Clay Mathematics Proceedings Volume 20 The Resolution of Singular Algebraic Varieties Clay Mathematics Institute Summer School 2012 The Resolution of Singular Algebraic Varieties Obergurgl Tyrolean Alps, Austria June 3–30, 2012 David Ellwood Herwig Hauser Shigefumi Mori Josef Schicho Editors American Mathematical Society Clay Mathematics Institute The Resolution of Singular Algebraic Varieties Clay Mathematics Proceedings Volume 20 The Resolution of Singular Algebraic Varieties Clay Mathematics Institute Summer School 2012 The Resolution of Singular Algebraic Varieties Obergurgl Tyrolean Alps, Austria, June 3–30, 2012 David Ellwood Herwig Hauser Shigefumi Mori Josef Schicho Editors American Mathematical Society Clay Mathematics Institute 2010 Mathematics Subject Classification. Primary 14-01, 14-06, 14Bxx, 14Exx, 13-01, 13-06, 13Hxx, 32Bxx, 32Sxx, 58Kxx. Cover photo of Obergurgl, Austria is by Alexander Zainzinger. Library of Congress Cataloging-in-Publication Data ClayMathematicsInstituteSummerSchool(2012: ObergurglCenter) The resolution of singular algebraic varieties: Clay Mathematics Institute Summer School, the resolutionofsingular algebraic varieties,June 3–30,2012,ObergurglCenter, TyroleanAlps, Austria/DavidEllwood,HerwigHauser,ShigefumiMori,JosefSchicho,editors. pagescm. —(Claymathematicsproceedings;volume20) Includesbibliographicalreferencesandindex. ISBN978-0-8218-8982-4(alk.paper) 1.Algebraicvarieties—Congresses. 2.Commutativealgebra—Congresses. I.Ellwood,David, 1966– II.Hauser,H.(Herwig),1956– III.Mori,Shigefumi. IV.Schicho,Josef,1964– V.Ti- tle. QA564.C583 2012 516.3(cid:2)5—dc23 2014031965 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 select pages for useinteachingorresearch. 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PublishedbytheAmericanMathematicalSociety,Providence,RI, fortheClayMathematicsInstitute,Cambridge,MA. PrintedintheUnitedStatesofAmerica. TheClayMathematicsInstituteretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. (cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ VisittheClayMathematicsInstitutehomepageathttp://www.claymath.org/ 10987654321 191817161514 A tribute to Shreeram Abhyankar and Heisuke Hironaka Contents Preface ix Blowups and Resolution Herwig Hauser 1 On the Behavior of the Multiplicity on Schemes: Stratification and Blow Ups Ana Bravo and Orlando Villamayor 81 A Simplified Game for Resolution of Singularities Josef Schicho 209 Resolution of Singularities in Characteristic p and Monomialization Steven D. Cutkosky 225 Resolution of Toric Varieties Santiago Encinas 239 Desingularization in Computational Applications and Experiments Anne Fru¨hbis-Kru¨ger 269 Introduction to the Idealistic Filtration Program with Emphasis on the Radical Saturation Hiraku Kawanoue 285 Algebraic Approaches to FlipIt Josef Schicho and Jaap Top 319 Higher Semple-Nash Blowups and F-Blowups Takehiko Yasuda 327 vii Preface A remote place in the Tyrolean Alps at 2000 meters of altitude, a venue with perfectworkingfacilitiesandaclearcutgoal: toexciteyoungmathematiciansinthe resolutionofsingularitiesofalgebraicvarietiesbyofferingthemafourweekprogram ofclassesandproblemsessions—thiswasthesceneofthe12thClaySummerSchool at Obergurgl, Austria. It was previewed from the outset that such a school should go beyond mere mathematicaleducation: itshouldrepresentadecisivestepinthecareerofpartici- pantsbyteachinghow tograspandincorporatethemainfeaturesofacomplicated theory, to evaluate and combine the many ideas involved in its proofs, and to develop an overall picture of what mathematics can be good for with respect to intellectual, cultural and personal development. This scientific intention was to be matched by social effects: the communication with colleagues and teachers, group work, mutual respect and stimulation, the dialectic of modesty versus ambition. The topic: Resolution of singularities consists in constructing for a given alge- braic variety X an algebraic manifold X(cid:2) together with a surjective map π :X(cid:2) −→ X. This map gives a parametrization of the singular variety by a smooth variety. Algebraically,thismeanstofind, foragivensystemofpolynomialequations,asys- tematic transformation of the polynomials by means of blowups which transform the system into one that satisfies the assumption of the implicit function theorem, so that certain variables can be expressed as functions of the remaining variables. This transformation allows one to interpret the solution set of the given system as the projection of a graph to the singular variety. The existence of resolutions is instrumental in many circumstances since it allows one to deduce properties of the variety from properties of the parametrizing manifold. Applications abound. The pioneer in this problem was Oscar Zariski. He introduced abstract alge- braic ideas and techniques to the field, and proved many important cases, both in small dimensions and, for more restrictive assertions, in arbitrary dimension. His perspective was mostly based on varieties defined over fields of characteristic zero. He recommended to his student Shreeram Abhyankar to abandon, after several vain attempts, the difficult positive characteristic case of surfaces. As a matter of protest and stubbornness, Abhyankar intensified his efforts and succeeded in his thesis to settle this case1. At that time, another of Zariski’s many students, Heisuke Hironaka, was a friendofAbhyankar,andtogethertheywerediscussingthissubjectattheendofthe fifties. Itseemsthattheseconversationsproducedthekeyideaforthecharacteristic 1ShreeramAbhyankar,Localuniformizationonalgebraicsurfacesovergroundfieldsofchar- acteristicp(cid:2)=0,Ann. Math. (2)63(1956),491–526. ix