COMPLEXALGEBRAICTHREEFOLDS
Thefirstbookontheexplicitbirationalgeometryofcomplexalgebraicthreefoldsarising fromtheminimalmodelprogram,thistextissuretobecomeanessentialreferencein thefieldofbirationalgeometry.Threefoldsremaintheinterfacebetweenlow-andhighdimensionalsettings,andagoodunderstandingofthemisnecessaryinthisactively evolvingarea.
Intendedforadvancedgraduatestudentsaswellasresearchersworkinginbirational geometry,thebookisasself-containedaspossible.Detailedproofsaregiventhroughout, andmorethan100exampleshelptodeepenunderstandingofbirationalgeometry.
Thefirstpartofthebookdealswiththreefoldsingularities,divisorialcontractions andflips.AfterathoroughexplanationoftheSarkisovprogram,thesecondpartis devotedtotheanalysisofoutputs,specificallyminimalmodelsandMorifibrespaces. Thelatteraredividedintoconicalfibrations,delPezzofibrationsandFanothreefolds accordingtotherelativedimension.
MasayukiKawakita isAssociateProfessorattheResearchInstituteforMathematical Sciences,KyotoUniversity.Hehasestablishedaclassificationofthreefolddivisorial contractionsandisaleadingexpertinalgebraicthreefolds.
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172T.Ceccherini-Silberstein,F.Scarabotti&F.Tolli DiscreteHarmonicAnalysis
173P.Garrett ModernAnalysisofAutomorphicFormsbyExample,I
174P.Garrett ModernAnalysisofAutomorphicFormsbyExample,II
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176P.Fleig,H.P.A.Gustafsson,A.Kleinschmidt&D.Persson EisensteinSeriesandAutomorphic Representations
177E.Peterson FormalGeometryandBordismOperators
178A.Ogus LecturesonLogarithmicAlgebraicGeometry
179N.Nikolski HardySpaces
180D.-C.Cisinski HigherCategoriesandHomotopicalAlgebra
181A.Agrachev,D.Barilari&U.Boscain AComprehensiveIntroductiontoSub-RiemannianGeometry
182N.Nikolski ToeplitzMatricesandOperators
183A.Yekutieli DerivedCategories
184C.Demeter FourierRestriction,DecouplingandApplications
185D.Barnes&C.Roitzheim FoundationsofStableHomotopyTheory
186V.Vasyunin&A.Volberg TheBellmanFunctionTechniqueinHarmonicAnalysis
187M.Geck&G.Malle TheCharacterTheoryofFiniteGroupsofLieType
188B.Richter CategoryTheoryforHomotopyTheory
189R.Willett&G.Yu HigherIndexTheory
190A.Bobrowski GeneratorsofMarkovChains
191D.Cao,S.Peng&S.Yan SingularlyPerturbedMethodsforNonlinearEllipticProblems
192E.Kowalski AnIntroductiontoProbabilisticNumberTheory
193V.Gorin LecturesonRandomLozengeTilings
194E.Riehl&D.Verity Elementsof ∞-CategoryTheory
195H.Krause HomologicalTheoryofRepresentations
196F.Durand&D.Perrin DimensionGroupsandDynamicalSystems
197A.Sheffer PolynomialMethodsandIncidenceTheory
198T.Dobson,A.Malnič&D.Marušič SymmetryinGraphs
199K.S.Kedlaya p-adicDifferentialEquations
200R.L.Frank,A.Laptev&T.Weidl SchrödingerOperators:EigenvaluesandLieb–ThirringInequalities
201J.vanNeerven FunctionalAnalysis
202A.Schmeding AnIntroductiontoInfinite-DimensionalDifferentialGeometry
203F.CabelloSánchez&J.M.F.Castillo HomologicalMethodsinBanachSpaceTheory
204G.P.Paternain,M.Salo&G.Uhlmann GeometricInverseProblems
205V.Platonov,A.Rapinchuk&I.Rapinchuk AlgebraicGroupsandNumberTheory,I(2ndEdition)
206D.Huybrechts TheGeometryofCubicHypersurfaces
207F.Maggi OptimalMassTransportonEuclideanSpaces
208R.P.Stanley EnumerativeCombinatorics,II(2ndedition)
209M.Kawakita ComplexAlgebraicThreefolds
210D.Anderson&W.Fulton EquivariantCohomologyinAlgebraicGeometry
Formyfamily
Prefacepage xi
1TheMinimalModelProgram 1
1.1Preliminaries2 1.2NumericalGeometry11 1.3TheProgram19
1.4LogarithmicandRelativeExtensions28 1.5ExistenceofFlips36
1.6TerminationofFlips46 1.7Abundance52
2Singularities 59
2.1AnalyticGerms60
2.2QuotientsandCoverings64
2.3TerminalSingularitiesofIndexOne76
2.4TerminalSingularitiesofHigherIndex86
2.5SingularRiemann–RochFormula96
2.6CanonicalSingularities107
3DivisorialContractionstoPoints 116
3.1IdentificationoftheDivisor117
3.2NumericalClassification121
3.3GeneralElephantsforExceptionalType128
3.4GeneralElephantsforOrdinaryType136
3.5GeometricClassification145
3.6Examples157
4DivisorialContractionstoCurves 162
4.1ContractionsfromGorensteinThreefolds162
4.2ContractionstoSmoothCurves168
4.3ContractionstoSingularCurves176
4.4ConstructionbyaOne-Parameter Deformation186
5Flips 199
5.1StrategyandFlops200
5.2NumericalInvariants207
5.3PlanarityofCoveringCurves213
5.4DeformationsofanExtremal Neighbourhood221
5.5GeneralElephants231
5.6Examples244
6TheSarkisovCategory 249
6.1LinksofMoriFibreSpaces250
6.2TheSarkisovProgram255
6.3RationalityandBirationalRigidity265
6.4Pliability275
7ConicalFibrations 285
7.1StandardConicBundles286
7.2 Q-ConicBundles294
7.3Classification303
7.4Rationality309
7.5BirationalRigidity319
8DelPezzoFibrations 324
8.1StandardModels325
8.2SimpleModelsandMultipleFibres333
8.3Rationality343
8.4BirationalRigidity348
9FanoThreefolds 359
9.1Boundedness360
9.2GeneralElephants371
9.3ClassificationinSpecialCases378
9.4ClassificationofPrincipalSeries388
9.5BirationalRigidityandK-Stability398
10MinimalModels 410
10.1Non-Vanishing411
10.2Abundance421
10.3BirationalMinimalModels428
Preface
Thepresentbooktreatsexplicitaspectsofthebirationalgeometryofcomplex algebraicthreefolds.Itisafundamentalprobleminbirationalgeometrytofind andanalyseagoodrepresentativeofeachbirationalclassofalgebraicvarieties. Theminimalmodelprogram,ortheMMPforshort,conjecturallyrealisesthis bycomparingthecanonicaldivisorsofvarieties.AccordingastheMMPhas developed,thebookhasarisenfromtwoperspectives.
