Introduction to Singularities
This book is an introduction to singularities for graduate students and researchers.Algebraic geometry is said to have originated in the seventeenth century with the famous workDiscours de la méthode pour bien conduire sa raison, et chercher la vérité dan
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Introduction to Singularities Second Edition
Introduction to Singularities
Shihoko Ishii
Introduction to Singularities Second Edition
123
Shihoko Ishii The University of Tokyo Tokyo, Japan
ISBN 978-4-431-56836-0 ISBN 978-4-431-56837-7 https://doi.org/10.1007/978-4-431-56837-7
(eBook)
Library of Congress Control Number: 2018942524 Original Japanese edition published by Springer-Verlag Tokyo, Tokyo, 1997 1st edition: © Springer Japan 2014 2nd edition: © Springer Japan KK, part of Springer Nature 2018 This work is subject to copyright. 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. The use of general descriptive names, registered names, trademarks, service marks, 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. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by the registered company Springer Japan KK part of Springer Nature The registered company address is: Shiroyama Trust Tower, 4-3-1 Toranomon, Minato-ku, Tokyo 105-6005, Japan
Preface to the Second Edition
In the preface to the first edition, the author wrote that there were three remarkable developments in singularity theory after 1997, the publication year of the Japanese version of this book. The three are: (1) applications of jet schemes to singularities; (2) big progress in the Minimal Model Problem in higher-dimensional varieties; (3) the positive characteristic method. In the first edition of the English version published in 2014, the author reflected the results of (2), but not (1) or (3). In this new version, the author introduces (1) and (3) very briefly but enough to show the basic ideas which would help the further studies of the reader. This new version also fixes some typographic and language problems of the first edition of the English version. The author would like to express her hearty thanks to Kohsuke Shibata and Nguyen Duc Tam for helping her in this process. Tokyo, Japan February 2018
Shihoko Ishii
v
Preface to the First Edition
“When I started working on mathematics, singularities were very fancy for me” said Lê Dung Tráng, a
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