal to the point at infinity
(0: 1 : 0) we may write the equation for E - {P} in affine form as
where the discriminant (a polynomial expression in the coefficients ai) does not
vanish. Conversely, any such regular plane cubic curve with the point P defines
an elliptic curve. For the theory of elliptic curves we refer to [64]. Two such curves
over an algebraically closed field k are isomorphic if and only if their j-invariants
(a certain rational expression in the ai) are the same: E ~ E' {=:::> j(E) = j(E').
So the isomorphism classes of such curves over k are in 1 - 1 correspondence with
the points of the affine line A,.1, the j-line. However, ideally one would like to have
much more, namely a universal family of elliptic curves X -r Al such that for
any (good) family of elliptic curves X -r S there is a morphism v : S -r Al such
=
that X v*(X).