Firstly,sincetheMMPindimensionthreewasestablishedaboutaquarter centuryago,itisdesirabletounderstandindividualthreefoldsexplicitlyby meansoftheMMP.Theinitialstepistodescribebirationaltransformationsin theMMP.Nowwehaveapracticalclassificationofthemindimensionthree. TheensuingimportantsubjectistoanalysethethreefoldsoutputbytheMMP. InthisdirectiononecanmentiontheSarkisovprogram,whichdecomposes everybirationalmapofMorifibrespaces.
Secondly,theMMPinhigherdimensionsisstillevolvingactivelyafterthe existenceofflipswasproved,astypifiedinthesettlementoftheBorisov–Alexeev–Borisovconjecture.Agoodknowledgeofthreefoldsisusefulandwill benecessaryinfurtherdevelopmentofhigherdimensionalbirationalgeometry. Thisiscomparabletothenaturethatmostresultsonthreefoldsarebasedupon theclassicaltheoryofsurfaces.
Thebookconcentratesontheexplicitstudyofalgebraicthreefoldsbythe MMP.Theauthorhastriedtoelucidatetheproofsrigorouslyandtomakethe bookasself-containedaspossible.Anumberofexampleswillhelptodeepen theunderstandingofthereader.Thereaderisstronglyencouragedtoverify thecomputationsinexamples.Thoughitdoesnotcoverimportanttopicssuch asaffinegeometry,derivedcategoriesandpositivecharacteristicaspects,the bookwillsupplyenoughknowledgeofthreefoldbirationalgeometrytoenter thefieldofhigherdimensionalbirationalgeometry.
Preface
Thebookisintendedforadvancedgraduatestudentswhoareinterestedin thebirationalgeometryofalgebraicvarieties.Itcanalsobeusedbyresearchers asareferencefortheclassificationresultsonthreefolds.Thereadershouldbe familiarwithbasicalgebraicgeometryatthelevelofHartshorne’stextbook [178].SomeknowledgeoftheMMPishelpfulbutitisnotaprerequisite.One canlearnthegeneraltheoryoftheMMPfromastandardbooksuchasthatby KollárandMori[277]orbyMatsuki[307].Whilstitroughlycorrespondsto Chapter1ofthebook,themainbodystartsfromChapter2andconcentrates onthreefolds.
ThevolumeeditedbyCortiandReid[97]isanoutstandingcollectionfrom thesamestandpoint.Itplayedaguidingroleintheexplicitstudyofalgebraic threefoldswhenitwaspublished.However,greatprogresshasbeenmadesince then.Thepresentbookaimsatanorganisedtreatmentofthreefolds,including recentresults.Itseekstobesomewhatofathreefoldversionofthebookon surfacesbyBarth,Hulek,PetersandVandeVen[30]orbyBeauville[35].
Chapter1summarisesthetheoryoftheMMPinanarbitrarydimensionbut excludesdetailedproofs.Thefirstpart,Chapters2to5,ofthemainbodydeals withobjectswhichappearinthecourseoftheMMP.Chapter2classifiesthreefoldsingularitiesintheMMPcompletely.ThenChapters3,4and5describe threefoldbirationaltransformationsintheMMP,thatis,divisorialcontractions andflips.Theycontainallthenecessaryargumentsbyomittingonlytheparts whichrepeattheprecedingarguments.
Thesecondpart,Chapters6to10,isdevotedtotheanalysisofoutputs oftheMMP,whichareMorifibrespacesandminimalmodels.AfterChapter6explainsthegeneraltheoryoftheSarkisovprogram,Chapters7,8and9 investigatethegeometryofthreefoldMorifibrespacesaccordingtotherelativedimensionofthefibrestructure.FinallyChapter10discussesminimal threefoldsfromthepointofviewofabundance.
Theauthorwouldliketoexpressthankstoallthecolleagueswithwhomhe helddiscussions.YujiroKawamataintroducedhimtothesubjectofbirational geometryashisacademicsupervisor.AlessioCortiandMilesReidcommunicatedtheirprofoundknowledgeofthreefoldstohimduringhisvisittothe UniversityofCambridge.WhenhestayedattheInstituteforAdvancedStudy, hereceivedwarmhospitalityfromJánosKollár.Healsolearntagreatdealfrom theregularseminarorganisedbyShigefumiMori,ShigeruMukaiandNoboru Nakayama.FinallyhewouldliketothankPhilipMeylerandJohnLingleiMeng atCambridgeUniversityPressfortheirsupportforthepublication.
TheMinimalModelProgram
Thischapteroutlinesthegeneraltheoryoftheminimalmodelprogram.Weshall studyalgebraicthreefoldsthoroughlyinthesubsequentchaptersinalignment withtheprogram.Thereaderwhoisnotfamiliarwiththeprogrammaygrasp thebasicnotionsatfirstandreferbacklater.
Blowingupasurfaceatapointisnotanessentialoperationfromthebirational pointofview.Itsexceptionalcurveischaracterisednumericallyasa (−1)-curve. Asisthecaseinthisobservation,theintersectionnumberisabasiclineartool inbirationalgeometry.Theminimalmodelprogram,ortheMMPforshort, outputsarepresentativeofeachbirationalclassthatisminimalwithrespectto thenumericalclassofthecanonicaldivisor.
TheMMPgrewoutofthesurfacetheorywithallowingmildsingularities. Foragivenvariety,itproducesaminimalmodeloraMorifibrespaceafter finitelymanybirationaltransformations,whicharedivisorialcontractionsand flips.Nowtheprogramisformulatedinthelogarithmicframeworkwherewe treatapairconsistingofavarietyandadivisor.
TheMMPfunctionssubjecttotheexistenceandterminationofflips.Hacon andMcKernanwithBirkarandCasciniprovedtheexistenceofflipsinan arbitrarydimension.Consideringafliptobetherelativecanonicalmodel,they establishedtheMMPwithscalinginthebirationalsetting.Theterminationof threefoldflipsfollowsfromthedecreaseinthenumberofdivisorswithsmall logdiscrepancy.Shokurovreducedtheterminationinanarbitrarydimension tocertainconjecturalpropertiesoftheminimallogdiscrepancy.
ItisalsoimportanttoanalysetherepresentativeoutputbytheMMP.The SarkisovprogramdecomposesabirationalmapofMorifibrespacesintoelementaryones.Foraminimalmodel,weexpecttheabundancewhichclaimsthe freedomofthelinearsystemofamultipleofthecanonicaldivisor.Itdefines amorphismtotheprojectivevarietyassociatedwiththecanonicalring,which weknowisfinitelygenerated.
