Ramified Surfaces: On Branch Curves and Algebraic Geometry in the 20th Century PDF
Preview Ramified Surfaces: On Branch Curves and Algebraic Geometry in the 20th Century
Frontiers in the History of Science Michael Friedman Ramified Surfaces On Branch Curves and Algebraic Geometry in the 20th Century Frontiers in the History of Science SeriesEditor VincenzoDeRisi,UniversitéParis-Diderot–CNRS,PARISCEDEX13,Paris,France Frontiers in the History of Science is designed for publications of up-to-date research resultsencompassingallareasofhistoryofscience,primarilywithafocusonthehistoryof mathematics, physics, and their applications. Graduates and post-graduates as well as scientists will benefit from the selected and thoroughly peer-reviewed publications at the research frontiers of history of sciences and at interdisciplinary “frontiers”: history of science crossing into neighboring fields such as history of epistemology, history of art, or history of culture. The series is curated by the Series Editor with the support of an internationalgroupofAssociateEditors. * * * SeriesEditor: VincenzodeRisi Paris,France AssociateEditors: KarineChemla Paris,France SvenDupré Utrecht,TheNetherlands MoritzEpple Frankfurt,Germany OrnaHarari TelAviv,Israel DanaJalobeanu Bucharest,Romania HenriqueLeitão Lisboa,Portugal DavidMarshalMiller Ames,Iowa,USA AurélienRobert Tours,France EricSchliesser Amsterdam,TheNetherlands Michael Friedman Ramified Surfaces On Branch Curves and Algebraic Geometry in the 20th Century MichaelFriedman TheCohnInstitutefortheHistoryandPhilosophyofScienceandIdeas,HumanitiesFaculty TelAvivUniversity TelAviv,Israel TheauthoracknowledgesthesupportoftheClusterofExcellence“MattersofActivity.ImageSpaceMaterial” fundedbytheDeutscheForschungsgemeinschaft(DFG,GermanResearchFoundation)underGemany’sExcel- lenceStrategy—EXC2025—390648296. ISSN2662-2564 ISSN2662-2572 (electronic) FrontiersintheHistoryofScience ISBN978-3-031-05719-9 ISBN978-3-031-05720-5 (eBook) https://doi.org/10.1007/978-3-031-05720-5 #TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG 2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whetherthe wholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now knownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublicationdoesnot imply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbookare believedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsortheeditorsgivea warranty,expressedorimplied,withrespecttothematerialcontainedhereinorforanyerrorsoromissionsthat mayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictionalclaimsinpublishedmapsand institutionalaffiliations. ThisbookispublishedundertheimprintBirkhäuser,www.birkhauser-science.combytheregisteredcompany SpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Acknowledgements Iwishtothankandacknowledgethemanyscholarswhohavehelpedmeintheprocessof writingthisbook.VincenzodeRisiaccompaniedtheconceptionandcreationofthisbook from the beginning till its completion. Both Vincenzo de Risi and Moritz Epple gave extremely valuable comments and advice concerning the content and structure of the book, which helped to bring clarity, shape and form to it. I am also extremely grateful to WolfgangSchäffner,whoprovidedmewiththespaceandtimetopursuetheresearchforthis bookattheClusterofExcellence‘MattersofActivity’,wherealargepartofitwaswritten. Moreover,theresearchonBorisMoishezonwouldnothavebeenpossiblewithoutthe helpofMinaTeicher; thenumerousconversationswith herledtothoughtfulinsightsnot only on his work and their mutual discoveries but also on the history of the research on branch curves in general, and I warmly thank her for that. I am also very grateful for the help given tome bythe Tel Aviv University Archives,andespecially from Ella Meirson andGedalyaZhagov.SpecialthanksmustalsogotoDonuArapura,RonLivneandVitali Milman for the conversations on the work and life of