Arrangements of Hyperplanes

336 Pages·1992·5.79 MB·English

by  Peter Orlik

Grundlehren der
mathematischen Wissenschaften 300
A Series o/Comprehensive Studies in Mathematics
Editors
M. Artin S. S. Chern J. Coates 1. M. Frohlich
H. Hironaka F. Hirzebruch L. Hormander
C. C. Moore 1. K. Moser M. Nagata W Schmidt
D. S. Scott Ya. G. Sinai 1. Tits M. Waldschmidt
S. Watanabe
Managing Editors
M. Berger B. Eckmann S. R. S. Varadhan
Peter Orlik Hiroaki Terao
Arrangements
of Hyperplanes
With 43 Figures
Springer-Verlag Berlin Heidelberg GmbH
Peter Orlik Hiroaki Terao
Department of Mathematics
University of Wisconsin
Madison, WI 53706, USA
Mathematics Subject Classification
05B35,32S25,57N65, 14F35, 14F40,20F36,20F55
ISBN 978-3-642-08137-8 ISBN 978-3-662-02772-1 (eBook)
DOI 10.1007/978-3-662-02772-1
Library of Congress Cataloging-in-Publication Data
Orlik, Peter, 1938-
Arrangements of hyperplanes / Peter Orlik, Hiroaki Terao.
p. cm. - (Grundlehren der mathematischen Wissenschaften ; 300)
Includes bibliographical references and index.
I. Combinatorial geometry. 2. Combinatorial enumeration problems.
3. Lattice theory. I. Terao, Hiroaki, 1951- . II. Title. III. Series.
QA167.067 1992 516'.13-dc20 92-6674
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© Springer-Verlag Berlin Heidelberg 1992
Originally published by Springer-Verlag Berlin Heidelberg New York in 1992.
Softcover reprint of the hardcover I st edition 1992
Data conversion: EDV-Beratung K. Mattes, Heidelberg
Typesetting output: Type 2000, Mill Valley, California, USA
41/3140-543210 -Printed on acid-free paper
To our parents
Preface
An arrangement of hyperplanes is a finite collection of codimension one affine
subspaces in a finite dimensional vector space. In this book we study arrange
ments with methods from combinatorics, algebra, algebraic geometry, topology,
and group actions. These first sentences illustrate the two aspects of our subject
that attract us most. Arrangements are easily defined and may be enjoyed at
levels ranging from the recreational to the expert, yet these simple objects lead
to deep and beautiful results. Their study combines methods from many areas
of mathematics and reveals unexpected connections.
The idea to write a book on arrangements followed three semesters of lec
tures on the topic at the University of Wisconsin, Madison. Louis Solomon
lectured on the combinatorial aspects in the fall of 1981. Peter Orlik continued
the course with the topological properties of arrangements in the spring of 1982.
Hiroaki Terao visited Madison for the academic year 1982-83 and gave a course
on free arrangements in the fall of 1982. The original project was to enlarge the
lecture notes into a book and a long but incomplete manuscript was typed up
in 1984.
The book was revived in 1988 when P. Orlik was invited to give a CBMS
lecture series on arrangements. The resulting lecture notes provided a core for
the present work. Also in 1988, H. Terao gave a course on arrangements at
Ohio State University. Parts of his lectures are used in this book. The fresh
start allowed us to take advantage of the recent 1EXnical innovations of word
processing and computer-aided typesetting.
P. Orlik had the opportunity to lecture on arrangements at the Swiss Sem
inar in Bern. He wishes to thank the mathematicians in Geneva for their hospi
tality and the participants of the seminar for many helpful comments. H. Terao
gave a graduate course on arrangements at International Christian University in
Tokyo in the fall of 1989. He wishes to thank all the participants of the course.
Time constraints forced L. Solomon to leave the completion to us. Much of
the material in the book is his joint work with one or both of us, and a large
part of Chapter 2 was written by him. We are grateful for permission to use his
work, and for his support and friendship.
We thank W. Arvola for permission to include a modified version of his
presentation of the fundamental group of the complement, P. Edelman and V.
Reiner for permission to include their example, and M. Falk and R. Randell for
help on many technical points. In addition, we owe thanks to C. Greene, L. Paris,
VIII Preface
M. Salvetti, S. Yuzvinsky, T. Zaslavsky, and G. Ziegler for valuable suggestions,
to V. 1. Arnold for references on the M-property, and to N. Spaltenstein for
references on connections with representation theory.
We thank A. B. Orlik for editing and proofreading the manuscript.
University of Wisconsin P. Orlik and H. Terao
February 1992
Table of Contents
1. Introduction 1
1.1 Introduction. 1
History .... 1
Recent Advances 8
