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Automorphisms in Birational and Affine Geometry Springer Proceedings in Mathematics & Statistics Ivan Cheltsov Ciro Ciliberto Hubert Flenner James McKernan Yuri G. Prokhorov Mikhail Zaidenberg Editors Automorphisms in Birational and A ne Geometry Levico Terme, Italy, October 2012 Springer Proceedings in Mathematics & Statistics Vo lu m e 7 9 For further volumes: http://www.springer.com/series/10533 Springer Proceedings in Mathematics & Statistics This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including OR and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today. Ivan Cheltsov • Ciro Ciliberto • Hubert Flenner • James McKernan • Yuri G. Prokhorov • Mikhail Zaidenberg Editors Automorphisms in Birational and Affine Geometry Levico Terme, Italy, October 2012 123 Editors Ivan Cheltsov Ciro Ciliberto School of Mathematics Department of Mathematics University of Edinburgh University of Rome Tor Vergata Edinburgh, United Kingdom Rome, Italy Hubert Flenner James McKernan Faculty of Mathematics Department of Mathematics Ruhr University Bochum University of California San Diego Bochum, Germany La Jolla, California, USA Yuri G. Prokhorov Mikhail Zaidenberg Steklov Mathematical Institute Institut Fourier de Mathématiques Russian Academy of Sciences Université Grenoble I Moscow, Russia Grenoble, France ISSN 2194-1009 ISSN 2194-1017 (electronic) ISBN 978-3-319-05680-7 ISBN 978-3-319-05681-4 (eBook) DOI 10.1007/978-3-319-05681-4 Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number: 2014941702 Mathematics Subject Classification (2010): 14L30, 14E07, 14R20, 14R10, 32M17 © Springer International Publishing Switzerland 2014 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. 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. 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. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. 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. 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. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface The conference brought together specialists from the birational geometry of pro- jective varieties, affine algebraic geometry, and complex algebraic geometry. Topics from these areas include Mori theory, Cremona groups, algebraic group actions, and automorphisms. The ensuing talks and the discussions have highlighted the close connections between these areas. The meeting allowed these groups to exchange knowledge and to learn methods from adjacent fields. For detailed information about this conference, see http://www.science.unitn.it/cirm/GABAG2012.html. The chapters in this book cover a wide area of topics from classical algebraic geometry to birational geometry and affine geometry, with an emphasis on group actions and automorphism groups. Among the total of 27 chapters, eight give an overview of an area in one of the fields. The others contain original contributions, or mix original contributions and surveys. In the birational part, there are chapters on Fano and del Pezzo fibrations, birational rigidity and superrigidity of Fano varieties, birational morphisms between threefolds, subgroups of the Cremona group, Jordan groups, real Cremona group, and a real version of the Sarkisov program. In the part on classical projective geometry, there are chapters devoted to algebraic groups acting on projective varieties, automorphism groups of moduli spaces, and the algebraicity problem for analytic compactifications of the affine plane. The topics in affine geometry include: different aspects of the automorphism groups of affine varieties, flexibility properties in affine algebraic and analytic geometry, automorphisms of affine spaces and Shestakov–Umirbaev theory, affine geometry in positive characteristic, the automorphism groups of configuration spaces and other classes of affine varieties and deformations of certain group actions on affine varieties. Hopefully these proceedings will be of help for both specialists, who will find information on the current state of research, and young researchers, who wish to learn about these fascinating and active areas of mathematics. v vi Preface Acknowledgments The editors would like to thank the contributors of this volume. They are grateful to the CIRM at Trento, its directors Marco Andreatta and Fabrizio Catanese, its secretary Augusto Micheletti, and the whole staff for their permanent support in the organization of the conference and the distribution of this volume. Our thanks are due also to Marina Reizakis, Frank Holzwarth, and the staff of the Springer publishing house for agreeable cooperation during the preparation of the proceedings. Edinburgh, UK Ivan Cheltsov Rome, Italy Ciro Ciliberto Bochum, Germany Hubert Flenner La Jolla, CA James McKernan Moscow, Russia Yuri G. Prokhorov Grenoble, France Mikhail Zaidenberg Contents Part I Birational Automorphisms Singular del Pezzo Fibrations and Birational Rigidity....................... 3 Hamid Ahmadinezhad Additive Actions on Projective Hypersurfaces................................ 17 Ivan Arzhantsev and Andrey Popovskiy Cremona Groups of Real Surfaces ............................................. 35 Jérémy Blanc and Frédéric Mangolte On Automorphisms and Endomorphisms of Projective Varieties .......... 59 Michel Brion Del Pezzo Surfaces and Local Inequalities .................................... 83 Ivan Cheltsov Fano Hypersurfaces and their Birational Geometry ......................... 103 Tommaso de Fernex On the Locus of Nonrigid Hypersurfaces ..................................... 121 Thomas Eckl and Aleksandr Pukhlikov On the Genus of Birational Maps Between Threefolds ...................... 141 Stéphane Lamy On the Automorphisms of Moduli Spaces of Curves ......................... 149 Alex Massarenti and Massimiliano Mella Normal Analytic Compactifications of C2 ..................................... 169 Pinaki Mondal Jordan Groups and Automorphism Groups of Algebraic Varieties ........ 185 Vladimir L. Popov vii viii Contents 2-Elementary Subgroups of the Space Cremona Group ..................... 215 Yuri Prokhorov Birational Automorphism Groups of Projective Varieties of Picard Number Two .......................................................... 231 De-Qi Zhang Part II Automorphisms of Affine Varieties Rational Curves with One Place at Infinity ................................... 241 Abdallah Assi The Jacobian Conjecture, Together with Specht and Burnside-Type Problems ................................................... 249 Alexei Belov, Leonid Bokut, Louis Rowen, and Jie-Tai Yu Equivariant Triviality of Quasi-Monomial Triangular 4 Ga-Actions on A ................................................................. 287 Adrien Dubouloz, David R. Finston, and Imad Jaradat Automorphism Groups of Certain Rational Hypersurfaces in Complex Four-Space.......................................................... 301 Adrien Dubouloz, Lucy Moser-Jauslin, and Pierre-Marie Poloni Laurent Cancellation for Rings of Transcendence Degree One Over a Field ....................................................................... 313 Gene Freudenburg Deformations of A1-Fibrations ................................................. 327 Rajendra V. Gurjar, Kayo Masuda, and Masayoshi Miyanishi Remark on Deformations of Affine Surfaces with A1-Fibrations........... 363 Takashi Kishimoto How to Prove the Wildness of Polynomial Automorphisms: An Example ....................................................................... 381 Shigeru Kuroda Flexibility Properties in Complex Analysis and Affine Algebraic Geometry ............................................................. 387 Frank Kutzschebauch Strongly
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