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Birationally Rigid Varieties Mathematical Surveys and Monographs Volume 190 Birationally Rigid Varieties Aleksandr Pukhlikov American Mathematical Society http://dx.doi.org/10.1090/surv/190 Birationally Rigid Varieties Mathematical Surveys and Monographs Volume 190 Birationally Rigid Varieties Aleksandr Pukhlikov American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair Benjamin Sudakov MichaelA.Singer MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 14E05, 14E07, 14J45, 14E08, 14E30, 14M22, 14M10, 14M20, 14J30, 14J40. For additional information and updates on this book, visit www.ams.org/bookpages/surv-190 Library of Congress Cataloging-in-Publication Data Pukhlikov, Aleksandr V., 1962– Birationally rigid varieties / Aleksandr Pukhlikov. pages cm. – (Mathematical surveys and monographs ; volume 190) Includes bibliographical references and index. ISBN 978-0-8218-9476-7 (alk. paper) 1. Geometry, Algebraic. I. American Mathematical Society. II. Title. QA564.P85 2013 516.35–dc23 2013001354 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2013 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 181716151413 Contents Introduction 1 Chapter 1. The Rationality Problem 9 1. The classical heritage 10 2. Towards the modern birational geometry 19 3. Rationally connected varieties 28 Chapter 2. The Method of Maximal Singularities 37 1. Canonical adjunction 37 2. Exclusion of maximal singularities 48 3. Three-dimensional quartics 59 4. The connectedness principle 73 5 5. Complete intersections V2·3 ⊂ P .I: Untwisting maximal singularities 84 5 6. Complete intersections V2·3 ⊂ P .II: Excluding maximal singularities 93 Notes and references 110 Chapter 3. Hypertangent Divisors 115 1. Definitions and examples 115 2. Fano complete intersections 123 3. Regular Fano varieties 134 4. K-trivial structures 142 Notes and references 150 Chapter 4. Rationally Connected Fibre Spaces 153 1. Fano fibre spaces 153 2. The Sarkisov program 162 3. Birational rigidity of Fano fibre spaces 176 Notes and references 186 Chapter 5. Fano Fibre Spaces Over P1 189 1. Sufficient conditions of birational rigidity. I 190 2. Pencils of Fano hypersurfaces 202 3. Pencils of double hypersurfaces 213 4. Sufficient conditions of birational rigidity. II 234 5. Pencils of Fano complete intersections 241 Notes and references 252 Chapter 6. Del Pezzo Fibrations 253 1. Explicit constructions and a summary of known results 253 v vi CONTENTS 2. Infinitely near maximal singularities 261 3. Completing the proof for the pencils of cubic surfaces 273 Notes and references 283 Chapter 7. Fano Direct Products 285 1. Fano direct products 285 2. Inversion of adjunction 302 3. Fano varieties with elementary singularities 309 Notes and references 322 Chapter 8. Double Spaces of Index Two 323 1. Half-anticanonical pencils 323 2. Centres of codimension two and three 331 3. Counting multiplicities 336 4. Infinitely near singularities 345 5. Generic double spaces 350 Notes and references 356 Bibliography 359 Index 367 Bibliography [AlgS] Algebraic surfaces. By the members of the seminar of I.R. Shafarevich. I.R. Shafare- vich, ed., Proc. Steklov Math. Inst. 75. 1965. English transl. by AMS, 1965. 281 pp. [Al87] Alexeev V.A., On conditions for the rationality of three-folds with a pencil of del Pezzo surfaces of degree 4, Math. Notes 41 (1987), 408-411. [Al93] Alexeev V. Two two-dimensional terminations. Duke Math. J. 69 (1993), 527-545. [ArMum] Artin M., Mumford D., Some elementary examples of unirational varieties which are not rational, Proc. London Math. Soc. 25 (1972), 75-95. [BaBeau00] Bayle L. and Beauville A., Birational involutions of P2.AsianJ.Math.4 (2000), No. 1, 11-17. [Bar] Bardelli F., Polarized mixed Hodge structures: On irrationality of threefolds via