Moishezon. Moreover, during my research on the Italian school of algebraic geometry I had various conversations with numerous scholars: Maria Dedò, Antonio Lanteri, Anatoly Libgober and Piera Manara helpedmegreatlyunderstandthevariousmathematicalconfigurationsthatwereshapednot only during this period but also during other periods. I would also like to thank warmly EliotBorenstein,LeoCorry,ChristopherHollings,FrançoisLê,KlausVolkert,Fernando Zalameaandtheanonymousrefereesforimportantinsightsandadvice.IfIhadforgotten anyonehere,Ideeplyapologize. Lastbutcertainlynotleast,IthankMichaelLorber,whosupportedmeconstantlyduring the creation and writing of this book. Without his support, my own thought would have probablyremainedramifiedandbranched.Thankyou. v Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 OnBranchPointsandBranchCurves. . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 DynamicsofaMathematicalObject. . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.1 EphemeralEpistemicConfigurationsandtheIdentityofthe MathematicalObjects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 10 1.2.2 OnBranchPoints,Again:onRiemann’sTerminologyandHow (Not)toTransferResults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.3 OnBranchCurves,Again:PluralityofNotations. . . . . . . . . . . . . 21 1.2.4 TransformationsBetweenEpistemicConfigurations. . . . . . . . . .. 24 1.3 AnOverview:HistoricalLiterature,StructureandArgument. . . . . . . . .. 25 1.3.1 OmittedTraditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.3.2 StructureoftheBook:TheTwentiethCentury. .. . . . . . . . . . . .. 29 2 Prologue:SeparateBeginningsDuringtheNineteenthCentury. . . . . . . . . 31 2.1 TheBeginningoftheNineteenthCentury:Mongeandthe“Contour Apparent”. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 32 2.2 1820s–1860s:ÉtienneBobillierandGeorgeSalmon. . . . . . . . . . . . . . .. 42 2.3 1890s–1900s:Wirtinger’sandHeegaard’sTurnTowardsKnotTheory. . 51 2.4 TheEndoftheNineteenthCentury:ARegressionTowardtheLocal. . . . 56 3 1900s–1930s:BranchCurvesandtheItalianSchoolofAlgebraic Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 59 3.1 Enriques:APluralityofMethodstoInvestigatetheBranchCurve. . . . .. 62 3.1.1 EnriquesonIntuitionandVisualization. . . . . . . . . . . . . . . . . ... 63 3.1.2 TheTurnofthenineteenthCentury:FirstAttemptsofClassification ofSurfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 69 3.1.2.1 OnDoubleCoversandBranchCurves. . . . . . . . . . . . . 71 3.1.2.2 Endofthe1890s:Enriques’sInitialConfigurations. . . . 76 3.1.3 TwoPapersfrom1912andtheCulminationoftheClassification Project. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 80 3.1.4 1923:AftertheClassificationProject. . . . . . . . . . . . . . . . . . . . . 86 vii viii Contents 3.2 ZariskiandSegre:NovelApproaches. . . . . . . . . . . . . . . . . . . . . . . . .. 91 3.2.1 TheLate1920s:ZariskionExistenceTheoremsandthe BeginningofaGroup-TheoreticApproach. . . . . . . . . . . . . . . . . 93 3.2.2 1930:SegreandSpecialPositionoftheSingularPoints. . . . . . . . 98 3.2.3 1930–1937:BeforeandAfterZariski’sAlgebraicSurfaces.. . . .. 101 3.2.3.1 1935:Zariski’sAlgebraicSurfaces. . . . . . . . . . . . . . .. 105 3.2.3.2 AfterAlgebraicSurfaces. . . . . . . . . . . . . . . . . . . . . .. 110 3.3 ReflectionsonRigor:ReassessmentandNewDefinitionsinthe1950s. . . 112 3.4 AppendixtoChap.3:BirationalMapsandGeneraofCurves andSurfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 4 1930s–1950s:Chisini’sBranchCurves:TheDeclineoftheClassical Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 123 4.1 The1930sandChisini’sFirstConjecture. . . . . . . . . . . . . . . . . . . . . . . . 125 4.1.1 The“CharacteristicBundle”. . . . . . . . . . . . . . . . . . . . . . . . . . .. 126 4.1.2 OnBraids,BranchCurvesandDegenerations. . . . . . . . . . . . . . . 129 4.1.2.1 Bernardd’OrgevalinOflagXB. . . . . . . . . . . . . . . . .. 