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Grundlehren Der Mathematischen Wissenschaften 334 a Series of Comprehensive Studies in Mathematics Grundlehren der mathematischen Wissenschaften 334 A Series of Comprehensive Studies in Mathematics Series editors M. Berger B. Eckmann P. de la Harpe F. Hirzebruch N. Hitchin L. Hörmander M.-A. Knus A. Kupiainen G. Lebeau M.Ratner D.Serre Ya.G.Sinai N.J.A. Sloane B. Totaro A. Vershik M. Waldschmidt Editor-in-Chief A. Chenciner J. Coates S.R.S. Varadhan Edoardo Sernesi Deformations of Algebraic Schemes ABC Edoardo Sernesi Universitá "Roma Tre" Department of Mathematics Largo San Leonardo Murialdo 1 00146 Roma, Italy e-mail: [email protected] Library of Congress Control Number: 2006924565 Mathematics Subject Classification (2000): 14D15, 14B12 ISSN 0072-7830 ISBN-10 3-540-30608-0 Springer Berlin Heidelberg New York ISBN-13 978-3-540-30608-5 Springer Berlin Heidelberg New York 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 Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the author and SPI Publisher Services using a Springer LTA EX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper SPIN: 11371335 41/SPI 543210 Preface In one sense, deformation theory is as old as algebraic geometry itself: this is because all algebro-geometric objects can be “deformed” by suitably varying the coefficients of their defining equations, and this has of course always been known by the classical geometers. Nevertheless, a correct understanding of what “deforming” means leads into the technically most difficult parts of our discipline. It is fair to say that such technical obstacles have had a vast impact on the crisis of the classical language and on the development of the modern one, based on the theory of schemes and on cohomological methods. The modern point of view originates from the seminal work of Kodaira and Spencer on small deformations of complex analytic manifolds and from its forma- lization and translation into the language of schemes given by Grothendieck. I will not recount the history of the subject here since good surveys already exist (e.g. [27], [138], [145], [168]). Today, while this area is rapidly developing, a self-contained text covering the basic results of what we can call “classical deformation theory” seems to be missing. Moreover, a number of technicalities and “well-known” facts are scattered in a vast literature as folklore, sometimes with proofs available only in the complex analytic category. This book is an attempt to fill such a gap, at least par- tially. More precisely, it aims at giving an account with complete proofs of the results and techniques which are needed to understand the local deformation theory of alge- braic schemes over an algebraically closed field, thus providing the tools needed, for example, in the local study of Hilbert schemes and moduli problems. The existing monographs, like [14], [93], [105], [109], [124], [163], [175], [176], [184], all aim at goals different from the above. For these reasons my approach has been to work exclusively in the category of locally noetherian schemes over a fixed algebraically closed field k, to avoid switch- ing back and forth between the algebraic and the analytic category. I tried to make the text self-contained as much as possible, but without forgetting that all the technical ideas and prerequisites can be found in [3] and [2]: therefore the reader is advised to keep a copy of them to hand while reading this text. In any case a good fami- liarity with [84] and with a standard text in commutative algebra like [48] or [127] VI Preface will be generally sufficient; the classical [167] and [190] will be also useful. A good acquaintance with homological algebra is assumed throughout. One of the difficulties of writing about this subject is that it needs a great num- ber of technical results, which make it hard to maintain a proper balance between generality and understandability. In order to overcome this problem I tried to keep the technicalities to a minimum, and I introduced the main deformation problems in an elementary fashion in Chapter 1; they are then reconsidered as functors of Artin rings in Chapter 2, where the main results of the theory are proved. The first two chapters therefore give a self-contained treatment of formal deformation theory via the “classical” approach; cotangent complexes and functors are not introduced, nor the method of differential graded Lie algebras. Another chapter treats in more detail the most important deformation functors, with the single exception of vector bun- dles; this was motivated by reasons of space and because good monographs on the subject are already available (e.g. [90], [118], [59]). Although they are not the central issue of the book, I considered it necessary to include a chapter on Hilbert schemes and Quot schemes, since it would be impossible to give meaningful examples and applications without them, and because of the lack of an appropriate reference. De- formation theory is closely tied with classical algebraic geometry because some of the issues which had remained controversial and unclear in the old language have found a natural explanation using the methods discussed here. I have included a sec- tion on plane curves which gives a good illustration of this point. Unfortunately, important topics and results have been omitted because of lack of space, energy and competence. In particular, I did not include the construction of any global moduli spaces/stacks, which would have taken me too far from the main theme. The book is organized in the following way. Chapter 1 starts with a concise treat- ment of algebra extensions which are fundamental in deformation theory. It then discusses locally trivial infinitesimal deformations of algebraic schemes. Chapter 2 deals with “functors of Artin rings”, the abstract tool for the study of formal defor- mation theory. The main result of this theory is Schlessinger’s theorem. A section on obstruction theory, an elementary but crucial technical point, is included. We discuss the relation between formal and algebraic deformations and the algebraization prob- lem. This part is not entirely self-contained since Artin’s algebraization theorem is not proved in general and the approximation theorem is only stated. The last section explains the role of automorphisms and the related notion of “isotriviality”. Chap- ter 3 is an introduction to the most important deformation problems. By applying Schlessinger’s theorem to them, we derive the existence of formal (semi)universal deformations. Many examples are discussed in detail so that all the basic principles of deformation theory become visible. This chapter can be used as a reference for several standard facts of deformation theory, and it can be also helpful in supplement- ing the study of the more abstract Chapter 2. Chapter 4 is devoted to the construction and general properties of Hilbert schemes, Quot schemes and their variants, the “flag Hilbert schemes”. It ends with a section on plane curves, where the main properties of Severi varieties are discussed. My approach to the proof of existence of nodal curves with any number of nodes uses multiple point schemes and is apparently new. Preface VII In the Appendices I have collected several topics which are well known and standard but I felt it would be convenient for the reader to have them available here. Acknowledgements. Firstly, I would like to express my deepest gratitude to D. Lieberman, who introduced me to the study of deformations of complex mani- folds a long time ago. More recently, R. Hartshorne, after a careful reading of a previous draft of this book, made a number of comments and suggestions which sig- nificantly contributed in improving it. I warmly thank him for his generous help. For many extremely useful remarks on another draft of the book I am also indebted to M. Brion. I gratefully acknowledge comments and suggestions from other colleagues and students, in particular: L. Badescu, I. Bauer, A. Bruno, M. Gonzalez, A. Lopez, M. Manetti, A. Molina Rojas, D. Tossici, A. Verra, A. Vistoli. I would also like to thank L. Caporaso, F. Catanese, C. Ciliberto, L. Ein, D. Laksov and H. Lange for encouragement and support which have been of great help. Contents Terminology and notation ........................................... 1 Introduction ....................................................... 3 1 Infinitesimal deformations ...................................... 9 1.1 Extensions................................................ 9 1.1.1 Generalities ......................................... 9 1.1.2 The module ExA(R, I ) ............................... 12 1.1.3 Extensions of schemes . ............................. 15 1.2 Locallytrivialdeformations.................................
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