Arrangements of Hyperplanes
An arrangement of hyperplanes is a finite collection of codimension one affine subspaces in a finite dimensional vector space. Arrangements have emerged independently as important objects in various fields of mathematics such as combinatorics, braids, con
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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, 1938Arrangements 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 n
