![]() Danilov for a series of recommendations on this subject. As an example we work out the theory of the Hilbert polynomial and the Hilbert scheme. The first of these is a discussion of the notion of the algebraic variety classifying algebraic or geometric objects of some type. For some questions it is only here that the natural and historical logic of the subject can be reasserted for example, differential forms were constructed in order to be integrated, a process which only makes sense for varieties over the (real or) complex fields.Ĭhanges from the First Edition As in the Book 1, there are a number of additions to the text, of which the following two are the most important. The theory of complex analytic manifolds leads to the study of the topology of projective varieties over the field of complex numbers. For example, it is within the framework of the theory of schemes and abstract varieties that we find the natural proof of the adjunction formula for the genus of a curve, which we have already stated and applied in Section 2.3, Chapter 4. Introducing them leads also to new results in the theory of projective varieties. They study schemes and complex manifolds, two notions that generalise in different directions the varieties in projective space studied in Book 1. Printed on acid-free paper Springer is part of Springer Science+Business Media (Preface to Books 2–3īooks 2–3 correspond to Chapters V–IX of the first edition. The publisher makes no warranty, express or implied, with respect to the material contained herein. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. 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. The use of general descriptive names, registered names, trademarks, service marks, etc. Violations are liable to prosecution under the respective Copyright Law. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. ![]() Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. ![]() MCCME, Moscow 2007, originally published in Russian in one volume © Springer-Verlag Berlin Heidelberg 1977, 1994, 2013 This work is subject to copyright. ISBN 978-9-9 ISBN 978-0-5 (eBook) DOI 10.1007/978-0-5 Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2013945857 Mathematics Subject Classification (2010): 14-01 Translation of the 3rd Russian edition entitled “Osnovy algebraicheskoj geometrii”. Translator Miles Reid Mathematics Institute University of Warwick Coventry, UK Shafarevich Algebra Section Steklov Mathematical Institute of the Russian Academy of Sciences Moscow, Russia Basic Algebraic Geometry 2 Schemes and Complex Manifolds Third Edition
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