1.1Preliminaries
Weshallfixthenotationandrecallthefundamentalsofalgebraicgeometry. Thebook[178]byHartshorneisastandardreference.
The naturalnumbers beginwithzero.Thesymbol �� ≥�� for �� = N, Z, Q or R standsforthesubset {�� ∈ �� | �� ≥ �� } andsimilarly ��>�� = {�� ∈ �� | ��>�� }
Forinstance, N = Z≥0.Thequotient Z�� = Z/��Z isthe cyclicgroup oforder �� The round-down �� ofarealnumber �� isthegreatestintegerlessthanorequal to ��,whilstthe round-up �� isdefinedas �� = − −��
Schemes A scheme isalwaysassumedtobeseparated.Itissaidtobe integral ifitisirreducibleandreduced.
Weworkoverthefield C ofcomplexnumbersunlessotherwisementioned. An algebraicscheme isaschemeoffinitetypeover Spec �� forthealgebraically closedgroundfield ��,whichistacitlyassumedtobe C.Wecallita complex scheme whenweemphasisethatitisdefinedover C.Analgebraicschemeis saidtobe complete ifitisproperover Spec ��.A point inanalgebraicscheme usuallymeansaclosedpoint.
A variety isanintegralalgebraicscheme.A complexvariety isavariety over C.A curve isavarietyofdimensiononeanda surface isavarietyof dimensiontwo.An ��-fold isavarietyofdimension ��.The affinespace A�� is Spec �� [��1,...,����] andthe projectivespace P�� is Proj �� [��0,...,����].The originof A�� isdenotedby ��.
The germ �� ∈ �� ofaschemeisconsideredataclosedpointunlessotherwise specified.Itisanequivalenceclassofthepair ( ��,��) ofascheme �� andapoint �� in �� where ( ��,��) isequivalentto ( �� ,�� ) ifthereexistsanisomorphism �� �� ofopenneighbourhoods �� ∈ �� ⊂ �� and �� ∈ �� ⊂ �� sending �� to ��
Bya singularity,wemeanthegermatasingularpointasarule.
Foralocallyfreecoherentsheaf E onanalgebraicscheme ��,the projective spacebundle P(E ) = Proj�� ��E over �� isdefinedbythesymmetric O�� -algebra ��E = �� ∈N ���� E of E .Itisa P��-bundle if E isofrank �� + 1.Inparticular,the projectivespace P�� = Proj ���� isdefinedforafinitedimensionalvectorspace �� .Itisregardedasthequotientspace (�� ∨ \ 0)/�� × ofthedualvectorspace �� ∨ minuszerobytheactionofthemultiplicativegroup �� × = �� \{0} oftheground field ��.Asusedabove,thesymbol ∨ standsforthedualand × forthegroupof units.
Morphisms Foramorphism �� : �� → �� ofschemes,the image �� ( ��) ofa closedsubset �� of �� andthe inverseimage �� 1 (��) ofaclosedsubset �� of �� areconsideredset-theoretically.When �� isproperand �� isaclosedsubscheme, weregard �� ( ��) asareducedscheme.Wealsoregard �� 1 (��) foraclosed
subscheme �� asareducedschemeanddistinguishitfromthescheme-theoretic fibre �� �� ��.
A rationalmap �� : �� �� ofalgebraicschemesisanequivalenceclassof amorphism �� → �� definedonadenseopensubset �� of ��.The image �� ( ��) of �� istheimage �� (Γ) ofthegraph Γ of �� asaclosedsubschemeof �� × �� bytheprojection �� : �� × �� → �� .Wesaythatamorphismorarationalmap is birational ifithasaninverseasarationalmap.Twoalgebraicschemesare birational ifthereexistsabirationalmapbetweenthem.Bydefinition,two varietiesarebirationalifandonlyiftheyhavethesamefunctionfield.
Let �� : �� → �� beamorphismofalgebraicschemes.Wesaythat �� is projective ifitisisomorphicto Proj�� R → �� byagraded O�� -algebra R = �� ∈N R�� generatedbycoherent R1,with R0 = O�� .When �� isquasi-projective,the projectivityof �� meansthatitisrealisedasaclosedsubschemeofarelative projectivespace P�� × �� → �� .Aninvertiblesheaf L on �� is relativelyvery ample (or veryample over �� or ��-veryample)ifitisisomorphicto O (1) byan expression �� Proj�� R asabove.Wesaythat L is relativelyample (��-ample) if L ⊗�� isrelativelyveryampleforsomepositiveinteger ��
Supposethat �� : �� → �� isproper.Wesaythat �� hasconnectedfibres ifthe naturalmap O�� → ��∗O�� isanisomorphism.Thisimpliesthatthefibre �� ×�� �� atevery �� ∈ �� isconnectedandnon-empty[160,IIIcorollaire4.3.2].Theproof foraprojectivemorphismisin[178,IIIcorollary11.3].Ingeneral, �� admits the Steinfactorisation �� = �� ◦ �� with �� : �� → �� and �� : �� → �� definedby �� = Spec�� ��∗O�� ,forwhich �� isproperwithconnectedfibresand �� isfinite.If �� isaproperbirationalmorphismfromavarietytoanormalvariety,thenthe factor �� intheSteinfactorisationisanisomorphismandhence �� hasconnected fibres.Thisisreferredtoas Zariski’smaintheorem.
Lemma1.1.1 Let �� : �� → �� and �� : �� → �� bemorphismsofalgebraic schemessuchthat �� isproperandhasconnectedfibres.Ifeverycurvein �� contractedtoapointby �� isalsocontractedby ��,then �� factorsthrough �� as �� = �� ◦ �� foramorphism �� : �� → ��.
Proof Let �� �� and �� �� denotethesetsofclosedpointsin �� and �� respectively. For �� ∈ �� ��,theinverseimage �� 1 (��) isconnectedand ��(�� 1 (��)) isonepoint. Define �� �� : �� �� → �� �� by �� �� (��) = ��(�� 1 (��)).Since �� isproperandsurjective, foranyclosedsubset �� of ��, �� (�� 1 (��)) isclosedin �� and ( �� ��) 1 (��| �� �� ) = �� (�� 1 (��))|�� �� .Thus �� �� extendstoacontinuousmap �� : �� → ��,whichisa morphismofschemesbythenaturalmap O�� → ��∗O�� = ��∗��∗O�� = ��∗O�� .
Chow’slemma [160,II§5.6]replacesthepropermorphism �� : �� → �� byaprojectivemorphism.Itassertstheexistenceofaprojectivebirational
TheMinimalModelProgram
morphism �� : �� → �� suchthat �� ◦ �� : �� → �� isprojective.The projection formula andthe Lerayspectralsequence,formulatedforringedspacesin[160, 0§12.2],willbefrequentlyused.Thereference[198,section3.6]explains spectralsequencesfromourperspective.