133 4.1.2.2 GuidoZappa’sdegenerations. . . . . . . . . . . . . . . . . . . . 134 4.1.3 Detour.1944:Chisini’sFirst‘Conjecture’. . . . . . . . . . . . . . . . .. 138 4.2 Chisini’sStudents:IsolationandAbandonment. . . . . . . . . . . . . . . . . . . 140 4.2.1 DedòandtheNewNotationofBraids. . . . . . . . . . . . . . . . . . . . 141 4.2.2 TibilettiandtheSecond‘Theorem’ofChisini. . . . . . . . . . . . . .. 148 4.3 Conclusion:Seclusion,IgnoranceandAbandonment. . . . . . . . . . . . . . . 154 4.4 AppendixtoChap.4:AShortIntroductiontotheBraidGroup. . . . . . . . 155 5 Fromthe1970sOnward:TheRiseofBraidMonodromyFactorization. .. 159 5.1 The1960s:GeneralizationandStagnationorthe“RisingSea”andthe SunkenBranchCurves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 5.1.1 Detour:Endofthe1950s:Abhyankar’sConjecture. . . . . . . . . . . 165 5.1.2 1971:TheNewEditionofZariski’sAlgebraicSurfaces. . . . . . . . 167 5.2 The1970s:LivneandMoishezononEquivalenceofFactorizations. . . . . 169 5.2.1 Livne’sMAThesisfrom1975. . . . . . . . . . . . . . . . . . . . . . . . . . 170 5.2.2 SeparationsofConfigurationsandShiftsofContexts. . . . . . . . .. 173 5.2.3 OnSurfaceswithc2 ¼3c andLivne’s1981PhDThesis. . . . . . 176 1 2 5.3 Moishezon’sProgram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 184 5.3.1 FromtheUSSRtoIsraelandtotheUSA. . . . . . . . . . . . . . . . .. 185 5.3.2 BeforeBraidMonodromy:TheShafarevichSchool,Moishezon andtheDecompositionofAlgebraicSurfaces. . . . . . . . . . . . . . . 192 5.3.3 From1981to1985:(Re)introducingBraidMonodromy. . . . . . .. 197 5.4 MoishezonandTeicherCrosstheWatershed. . . . . . . . . . . . . . . . . . . .. 209 5.4.1 Coda:TheGroup-TheoreticalApproachofthe1990s. . . . . . . . . 214 Contents ix 6 Epilogue:OnRamifiedandIgnoredSpaces. . . . . . . . . . . . . . . . . . . . . . .. 221 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Index. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 1 Introduction Monge,1785:“Theprojectionofabody’sshadowonanysurfaceis[...]thefigurethatthe extensionsoftheraysoflighttangenttothebody’ssurfaceendonthatsurface.[...]Inthe followingoperationswewillgeometricallydetermineonlytheprojectionsofthecontoursof thepureshadows,theyaretheonlyonesthatitisnecessarytohaveexactlyinthedrawings.”1 Attheendoftheeighteenthcentury,themathematicianGaspardMongeemphasizedthatto investigatesurfacesproperly,the“projectionsofthecontoursofthepureshadows”arethe onlycurvesnecessarytodrawaccurately,andinacertainsense,whosepropertiesarethe only ones one should know exactly. An example of what should be drawn is given by Monge,ascanbeseeninFig.1.1,whenthesourceof“light”iseitherapointoraspherical body.Monge’ssurfaceswererealsurfaces,thatis,definedoverthereal numbers,andhe alsotermedthe“contourofthepureshadow”as“apparentcontour.” Jumping to the twentieth century, when one takes these “surfaces” as complex and algebraicsurfaces,embeddedinathree-dimensionalcomplexspace,thentheprojectionof this contour is called the ‘branch curve.’ Taking into consideration the fact that in the twenty-firstcentury,Monge’srequirementsseemalmostirrelevant,lookingatthecurrent researchofcomplexalgebraicsurfaces,thequestionarises:Whathappened?Howwasthis curveresearchedoverdecades,andhowdiditsepistemicstatuschange,especiallyduring thetwentiethcentury,inthethenflourishingdomainofalgebraicgeometry? 1“Laprojectiondel’ombred’uncorpssurunesurfacequelconqueestdonclafigurequeterminentsur cettesurfacelesprolongementsdesrayonsdelumièretangentsàlasurfaceducorps[...]Dansles opérationssuivantes nousnedétermineronsgéométriquement queles projectionsdescontours des ombrespures,cesontlesseulesqu’ilsoitnécessaired’avoirexactementdanslesdessins.”(Monge 1847[1785],p.27,29). #TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2022 1 M.Friedman,RamifiedSurfaces,FrontiersintheHistoryofScience, https://doi.org/10.1007/978-3-031-05720-5_1
The list of books you might like
