Classification of Higher Dimensional Algebraic Varieties
This book focuses on recent advances in the classification of complex projective varieties. It is divided into two parts. The first part gives a detailed account of recent results in the minimal model program. In particular, it contains a complete proof o
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Christopher D. Hacon Sándor J. Kovács
Classification of Higher Dimensional Algebraic Varieties
Birkhäuser
Authors: Christopher D. Hacon Department of Mathematics University of Utah Salt Lake City, UT 84112-0090 USA e-mail: [email protected]
Sándor Kovács Department of Mathematics University of Washington Box 354350 Seattle, WA 98195 USA e-mail: [email protected]
2000 Mathematics Subject Classification: Primary 14E30, 14D22; Secondary 14B05, 14C20, 14D20, 14E05, 14E15, 14F17, 14F18, 14J17
Library of Congress Control Number: 2010924096
Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at .
ISBN 978-3-0346-0289-1 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, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.
© 2010 Springer Basel AG Basel · Boston · Berlin P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF ∞ Printed in Germany ISBN 978-3-0346-0289-1
e-ISBN 978-3-0346-0290-7
987654321
www.birkhauser.ch
Preface
This book grew out of the Oberwolfach-Seminar Higher Dimensional Algebraic Geometry organized by the two authors in October 2008. The aim of the seminar was to introduce advanced PhD students and young researchers to recent advances and research topics in higher dimensional algebraic geometry. The main emphasis was on the minimal model program and on the theory of moduli spaces. The authors would like to thank the Mathematishes Forshunginstitut Oberwolfach for its hospitality and for making the above mentioned seminar possible, the participants to the seminar for their useful comments, and Alex Küronya, Max Lieblich, and Karl Schwede for valuable suggestions and conversations. The first named author was partially supported by the National Science Foundation under grant number DMS-0757897 and would like to thank Aleksandra, Stefan, Ana, Sasha, Kristina and Daniela Jovanovic-Hacon for their love and continuos support. The second named author was partially supported by the National Science Foundation under grant numbers DMS-0554697 and DMS-0856185, and the Craig McKibben and Sarah Merner Endowed Professorship in Mathematics at the University of Washington. He would also like to thank Tímea Tihanyi for her enduring love and support throughout and beyond this project and his other co-authors for their patience and understanding.
Contents
I Basics
1
1
I NTRODUCTION
3
C LASSIFICATION
3
1.A.
2
P RELIMINARIES
17
2.A.
N OTATION
17
2.B.
D IVISORS
18
2.C.
R EFLEXIVE SHEAVES
20
2.D.
C YCLIC COVERS
21
2.E.
R- DIVISORS IN THE RELATIVE SETTING
22
2.F.
FAMILIES AND BASE CHANGE
2
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