Theorem1.1.2 (Projectionformula) Let �� : �� → �� beamorphismofringed spaces.Let F bean O�� -moduleandlet E beafinitelocallyfree O�� -module. Thenthereexistsanaturalisomorphism ���� ��∗F ⊗ E ���� ��∗ (F ⊗ ��∗E )
Theorem1.1.3 (Lerayspectralsequence) Let �� : �� → �� and �� : �� → �� bemorphismsofringedspaces.Let F bean O�� -module.Thenthereexistsa spectralsequence
Inpracticeforaspectralsequence ��
,weassumethat �� ��,�� 2 is zerowhenever �� or �� isnegative.Thenthereexistsanexactsequence
Iffurther �� ��,�� 2 = 0 forall �� ≥ 0
Cohomologies Wewrite ���� (F ) forthecohomology ���� ( ��, F ) ofasheaf F ofabeliangroupsonatopologicalspace �� whenthereisnoconfusion.If �� is noetherian,then ���� (F ) vanishesforall �� greaterthanthedimensionof ��.
Let F beacoherentsheafonanalgebraicscheme ��.If �� isaffine,then ���� (F ) = 0 forall �� ≥ 1.If �� : �� → �� isapropermorphism,thenthehigher directimage ���� ��∗F iscoherent[160,IIIthéorème3.2.1].Inparticularif �� is complete,then ���� (F ) isafinitedimensionalvectorspace.Thedimensionof ���� (F ) isdenotedby ℎ�� (F ).Thealternatingsum ��(F ) = �� ∈N (−1)�� ℎ�� (F ) iscalledthe Eulercharacteristic of F .
Let �� beacompleteschemeofdimension ��.Foracoherentsheaf F and aninvertiblesheaf L on ��,the asymptoticRiemann–Rochtheorem definesthe intersectionnumber (L �� · F )∈ Z bytheexpression ��(L ⊗�� ⊗ F ) = (L �� F ) ��! �� �� + �� (�� �� 1), whereby Landau’ssymbol ��, �� (��) = �� (��(��)) meanstheexistenceofaconstant �� suchthat | �� (��)|≤ ��|��(��)| foranylarge ��.Bythis,Grothendieck’s dévissage yieldstheestimate ℎ�� (F ⊗ L ⊗�� ) = �� (�� ��) forall �� [266,sectionVI.2].
If �� isprojectivewithaveryamplesheaf O�� (1),thentheEulercharacteristic ��(F ⊗ O�� (��)) isdescribedasapolynomialin Q[��],calledthe Hilbert
polynomial of F .Thevanishingof ���� (F ⊗ O�� (��)) belowisknownas Serre vanishing.
Theorem1.1.4 (Serre) Let F beacoherentsheafonaprojectivescheme ��.Thenforanysufficientlylargeinteger ��,thetwistedsheaf F ⊗ O�� (��) is generatedbyglobalsectionsandsatisfies ���� (F ⊗ O�� (��)) = 0 forall �� ≥ 1.
Wehavethe cohomologyandbasechangetheorem forflatfamiliesofcoherentsheaves[160,III§§7.6–7.9],[361,section5].Seealso[178,sectionIII.12].
Theorem1.1.5 (Cohomologyandbasechange) Let �� : �� → �� beaproper morphismofalgebraicschemes.Let F beacoherentsheafon �� flatover �� . Taketherestriction F�� of F tothefibre ���� = �� ×�� �� ataclosedpoint �� in �� andconsiderthenaturalmap
, where �� (��) istheskyscrapersheafoftheresiduefieldat ��.
(i) Thedimension ℎ�� (F�� ) isuppersemi-continuouson �� andtheEulercharacteristic ��(F�� ) islocallyconstanton �� .
(ii) Fix �� and �� andsupposethat ���� �� issurjective.Then ���� �� isanisomorphism forall �� inaneighbourhoodat ��.Further, ���� ��∗F islocallyfreeat �� if andonlyif ���� 1 �� issurjective.
(iii) (Grauert) Supposethat �� isreduced.Fix ��.If ℎ�� (F�� ) islocallyconstant, then ���� ��∗F islocallyfreeand ���� �� isanisomorphism.
Divisors Let �� beanalgebraicscheme.Wewrite K�� forthesheafoftotal quotientringsof O�� .If �� isavariety,thenitistheconstantsheafofthe functionfield �� ( ��) of ��.A Cartierdivisor �� on �� isaglobalsectionofthe quotientsheaf K × �� /O × �� ofmultiplicativegroupsofunits.Itisassociatedwith aninvertiblesubsheaf O�� (��) of K�� .If �� isrepresentedbylocalsections ���� ∈ K × ���� with ���� �� 1 �� ∈ O × ���� ∩�� �� ,then O�� (��)|���� = �� 1 �� O���� .Wesaythat �� is principal ifitisdefinedbyaglobalsectionof K × �� orequivalently O�� (��) O�� .Theprincipaldivisorgivenby �� ∈ Γ( ��, K × �� ) isdenotedby ( �� )�� .If ���� belongsto O���� ∩ K × ���� forall ��,then �� definesaclosedsubscheme of �� andwesaythat �� is effective.
The Picardgroup Pic �� of �� isthegroupofisomorphismclassesofinvertible sheaveson ��.Ithasanisomorphism
Pic �� ��1 (O × �� )
InfactthisholdsforanyringedspaceviaČechcohomology.Theproofisfound in[440,section5.4].Theisomorphismforavariety �� isderivedatoncefrom thevanishingof ��1 (K × �� ) fortheflasquesheaf K × ��
By Serre’scriterion,analgebraicscheme �� isnormalifandonlyifitsatisfies theconditions ��1 and ��2 definedas
(���� ) forany �� ∈ ��, O��,�� isregularif O��,�� isofdimensionatmost �� and (���� ) forany �� ∈ ��, O��,�� isCohen–Macaulayif O��,�� isofdepthlessthan ��, inwhichweconsiderscheme-theoreticpoints �� ∈ ��.Let �� beanormalvariety.
Aclosedsubvarietyofcodimensiononein �� iscalleda primedivisor.A Weil divisor �� on ��,orsimplycalleda divisor,isanelementinthefreeabelian group �� 1 ( ��) generatedbyprimedivisorson ��.ACartierdivisoronanormal varietyisaWeildivisor.EveryWeildivisoronasmoothvarietyisCartier.The divisor �� isexpressedasafinitesum �� = �� ���� ���� ofprimedivisors ���� with non-zerointegers ���� .The support of �� istheunionof ���� .Thedivisor �� is effective ifall ���� arepositive,anditis reduced ifall ���� equalone.Wewrite �� ≤ �� if �� �� iseffective.The linearequivalence �� ∼ �� ofdivisors meansthat �� �� isprincipal.