degeneration. Ann. Mat. Pura et Appl. 137 (1984), 287-369. [Beau77] Beauville A., Vari´et´es de Prym et jacobiennes interm´ediares. Ann. Sci. Ecole´ Norm. Sup. 10 (1977), No. 3, 309-391. [Beau82] Beauville A., Les singularit´es du diviseur Θ de la jacobienne interm´ediare de l’hypersurface cubique dans P4. In: Lecture Notes in Math. 947 (1982), 190-208. [Beau08] Beauville A., p-elementary subgroups of the Cremona group. J. Algebra 314 (2007), No. 2, 553-564. [Beau11] Beauville A., Non-rationality of the symmetric sextic Fano three-fold. arXiv:1102.1255 [math.AG], 3 pp. [BeBl04] Beauville A. and Blanc, J. On Cremona transformations of prime order. C.R. Math. Acad. Sci. 339 (2004), No. 4, 257-259. [BCTSSD] Beauville A., Colliot-Th´el`ene J.-L., Sansuc J.-J. and Swinnerton-Dyer P., Vari´et´es stablement rationnelles non rationnelles. Ann. of Math. 121 (1985), No. 2, 283-318. [BeS] Beheshti R. and Starr J. M., Rational surfaces in index-one Fano hypersurfaces, arXiv:math.AG/0601710, 13 pp. [BCHM] Birkar C., Cascini P., Hacon C.D. and McKernan J., Existence of minimal models for varieties of log general type. J. Amer. Math. Soc. 23 (2010), No. 2, 405-468. [Bi07] Birkar C., Ascending chain condition for log canonical thresholds and termination of log flips. Duke Math. J. 136 (2007), 173-180. [Bl07] Blanc J., Finite abelian subgroups of the Cremona group of the plane. C. R. Math. Acad. Sci. Paris 344 (2007), No. 1, 21-26. [Bl10] Blanc J., Groupes de Cremona, connexit´e et simplicit´e. Ann. Sci. Ec.´ Norm. Sup´er. 43 (2010), No. 2, 357-364. [Bl11] Blanc J., Elements and cyclic subgroups of finite order of the Cremona group. Com- ment. Math. Helv. 86 (2011), No. 2, 469-497. [BCZ] Brown G., Corti A. and Zucconi F. Birational geometry of 3-fold Mori fibre spaces. In: The Fano conference. Univ. Torino, Turin. 2004. 235-275. [CL] Call F. and Lyubeznik G., A simple proof of Grothendieck’s theorem on the parafac- toriality of local rings, Contemp. Math. 159 (1994), 15-18. [Cam92] Campana F., Connexit´e rationnelle des vari´et´es de Fano. Ann. Sci. Ec.´ Norm. Sup´er., 1992, IV. S´er. 25, 539-545. [CPP] Campana F., Peternell Th. and Pukhlikov A. V., Generalized Tsen theorem and rationally connected Fano fibrations. Sbornik: Mathematics. 193 (2002), No. 9-10, 1443-1468. [CaL10] Cantat S. and Lamy S., Normal subgroups in the Cremona group. arXiv: math1007.0895, 53 p. 359 360 BIBLIOGRAPHY [Cas] Castravet A.-M., Examples of Fano varieties of index one that are not birationally rigid, Proc. Amer. Math. Soc. 135 (2007), No. 12, 3783-3788. [Ch00] Cheltsov I. A., On a smooth four-dimensional quintic. Sbornik: Mathematics. 191 (2000), No. 9-10, 1399-1419. [Ch00b] Cheltsov I.A., Log pairs on hypersurfaces of degree N in PN ,Math.Notes.68 (2000), No.1-2, 113-119. [Ch01] Cheltsov I. A., Log canonical thresholds on hypersurfaces. Sbornik: Mathematics. 192 (2001), No. 8, 155-172. [Ch03] Cheltsov I. A., Non-rationality of a four-dimensional smooth complete intersection of a quadric and a quartic, not containing a plane, Sbornik: Mathematics, 194 (2003), 1679-1699. [Ch04a] Cheltsov I. A., Conic bundles with a large discriminant. Izvestiya: Mathematics 68 (2004), No. 2, 429-434. [Ch04b] Cheltsov I. A., A double space with a double line. Sbornik: Mathematics 195 (2004), No. 9-10, 1503-1544. [Ch05a] Cheltsov I. A., Nonexistence of elliptic structures on general Fano complete intersec- tions of index one. Moscow Univ. Math. Bull. 60 (2005), No. 3, 30-33. [Ch05b] Cheltsov I. A., Birationally rigid del Pezzo fibrations. Manuscripta Math. 116 (2005), No. 4, 385-396. [Ch05c] Cheltsov I. A., Birationally rigid Fano varieties. Russian Math. Surveys. 60 (2005), No. 5, 875-965. [Ch06a] Cheltsov I., Double cubics and double quartics, Math. Z. 253 (2006), no. 1, 75-86. 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