The basic problem of deformation theory in algebraic geometry involves watching a small deformation of one member of a family of objects, such as varieties, or subschemes in a fixed space, or vector bundles on a fixed scheme. In this new book, Robin Hartshorne studies first what happens over small infinitesimal deformations, and then gradually builds up to more global situations, using methods pioneered by Kodaira and Spencer in the complex analytic case, and adapted and expanded in algebraic geometry by Grothendieck.
* infinitesimal study of moduli spaces such as the Hilbert scheme, Picard scheme, moduli of curves, and moduli of stable vector bundles.
The author includes numerous exercises, as well as important examples illustrating various aspects of the theory. This text is based on a graduate course taught by the author at the University of California, Berkeley.
Graduate Texts in Mathematics 257 Editorial Board S. Axler K.A. Ribet For other titles published in this series, go to http://www.springer.com/series/136 Robin Hartshorne Deformation Theory 123 Robin Hartshorne Department of Mathematics University of California Berkeley, CA 94720-3840 USA
[email protected] Editorial Board S. Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA
[email protected] K.A. Ribet Mathematics Department University of California at Berkeley Berkeley, CA 94720-3840 USA
[email protected] ISSN 0072-5285 ISBN 978-1-4419-1595-5 e-ISBN 978-1-4419-1596-2 DOI 10.1007/978-1-4419-1596-2 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2009939327 Mathematics Subject Classification (2000): 14B07, 14B12, 14B10, 14B20, 13D10, 14D15, 14H60, 14D20 c Robin Hartshorne 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com). Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 First-Order Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1. The Hilbert Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2. Structures over the Dual Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 9 3. T