Thedivisor �� isassociatedwithadivisorialsheaf O�� (��) on ��.A divisorial sheafisareflexivesheafofrankone,whereacoherentsheaf F issaidtobe reflexive ifthenaturalmap F → F ∨∨ tothedoubledualisanisomorphism. Thesheaf O�� (��) isthesubsheafof K�� definedby Γ(��, O�� (��)) = { �� ∈ �� ( ��)|( �� )�� + �� |�� ≥ 0}, inwhichzeroiscontainedinthesetontherightbyconvention.The divisor classgroup Cl �� isthequotientofthegroup �� 1 ( ��) ofWeildivisorsdividedby thesubgroupofprincipaldivisors.Itisregardedasthegroupofisomorphism classesofdivisorialsheaveson �� andhasaninjection Pic ��↩→ Cl ��.
Linearsystems Let �� beanormalcompletevariety.Let �� beaWeildivisor on �� andlet �� beavectorsubspaceofglobalsectionsin ��0 (O�� (��)).The projectivespace Λ= P�� ∨ = (�� \ 0)/�� × where �� isthegroundfieldiscalleda linearsystem on ��.Itdefinesarationalmap �� P�� .When �� = ��0 (O�� (��)), wewrite |�� | = P��0 (O�� (��))∨ andcallita complete linearsystem.Bythe inclusion O�� (��)⊂ K�� ,thelinearsystem |�� | isregardedasthesetofeffective divisors �� linearlyequivalentto ��,and Λ isasubsetof |�� |.Thatis, Λ ⊂|�� | = {�� ≥ 0 | �� ∼ �� }
The baselocus of Λ meansthescheme-theoreticintersection �� = �� ∈Λ �� in ��.Wesaythatthelinearsystem Λ is free if �� isempty.Wesaythat Λ is mobile if �� isofcodimensionatleasttwo.Thedivisor �� issaidtobe free (resp.
mobile)if |�� | isfree(resp.mobile).Bydefinition,afreeWeildivisorisCartier. When ∅ ≠Λ ⊂|�� |, Λ isdecomposedas Λ=Λ + �� withamobilelinear system Λ ⊂|�� �� | andthemaximaleffectivedivisor �� suchthat �� ≤ ��1
forall ��1 ∈ Λ.Theconstituents Λ and �� arecalledthe mobilepart andthe fixedpart of Λ respectively.Therationalmapdefinedby Λ isisomorphicto �� P�� .Thelinearsystem Λ ismobileifandonlyif �� iszero.
Evenif �� isnotcomplete,thelinearsystem Λ= P�� ∨ isdefinedforafinite dimensionalvectorsubspace �� of ��0 (O�� (��)).Weconsider |�� | tobethedirect limit lim −−→�� Λ oflinearsystems.
A general pointinavariety �� meansapointinadenseopensubset �� of ��.A verygeneral pointin �� meansapointintheintersection �� ∈N ���� ofcountably manydenseopensubsets ���� .Thusbythegeneralmemberofthelinearsystem Λ,wemeanageneralpointin Λ asaprojectivespace.Bertini’stheoremasserts thatafreelinearsystemonasmoothcomplexvarietyhasasmoothmember. Thestatementforthehyperplanesectionholdseveninpositivecharacteristic.
Theorem1.1.6 (Bertini’stheorem) Let Λ= P�� ∨ beafreelinearsystemona smoothvariety �� andlet �� : �� → P�� betheinducedmorphism.Supposethat �� isaclosedembeddingorthegroundfieldisofcharacteristiczero.Thenthe generalmember �� of Λ isasmoothdivisoron ��,andiftheimage ��( ��) isof dimensionatleasttwo,then �� isasmoothprimedivisor.
Thecanonicaldivisor Itisthecanonicaldivisorthatplaysthemostimportant roleinbirationalgeometry.The sheafofdifferentials onanalgebraicscheme �� isdenotedby �� .When �� issmooth, �� �� denotesthe ��-thexteriorpower �� �� .
Definition1.1.7 The canonicaldivisor ���� onanormalvariety �� isthe divisordefineduptolinearequivalencebytheisomorphism O�� (���� )|�� �� �� onthesmoothlocus �� in ��,where �� isthedimensionof ��.
Example1.1.8 Theprojectivespace P�� hasthecanonicaldivisor ��P�� ∼ −(�� + 1)�� forahyperplane ��.Thisfollowsfromthe Eulersequence 0 → ΩP�� → OP�� (−1) ⊕(��+1) → OP�� → 0.
Onecandescribe ��P�� inanexplicitway.Takehomogeneouscoordinates ��0,...,���� of P��.Let ���� A�� denotethecomplementofthehyperplane ���� definedby ���� .Thechart ��0 admitsanowherevanishing ��-form ����1 ∧···∧ ������ withcoordinates ��1,...,���� for ���� = ���� �� 1 0 .Itisexpressedonthechart ��1 havingcoordinates ��0,��2,...,���� for ���� = ���� �� 1 1 astherational ��-form ���� 1 0 ∧ �� (��2 �� 1 0 )∧···∧ �� (���� �� 1 0 ) = ��−(��+1) 0 ����0 ∧ ����2 ∧···∧ ������,whichhas poleoforder �� + 1 along ��0.Thus ��P�� ∼−(�� + 1)��0.
Inspiteoftheambiguityconcerninglinearequivalence,itisstandardtotreat thecanonicaldivisorasifitwereaspecifieddivisor.
Foraclosedsubscheme �� ofanalgebraicscheme ��,thereexistsanexact sequence I /I 2 → Ω�� ⊗ O�� → Ω�� → 0,where I istheidealsheafin O�� defining ��.Thisinducesthe adjunctionformula,whichconnectsthecanonical divisortothatonaCartierdivisor.
Theorem1.1.9 (Adjunctionformula) Let �� beanormalvarietyandlet �� be areducedCartierdivisoron �� whichisnormal.Then ���� = (���� + ��)|�� in thesensethat O�� (���� ) O�� (���� + ��)⊗ O��
Duality AlbeitGrothendieck’sdualitytheoryworksinthederivedcategoryfor propermorphisms[177],itisextremelyhardtoobtainthedualisingcomplex andatracemapinacompatiblemanner.Thetheorybecomesefficientif itisrestrictedtotheCohen–Macaulayprojectivecaseasexplainedin[178, sectionIII.7]and[277,section5.5].Forexample,thedualisingcomplexona Cohen–Macaulayprojectivescheme �� ofpuredimension �� istheshift ���� [��] ofthedualisingsheaf ���� .
Definition1.1.10 Let �� beacompleteschemeofdimension �� overanalgebraicallyclosedfield ��.The dualisingsheaf ���� for �� isacoherentsheafon �� endowedwitha tracemap �� : �� �� (���� )→ �� suchthatforanycoherentsheaf F on ��,thenaturalpairing
(F ,���� )×
inducesanisomorphism Hom(F ,���� ) �� �� (F )∨ .