1.2 Definitions and Examples 10
Examples · ........ 11
Basic Constructions. . . . 13
The Module of A-Derivations 15
The Complement of a Complex Arrangement . 16
Reflection Arrangements 16
1.3 Outline ..... 17
Combinatorics . . . . . . 17
Algebras ......... 17
The Module of A-Derivations 18
Topology ......... 19
Reflection Arrangements 20
2. Combinatorics 23
2.1 The Poset L(A) 23
Definitions . . . 23
Examples · .. 26
Oriented Matroids 28
Supersolvable Arrangements 30
2.2 The Mobius Function . 32
The Mobius Function. 32
Mobius Inversion 34
The Function J1(X) .. 35
Historical Notes .... 38
2.3 The Poincare Polynomial . 42
Examples · ........ 42
The Deletion-Restriction Theorem 46
Supersolvable Arrangements 48
Nice Partitions .. 50
Counting Functions . . . . . 51
X Table of Contents
2.4 Graphic Arrangements 52
Definitions . . . . . . 52
Deletion-Contraction 54
Acyclic Orientations 57
3. Algebras ........ . 59
3.1 A(A) for Central Arrangements 60
Construction of A(A) . 60
An Acyclic Complex . . . . . . 62
The Structure of A(A) ..... 63
The Injective Map A(Ax) --t A(A) 65
The Broken Circuit Basis. . . 67
3.2 A(A) for Affine Arrangements 70
Construction of A(A) ... 70
The Broken Circuit Basis . 72
Deletion and Restriction 74
The Structure of A(A) 77
A-equivalence . . . . . . 78
3.3 Algebra Factorizations 79
Supersolvable Arrangements 80
Nice Partitions of Central Arrangements 82
Nice Partitions of Affine Arrangements 85
3.4 The Algebra B(A) 86
The Shuffle Product .... . 86
The Algebra B(A) ..... . 88
The Isomorphism of B and A 89
3.5 Differential Forms .... 92
The de Rham Complex . 92
The Algebra R(A) ... 93
Deletion and Restriction 95
The Isomorphism of R and A 97
4. Free Arrangements . . . 99
4.1 The Module D(A) 100
Derivations .... 100
Basic Properties. . 102
4.2 Free Arrangements 104
Saito's Criterion. 104
Exponents .... . 107
Examples .... . 111
4.3 The Addition-Deletion Theorem. 113
Basis Extension . . . . . . . . . . 114
The Map from D(A) to D(A") 115
The Addition-Deletion Theorem. 117
Inductively Free Arrangements. 119
Supersolvable Arrangements . . . 121
Table of Contents XI
Factorization Theorem 122
4.4 The Modules np(A) 123
Definition of np(A) .. 124
Basic Properties of np(A) 124
The Acyclic Complex (.f.?"(A), 0) . 133
The ry-Complex (n·(A), Oh) 133
4.5 Lattice Homology ... 135
The Order Complex. . 136
The Folkman Complex 137
The Homology Groups 140
The Homotopy Type . 141
Whitney Homology . . 142
Connection with the Folkman Complex . 144
4.6 The Characteristic Polynomial . . . 145
The Order Complex with Functors 145
Local Functors ..... . 147
The Homology Hp(A, F) .. 148
The Polynomial IJi(A, x, t) 150
The Factorization Theorem 154
5. Topology .......... . 157
5.1 The Complement M(A) 158
K (7 f, I)-Arrangements 159
Free Arrangements . . 163
Generic Arrangements 164
Deformation . . . . . . 166
Arnold's Conjectures . 167
5.2 The Homotopy Type of M(A) 168
Real Arrangements . . . . . . 168
The Homotopy Type . . . . . 171
Complexified Real Arrangements 173
Salvetti's Complex . . . . . 175
The Homotopy Equivalence 176
5.3 The Fundamental Group 177
Admissible Graphs . . . . 179
Arvola's Presentation . . . 184
5.4 The Cohomology of M(A) 190
The Thorn Isomorphism 191
Brieskorn's Lemma ... 195
5.5 The Fibration Theorem. 196
Horizontal Subspaces 197
Good Subspaces . 198
Good Lines ... 199
5.6 Related Research 202
Minimal Models . 202
XII Table of Contents
Discriminantal Arrangements ....... . 205
Alexander Duality ............. . 207
The Milnor Fiber of a Generic Arrangement 209
Arrangements of Subspaces 211
6. Reflection Arrangements 215
6.1 Equivariant Theory 216
The Action of G. . . 216
Matrices ...... . 218
Character Formulas . 219
Topological Interpretation 222
6.2 Reflection Arrangements 223
Basic Properties. . 223
Examples ....... . 225
Relative Invariants . . . 228
Jacobian and Discriminant 229
Classification ... 231
6.3 Free Arrangements 232
Invariant Theory 232
The Hessian . 234
DR(8) Is Free .. 235
D(A) Is Free .. 237
The Discriminant Matrix . 238
A Character Formula . 241
6.4 The Structure of L(A) .. 243
The Symmetric Group .. 243
The Full Monomial Group 244
The Monomial Group G(r, r, £) 247
The Exceptional Groups 251
6.5 Restrictions .......... . 254
The Cardinality of AH .... . 254
AH Is Free In Coxeter Arrangements 256
6.6 Topology ............. . 259
Stratification of the Discriminant 259
Shephard Groups . . 265
The K(w, 1) Problem. 267
A. Some Commutative Algebra 271
A.l Free Modules . . 271
A.2 Krull Dimension. . . . 272
A.3 Graded Modules. . . . 274
A.4 Associated Primes and Regular Sequences 276