Table Of ContentFrontiers in the History of Science
Michael Friedman
Ramified
Surfaces
On Branch Curves and Algebraic
Geometry in the 20th Century
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Michael Friedman
Ramified Surfaces
On Branch Curves and Algebraic Geometry
in the 20th Century
MichaelFriedman
TheCohnInstitutefortheHistoryandPhilosophyofScienceandIdeas,HumanitiesFaculty
TelAvivUniversity
TelAviv,Israel
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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).
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