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

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer­Verlag. Violations are liable for prosecution under the German Copyright Law.

© 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 arrange­ments 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 lec­tures 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 Sem­inar in Bern. He wishes to thank the mathematicians in Geneva for their hospi­tality 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 February 1992

P. Orlik and H. Terao

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 Definitions . . . . . . Deletion-Contraction Acyclic Orientations

3. Algebras ........ . 3.1 A(A) for Central Arrangements

Construction of A(A) . An Acyclic Complex . . . . . . The Structure of A(A) ..... The Injective Map A(Ax) --t A(A) The Broken Circuit Basis. . .

3.2 A(A) for Affine Arrangements Construction of A(A) ... The Broken Circuit Basis . Deletion and Restriction The Structure of A(A) A-equivalence . . . . . .

3.3 Algebra Factorizations Supersolvable Arrangements Nice Partitions of Central Arrangements Nice Partitions of Affine Arrangements

3.4 The Algebra B(A) The Shuffle Product .... . The Algebra B(A) ..... . The Isomorphism of B and A

3.5 Differential Forms .... The de Rham Complex . The Algebra R(A) ... Deletion and Restriction The Isomorphism of R and A

4. Free Arrangements . . . 4.1 The Module D(A)

Derivations .... Basic Properties. .

4.2 Free Arrangements Saito's Criterion. Exponents .... . Examples .... .

4.3 The Addition-Deletion Theorem. Basis Extension . . . . . . . . . . The Map from D(A) to D(A") The Addition-Deletion Theorem. Inductively Free Arrangements. Supersolvable Arrangements . . .

52 52 54 57

59 60 60 62 63 65 67 70 70 72 74 77 78 79 80 82 85 86 86 88 89 92 92 93 95 97

99 100 100 102 104 104 107 111 113 114 115 117 119 121

Factorization Theorem 4.4 The Modules np(A)

Definition of np(A) .. Basic Properties of np(A) The Acyclic Complex (.f.?"(A), 0) . The ry-Complex (n·(A), Oh)

4.5 Lattice Homology ... The Order Complex. . The Folkman Complex The Homology Groups The Homotopy Type . Whitney Homology . . Connection with the Folkman Complex .

4.6 The Characteristic Polynomial . . . The Order Complex with Functors Local Functors ..... . The Homology Hp(A, F) .. The Polynomial IJi(A, x, t) The Factorization Theorem

5. Topology .......... . 5.1 The Complement M(A)

K ( 7f, I)-Arrangements Free Arrangements . . Generic Arrangements Deformation . . . . . . Arnold's Conjectures .

5.2 The Homotopy Type of M(A) Real Arrangements . . . . . . The Homotopy Type . . . . . Complexified Real Arrangements Salvetti's Complex . . . . . The Homotopy Equivalence

5.3 The Fundamental Group Admissible Graphs . . . . Arvola's Presentation . . .

5.4 The Cohomology of M(A) The Thorn Isomorphism Brieskorn's Lemma ...

5.5 The Fibration Theorem. Horizontal Subspaces Good Subspaces . Good Lines ...

5.6 Related Research Minimal Models .

Table of Contents XI

122 123 124 124 133 133 135 136 137 140 141 142 144 145 145 147 148 150 154

157 158 159 163 164 166 167 168 168 171 173 175 176 177 179 184 190 191 195 196 197 198 199 202 202

XII Table of Contents

Discriminantal Arrangements ....... . Alexander Duality ............. . The Milnor Fiber of a Generic Arrangement Arrangements of Subspaces

6. Reflection Arrangements 6.1 Equivariant Theory

The Action of G. . . Matrices ...... . Character Formulas . Topological Interpretation

6.2 Reflection Arrangements Basic Properties. . Examples ....... . Relative Invariants . . . Jacobian and Discriminant Classification ...

6.3 Free Arrangements Invariant Theory The Hessian . DR(8) Is Free .. D(A) Is Free .. The Discriminant Matrix . A Character Formula .