Thedualisingsheafisuniqueuptoisomorphismifitexists.Theprojective space P�� hasthedualisingsheaf ��P�� �� ΩP�� .ThiswithLemma1.1.11yields theexistenceof ���� foreveryprojectivescheme �� bytakingafinitemorphism �� → P�� knownasprojective Noethernormalisation.If �� isembeddedinto aprojectivespace �� withcodimension ��,then ���� Ext�� �� (O�� ,���� ) [178,III proposition7.5].If �� isanormalprojectivevariety,then ���� coincideswith thesheaf O�� (���� ) associatedwiththecanonicaldivisor.
Forafinitemorphism �� : �� → �� ofalgebraicschemes,thepush-forward ��∗ definesanequivalenceofcategoriesfromthecategoryofcoherent O�� -modules tothatofcoherent ��∗O�� -modules.Thisassociateseverycoherentsheaf G on �� functoriallywithacoherentsheaf ��!G on �� satisfying ��∗ H om�� (F ,��!G )
H om�� (��∗F , G ) foranycoherentsheaf F on ��.
Lemma1.1.11 Let �� : �� → �� beafinitemorphismofcompleteschemesof thesamedimension.Ifthedualisingsheaf ���� for �� exists,then ���� = ��!���� is thedualisingsheaffor ��
Proof Let �� denotethecommondimensionof �� and �� .Foracoherentsheaf F on ��, Hom�� (F ,��!���� ) = Hom�� (��∗F ,���� ) isdualto �� �� (F ) = �� �� (��∗F ) bythepropertyof ���� ,wherethelatterequalityfollowsfromtheLerayspectral sequence �� �� (���� ��∗F )⇒ �� ��+�� (F )
ThedualityforCohen–Macaulaysheavesonaprojectiveschemeisderived fromthatontheprojectivespaceviaprojectiveNoethernormalisation.See [277,theorem5.71].
Theorem1.1.12 (Serreduality) Let �� beaprojectiveschemeofdimension ��.Let F beaCohen–Macaulaycoherentsheafon �� withsupportofpure dimension ��.Then ���� (H om�� (F ,���� )) isdualto �� �� �� (F ) forall ��
The adjunctionformula ���� ���� ⊗ O�� (��)⊗ O�� holdsforaCohen–Macaulayprojectivescheme �� ofpuredimensionandaneffectiveCartier divisor �� on ��.CompareitwithTheorem1.1.9.
Resolutionofsingularities Aprojectivebirationalmorphismisdescribedasa blow-up.The blow-up ofanalgebraicscheme �� alongacoherentidealsheaf I in O�� ,oralongtheclosedsubschemedefinedby I ,istheprojectivemorphism �� : �� = Proj�� �� ∈N I �� → ��.Thepull-back IO�� = �� 1I · O�� in O�� isan invertibleidealsheaf.Noticethat IO�� isdifferentfrom ��∗I .Theblow-up �� hastheuniversalpropertythateverymorphism �� : �� → �� thatmakes IO�� invertiblefactorsthrough �� as �� = �� ◦ �� foramorphism �� : �� → ��
Let �� : �� �� beabirationalmapofvarieties.The exceptionallocus of �� isthelocusin �� where �� isnotbiregular.Let �� beaclosedsubvarietyof �� notcontainedintheexceptionallocusof �� .The stricttransform ��∗ �� in �� of �� istheclosureoftheimageof �� �� .When �� and �� arenormal,the strict transform ��∗ �� in �� ofanarbitraryprimedivisor �� on �� isdefinedasadivisor insuchamannerthat ��∗ �� iszeroif �� isintheexceptionallocusof �� .Bylinear extension,wedefinethestricttransform ��∗ �� in �� foranydivisor �� on ��.
Resolutionofsingularitiesisafundamentaltoolincomplexbirationalgeometry.Wesaythatareduceddivisor �� onasmoothvariety �� is simplenormal crossing,or snc forshort,if �� isdefinedateverypoint �� in �� bytheproduct ��1 ���� ofapartofaregularsystem ��1,...,���� ofparametersin O��,�� .
Definition1.1.13 A resolution ofavariety �� isaprojectivebirationalmorphism �� : �� → �� fromasmoothvariety.Theresolution �� issaidtobe strong ifitisisomorphiconthesmoothlocusin ��.
Definition1.1.14 Let �� beanormalvariety,let Δ beadivisoron �� andlet I beacoherentidealsheafin O�� .A logresolution of ( ��, Δ, I ) isaresolution �� : �� → �� suchthat
• theexceptionallocus �� of �� isadivisoron �� ,
• thepull-back IO�� isinvertibleandhencedefinesadivisor �� and
• �� + �� + �� 1 ∗ �� hassncsupportforthesupport �� of Δ.
Thelogresolution �� issaidtobe strong ifitisisomorphiconthemaximal locus �� in �� suchthat �� issmooth, I |�� definesadivisor ���� and ���� + ��|�� hassncsupport.A(strong)logresolutionof �� meansthatof ( ��, 0, O�� ),and thoseof ( ��, Δ) and ( ��, I ) arelikewisedefined.
TheexistenceoftheseresolutionsforcomplexvarietiesisduetoHironaka. Theitems(i)and(ii)belowarederivedfromthemaintheoremsIandIIin [187]respectively.
Theorem1.1.15 (Hironaka[187]) (i) Astrongresolutionexistsforevery complexvariety.
(ii) Astronglogresolutionexistsforeverypair ( ��, I ) ofasmoothcomplex variety �� andacoherentidealsheaf I in O��
Hironaka’sconstructionincludestheexistenceofastronglogresolution �� → �� equippedwithaneffectiveexceptionaldivisor �� on �� suchthat O�� (−�� ) isrelativelyample.
Analyticspaces Weshalloccasionallyconsideracomplexschemetobean analyticspaceintheEuclideantopology.Whilstanalgebraicschemeisobtained bygluingaffineschemesin A��,ananalyticspaceisconstructedbygluing analyticmodelsinadomainin C��.Areferenceis[151].Theringofconvergent complexpowerseriesisdenotedby C{��1,...,���� }
Let �� beadomaininthecomplexmanifold C��.Let O�� denotethesheafof holomorphicfunctionson ��.Let I beanidealsheafin O�� generatedbya finitenumberofglobalsections.Thelocally C-ringedspace (��, (O�� /I )|��) forthesupport �� ofthequotientsheaf O�� /I iscalledan analyticmodel,where being C-ringed meanshavingthestructuresheafof C-algebras.An analytic space isalocally C-ringedHausdorffspacesuchthateverypointhasanopen neighbourhoodisomorphictoananalyticmodel.
