Deformations of Algebraic Schemes
The study of small and local deformations of algebraic varieties originates in the classical work of Kodaira and Spencer and its formalization by Grothendieck in the late 1950's. It has become increasingly important in algebraic geometry in every context
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deformations of algebraic schemes
Grundlehren der mathematischen Wissenschaften 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
334
Edoardo Sernesi
Deformations of Algebraic Schemes
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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. A EX macro package Typesetting: by the author and SPI Publisher Services using a Springer LT Cover design: design & production GmbH, Heidelberg
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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 formalization 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-co
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