6.4 The Structure of L(A) .. The Symmetric Group .. The Full Monomial Group The Monomial Group G(r, r, £) The Exceptional Groups

6.5 Restrictions .......... . The Cardinality of AH .... . AH Is Free In Coxeter Arrangements

6.6 Topology ............. . Stratification of the Discriminant Shephard Groups . . The K(w, 1) Problem.

A. Some Commutative Algebra A.l Free Modules . . A.2 Krull Dimension. . . . A.3 Graded Modules. . . . A.4 Associated Primes and Regular Sequences

205 207 209 211

215 216 216 218 219 222 223 223 225 228 229 231 232 232 234 235 237 238 241 243 243 244 247 251 254 254 256 259 259 265 267

271

271

272 274 276

B. Basic Derivations . . . . . . . . . . B.1 The Infinite Families ..... B.2 Exceptional Groups of Rank 2 B.3 Exceptional Groups of Rank ~ 3 B.4 The Coexponents

C. Orbit Types

D. Three-Dimensional Restrictions

References

Index ...

Index of Symbols

Table of Contents XIII

279 279 280 280 286

289

301

303

315

323

List of Figures

1.1 An illustration of chamber counting . . . . . . 1.2 Q(A) = xy(x + y) and Q(A) = xy(x + y - 1) 1.3 The B3-arrangement . . . . . . . . . . . . .

2.1 The Hasse diagram of Q(A) = xy(x + y) .. 2.2 The Hasse diagram of Q(A) = xy(x + y - 1) 2.3 The Hasse diagram of the B3-arrangement 2.4 The chambers of Q(A) = xy(x + y - 1) . 2.5 The face poset of Q(A) = xy(x + y - 1) 2.6 Different face posets ........... . 2.7 The values of J.1.(X) for the B3-arrangement 2.8 1l'-equivalent but not L-equivalent . 2.9 A graph with three vertices 2.10 A complete graph .... 2.11 Deletion and contraction 2.12 An oriented graph .... 2.13 Not acyclic orientations.

3.1 A-equivalent but not L-equivalent

4.1 Free but not inductively free .... 4.2 Subdivision of ..12 x I . . . . . . . . 4.3 Folkman complexes for Q = xyz(x + y)(x + y - z) . 4.4 Complexes for the Boolean arrangement 4.5 1l'(A, t) factors, but A is not free .

5.1 A braid on three strands ... 5.2 A pure braid on three strands 5.3 The generator ai ...... .

5.4 K(1l', 1), but not free .... . 5.5 Three lines in general position 5.6 The critical half-line ..... 5.7 Dual cells . . . . . . . . . . . 5.8 Three concurrent lines in IR? and in (C2

5.9 A 3-graph . . . . . . . . . . . . . . . .

4 12 13

25 25 25 28 29 30 37 48 52 53 55 57 58

79

122 137 138 139 155

160 160 161 164 166 169 171 174 181

XVI List of Figures

5.10 A flat graphing map 5.11 An admissible graph 5.12 Generators .... 5.13 Loop passing .. 5.14 A pencil of lines. 5.15 Linking ..... 5.16 The arrangement C*(5) 5.17 Falk's Linking .....

6.1 The lattice £ = £(1) ..

6.2 The lattices £(12), £(123), £(12)(34), £(1234)

6.3 The Hessian configuration 6.4 A tetrahedron in the cube . . . . . . . .

182 184 186 187 188 189 206 208

222 222 227 268

List of Tables

4.1 Induction table .... 4.2 The braid arrangement

5.1 A subspace arrangement

6.1 Poincare polynomials 6.2 Restrictions . . . . . 6.3 Orbits in G25 •.••

6.4 Induction table for G25

6.5 The associated pairs (G, W)

B.1 Exponents and coexponents B.2 The monomial groups.

C.1 The rank 2 groups (I) . C.2 The rank 2 groups (II) C.3 Orbits in GI5

C.4 Orbits in H3 .

C.5 Orbits in G24

C.6 Orbits in G25

C.7 Orbits in G26

C.8 Orbits in G27

C.9 Orbits in F4 . C.lO Orbits in G29

C.11 Orbits in H4 . C.12 Orbits in G31

C.13 Orbits in G32

C.14 Orbits in G33

C.15 Orbits in G34 (I) C.16 Orbits in E6 (I) . C.17 Orbits in G34 (II) C.18 Orbits in E6 (II) C.19 Orbits in E7 (I) . C.20 Orbits in E7 (II) C.21 Orbits in Es (I) .

119 121

212

223 249 253 254 267

287 288

289 290 290 290 291 291 291 291 292 292 292 293 293 293 294 294 295 295 296 297 298

XVIII List of Tables

C.22 Orbits in Es (II) C.23 Orbits in Es (III)

D.l The infinite families. D.2 Coxeter groups ... D.3 The remaining groups

299 300

302 302 302