Everycomplexscheme �� isassociatedwithananalyticspace ��ℎ .This definesafunctor ℎ fromthecategoryofcomplexschemestothecategoryof analyticspaces.Thereexistsanaturalmorphism ��ℎ → �� oflocally C-ringed spaceswhichmaps ��ℎ bijectivelytothesetofclosedpointsin ��.Itpullsback acoherentsheaf F on �� toacoherentsheaf Fℎ on ��ℎ .When �� iscomplete,it
inducesanequivalenceofcategories.Thisisknownasthe GAGAprinciple, whichtakestheacronymfromthetitleofSerre’spaper[414].
Theorem1.1.16 (GAGAprinciple[163,exposéXII],[414]) Let �� bea completecomplexschemeandlet ��ℎ betheanalyticspaceassociatedwith �� Thenthefunctor ℎ inducesanequivalenceofcategoriesfromthecategoryof coherentsheaveson �� tothecategoryofcoherentsheaveson ��ℎ .
Forananalyticspace �� ,theexponentialfunction exp(2��√ 1��) definesa grouphomomorphism O�� → O × �� .Theinducedexactsequence
iscalledthe exponentialsequence
Inprinciple,onecandealwithanalyticspacesanalogouslytocomplex schemesasin[29].Forananalyticspace �� ,the Oka–Cartantheorem assertsthecoherenceofeveryidealsheafin O�� thatdefinesananalyticsubspace of �� .Forapropermap �� : �� → �� ofanalyticspaces,thehigherdirectimage ���� ��∗F ofacoherentsheaf F on �� iscoherenton �� .Inparticular,theimage �� (�� ) isthesupportoftheanalyticsubspaceof �� definedbythekernelofthe map O�� → ��∗O�� ,whichisreferredtoasthe propermappingtheorem.
Thecanonicaldivisoronanormalanalyticspacemaynotbedefinedasa finitesumofprimedivisors.Somenotionssuchasprojectivityofresolutionof singularitiesonlymakesenseonasmallneighbourhoodaboutafixedcompact subsetofananalyticspace.Thesewillposenoobstaclesaswemainlyworkon thegermatapointintheanalyticcategory.
Notation1.1.17 Thesymbol ���� denotesadomaininthecomplexspace C�� whichcontainstheorigin ��.Forexample,wewrite �� ∈ ���� foragermofa complexmanifold.
1.2NumericalGeometry
Theintersectionnumberisabasiclineartoolinbirationalgeometry.Weshall defineitintherelativesettingofapropermorphism �� → ��.Thissectionworks overanalgebraicallyclosedfield �� ofanycharacteristic.
Oneencountersdivisorswithrationalcoefficientsnaturally.Forexamplefor afinitesurjectivemorphism �� → �� ofsmoothvarietiestamelyramifiedalong asmoothprimedivisor �� on �� ,theramificationformulawhichwillbeproved inTheorem2.2.20expresses ���� asthepull-backof ���� +(1 1/��) �� with theramificationindex �� along ��.Onealsohasdivisorswithrealcoefficients takinglimits.Webeginwithformulationofthesenotions.
Let �� beanormalvariety.Let �� 1 ( ��) denotethegroupofWeildivisorson ��.A Q-divisor isanelementintherationalvectorspace �� 1 ( ��)⊗ Q.Inlike manner,an R-divisor isanelementin �� 1 ( ��)⊗ R.An R-divisor �� isexpressed asafinitesum �� = �� ���� ���� ofprimedivisors ���� withrealcoefficients ���� ,and �� isa Q-divisorif ���� arerational.Itis effective if ���� ≥ 0 forall �� and �� ≤ �� meansthat �� �� iseffective.The round-down �� andthe round-up �� are definedas �� = �� ���� ���� and �� = �� ���� ���� .Wesometimessaythata usualdivisoris integral todistinguishitfroma Q-divisorandan R-divisor.
Let ��1 ( ��) denotethesubgroupof �� 1 ( ��) generatedbyCartierdivisorson ��.A Q-Cartier Q-divisorisanelementintherationalvectorspace ��1 ( ��)⊗ Q
Inotherwords,a Q-divisor �� is Q-Cartierifandonlyifthereexistsanon-zero integer �� suchthat ���� isintegralandCartier.Likewisean R-Cartier R-divisor isanelementin ��1 ( ��)⊗ R.An R-Cartier Q-divisorisalways Q-Cartierbuta Q-CartierintegraldivisorisnotnecessarilyCartier.
Example1.2.1 Considertheprimedivisor �� = (��1 = ��2 = 0) onthesurface �� = (��2 1 ��2��3 = 0)⊂ A3 withcoordinates ��1,��2,��3.Then 2�� istheCartier divisordefinedby ��2 andthescheme-theoreticintersection 2�� ∩ �� withtheline �� = (��1 = ��2 = ��3) in �� isoflengthone.Itfollowsthat �� isnotCartier.
Let �� : �� → �� beamorphismofnormalvarieties.The pull-back ��∗ �� ofan R-Cartier R-divisor �� on �� isdefinedasan R-Cartier R-divisoron �� bythe naturalmap ��∗ : ��1 ( ��)⊗ R → ��1 (�� )⊗ R.If �� isa Q-divisor,thensois ��∗ ��.
Definition1.2.2 Let �� beanormalvariety.Wesaythat �� is Q-Gorenstein if thecanonicaldivisor ���� is Q-Cartier.Wesaythat �� is Q-factorial ifalldivisors on �� are Q-Cartier,thatis, Cl ��/Pic �� istorsion.Itissaidtobe factorial ifall divisorsareCartier,thatis, Pic �� = Cl ��
The Q-factorialityisnotananalyticallylocalproperty.
Example1.2.3 Thealgebraicgerm �� ∈ �� = (��1��2 + ��3��4 = 0)⊂ A4 isnot Q-factorial.Theprimedivisor �� = (��1 = ��4 = 0) on �� isnot Q-Cartierand thedivisorclassgroup Cl �� is Z[��] Z.Indeed,theblow-up �� of �� at �� resolvestheprojection �� P3 from �� asamorphism �� → P3 andityields alinebundle �� → �� overthesurface �� = (��1��2 + ��3��4 = 0) P1 × P1 ⊂ P3 . Bythisstructure, Pic �� isgeneratedbythestricttransforms �� �� and �� �� of �� and �� = (��2 = ��4 = 0).Theysatisfytherelation �� �� + �� �� + �� ∼ 0 forthe exceptionaldivisor �� of �� → ��.Thus Cl �� Pic(�� \ ��) = Z[�� �� \ ��].
Ontheotherhand,thealgebraicgerm �� ∈ �� = (��1��2 + ��3��4 + �� = 0)⊂ A4 isfactorialforageneralcubicform �� in ��1,...,��4.Toseethis,wecompactify �� to �� = (��0 (��1��2 + ��3��4)+ �� = 0)⊂ P4.Theblow-up �� of �� at �� resolves
theprojectionfrom �� as �� → P3,andthisistheblow-upof P3 alongthe sexticcurve (��1��2 + ��3��4 = �� = 0).Bythisstructure, Pic �� isgeneratedbythe exceptionaldivisor �� of �� → �� andthestricttransform ���� of �� = �� \ �� .Thus
Cl �� Pic(�� \(�� + ���� )) = 0
Thetwogerms �� ∈ �� and �� ∈ �� becomeisomorphicintheanalyticcategory aswillbeseeninProposition2.3.3.SeeRemark3.1.11forfurtherdiscussion.
Weshallfixthebasescheme �� andwork relatively onapropermorphism �� : �� → �� ofalgebraicschemes,whichisfrequentlydenotedby ��/��.Every terminologyisaccompaniedbythereferencetotherelativesetting.Thereferenceisomittedwhenweconsideracompletescheme �� withthestructure morphism �� → �� = Spec ��.
A relativesubvariety �� of ��/�� meansaclosedsubvarietyof �� suchthat �� (��) isapointin ��.A relative ��-cycle on ��/�� isanelementinthefreeabelian group ���� ( ��/��) generatedbyrelativesubvarietiesofdimension �� in ��/��.For invertiblesheaves L1,..., L�� andarelative ��-cycle �� on ��,the intersection number (L1 ··· L�� · ��) isdefinedbythemultilinearmap (Pic ��) ⊕�� × ���� ( ��/��)→ Z suchthat (L �� ��) forarelativesubvariety �� coincideswith (L �� O�� ) in theasymptoticRiemann–Rochtheorem ��(L ⊗�� ⊗ O�� ) = (L �� O�� )�� ��/��! + �� (�� �� 1).The intersectionnumber (��1 �� �� ��) withCartierdivisors ���� on �� isdefinedas (O�� (��1)··· O�� (�� ��)· ��).If ���� areeffectiveandintersect properlyonarelativesubvariety ��,then (��1 �� �� ��) equalsthelengthofthe structuresheaf O�� oftheartinianscheme �� = ��1 ∩···∩ �� �� ∩ ��.Thelengthof O��,�� for �� ∈ �� isreferredtoasthe localintersectionnumber at �� anddenoted by (��1 ··· �� �� · ��)�� .When �� isacompletevarietyofdimension �� withthe structuremorphism �� → �� = Spec ��,wewrite (L1 ··· L��) = (L1 ··· L�� · ��) and ��1 ··· �� �� = (��1 ··· �� ��)�� = (��1 ··· �� �� · ��)
Bytheextension (Pic �� ⊗ R)×(��1 ( ��/��)⊗ R)→ R,the relativenumerical equivalence ≡�� isdefinedinboththerealvectorspaces Pic �� ⊗R and ��1 ( ��/��)⊗ R insuchawaythatitinducesaperfectpairing �� 1 ( ��/��)× ��
( ��/��)→ R ofvectorspacesonthequotients �� 1 ( ��/��) = (Pic �� ⊗ R)/≡�� and ��1 ( ��/��) = (��1 ( ��/��)⊗ R)/≡�� .When �� = Spec ��,wejustwrite ≡, �� 1 ( ��) and ��1 ( ��) withoutreferenceto �� asremarkedabove.
Definition1.2.4 Thespaces �� 1 ( ��/��) and ��1 ( ��/��) arefinitedimensional [254,IV§4,proposition3].Theequaldimensionof �� 1 ( ��/��) and ��1 ( ��/��) iscalledthe relativePicardnumber of ��/�� anddenotedby �� ( ��/��).When
�� = Spec ��,thisnumberiscalledthe Picardnumber ofthecompletescheme �� anddenotedby �� ( ��).
Let �� : �� → �� beapropermorphism.Itinducesthe pull-back ��∗ :Pic �� → Pic �� andthe push-forward ��∗ : ���� (�� /��)→ ���� ( ��/��) asgrouphomomorphisms.Thepush-forward ��∗ �� ofarelativesubvariety �� of �� /�� is ����(��) if themorphism �� → ��(��) isgenericallyfiniteofdegree ��,and ��∗ �� iszeroif ��(��) isofdimensionlessthanthatof ��.Thesesatisfythe projectionformula (�� ∗L1 �� ∗L�� ��) = (L1 L�� ��∗ ��) forinvertiblesheaves L�� on �� andarelative ��-cycle �� on �� .Theyyield ��∗ : �� 1 ( ��/��)→ ��
(�� /��) anddually ��
(�� /��)→ ��1 ( ��/��).Onealso hasthenaturalsurjection �� 1 (�� /��) �� 1 (�� /��) andinjection ��1 (�� /��) ↩→ ��1 (�� /��).If �� issurjective,then ��∗ : �� 1 ( ��/��)→ �� 1 (�� /��) isinjectiveand ��∗ : ��1 (�� /��)→ ��1 ( ��/��) issurjective. Henceforthwefixapropermorphism �� : �� → �� fromanormalvarietyto avarietyandmakebasicdefinitionsforan R-Cartier R-divisor �� on ��.We saythatintegraldivisors �� and �� on �� are relativelylinearlyequivalent and write �� ∼�� �� ifthedifference �� �� islinearlyequivalenttothepull-back ��∗ �� ofsomeCartierdivisor �� on ��.Namely, �� �� iszerointhequotient Cl ��/��∗ Pic ��.For R-divisors �� and �� on ��,the relative R-linearequivalence �� ∼R,�� �� meansthat �� �� iszeroin (Cl ��/��∗ Pic ��)⊗ R.When �� and �� are Q-divisors,thisisreferredtoasthe relative Q-linearequivalence and denotedby �� ∼Q,�� �� .Thespace Pic �� ⊗ R isregardedasthatof R-linear equivalenceclassesof R-Cartier R-divisorson ��.Theintersectionnumber (�� · ��) isdefinedforapairofan R-Cartier R-divisor �� on �� andarelative one-cycle �� on ��/��.Thismakesthenotionof relativenumericalequivalence �� ≡�� �� for R-Cartier R-divisors �� and �� on ��.
Definition1.2.5 An R-Cartier R-divisor �� on ��/�� issaidtobe relativelynef (or nef over �� or ��-nef )if (�� · ��)≥ 0 foranyrelativecurve �� in ��/��.When �� = Spec ��,wejustsaythat �� is nef asusual.
ACartierdivisor �� on �� issaidtobe relativelyample (��-ample)if O�� (��) is arelativelyampleinvertiblesheaf.Itissaidtobe relativelyveryample (��-very ample)if O�� (��) isarelativelyveryampleinvertiblesheaf.Inspiteofthe geometricdefinition,theamplenessischaracterisednumerically.
Theorem1.2.6 (Nakai’scriterion) Let �� : �� → �� beapropermorphismof algebraicschemes.Aninvertiblesheaf L on �� isrelativelyampleifandonly if (L dim �� ��) > 0 foranyrelativesubvariety �� of ��/��.
