内容简介

  Thisbookwillbeveryusefulasareferenceandresearchguideforresearchersandgraduatestudentsinalgebraicgeometry.Theaimofthissurvey,writtenbyV.A.lskovskikhandYu.G.Prokhorov,istoprovideanexpositionofthestructuretheoryofFanovarieties,i.e.algebraicvarietieswithanampleanticanonicaldivisor.SuchvarietiesnaturallyappearinthebirationalclassificationofvarietiesofnegativeKodairadimension,andtheyareveryclosetorationalones.ThisEMSvolumecoversdifferentapproachestotheclassificationofFanovarietiessuchastheclassicalFanolskovskikh"doubleprojection"methodanditsmodifications,thevectorbundlesmethodduetoS.Mukai,andthemethodofextremalrays.TheauthorsdiscussunirulednessandrationalconnectednessaswellasrecentprogressinrationalityproblemsofFanovarieties.TheappendixcontainstablesofsomeclassesofFanovarieties.

目录

Introduction

Chapter1.Preliminaries
1.1.Singularities
1.2.OnNumericalGeometryofCycles
1.3.OntheMoriMinimalModelProgram
1.4.ResultsonMinimalModelsinDimensionThree

Chapter2.BasicPropertiesofFanoVarieties
2.1.Definitions,ExamplesandtheSimplestProperties
2.2.SomeGeneralResults
2.3.ExistenceofGoodDivisorsintheFundamentalLinearSystem
2.4.BasePointsintheFundamentalLinearSystem

Chapter3.DelPezzoVarietiesandFanoVarietiesofLargeIndex
3.1.OnSomePreliminaryResultsofFujita
3.2.DelPezzoVarieties.DefinitionandPreliminaryResults
3.3.NonsingulardelPezzoVarieties.StatementoftheMainTheoremandBeginningoftheProof
3.4.DelPezzoVarietieswithPicardNumberp=1
ContinuationoftheProofoftheMainTheorem
3.5.DelPezzoVarietieswithPicardNumberp≥2
ConclusionoftheProofoftheMainTheorem

Chapter4.FanoThreefoldswithp=1
4.1.ElementaryRationalMaps:PreliminaryResults
4.2.FamiliesofLinesandConicsonFanoThreefolds
4.3.ElementaryRationalMapswithCenteralongaLine
4.4.ElementaryRationalMapswithCenteralongaConic
4.5.ElementaryRationalMapswithCenterataPoint
4.6.SomeOtherRationalMaps

Chapter5.FanoVarietiesofCoindex3withp=1
TheVectorBundleMethod
5.1.FanoThreefoldsofGenus6and8:Gushel'sApproach
5.2.AReviewofMukai'sResultsontheClassificationofFanoManifoldsofCoindex3

Chapter6.BoundednessandRationalConnectednessofFanoVarieties
6.1.Uniruledness
6.2.RationalConnectednessofFanoVarieties

Chapter7.FanoVarietieswithp≥2
7.1.FanoThreefoldswithPicardNumberp≥2(SurveyofResultsofMoriandMukai
7.2.ASurveyofResultsaboutHigher-dimensionalFanoVarietieswithPicardNumberp≥2

Chapter8.RationalityQuestionsforFanoVarietiesI
8.1.IntermediateJacobianandPrymVarieties
8.2.IntermediateJacobian:theAbel-JacobiMap
8.3.TheBrauerGroupasaBirationalInvariant

Chapter9.RationalityQuestionsforFanoVarietiesII
9.1.BirationalAutomorphismsofFanoVarieties
9.2.DecompositionofBirationalMapsintheContextofMoriTheory

Chapter10.SomeGeneralConstructionsofRationalityandUnirationality
10.1.SomeConstructionsofUnirationality
10.2.UnirationalityofCompleteIntersections
10.3.SomeGeneralConstructionsofRationality

Chapter11.SomeParticularResultsandOpenProblems
11.1.OntheClassificationofThree-dimensional-FanoVarieties
11.2.Generalizations
11.3.SomeParticularResults
11.4.SomeOpenProblems

Chapter12.Appendix:Tables
12.1.DelPezzoManifolds
12.2.FanoThreefoldswithp=1
12.3.FanoThreefoldswithp=2
12.4.FanoThreefoldswithp=3
12.5.FanoThreefoldswithp=4
12.6.FanoThreefoldswithp≥5
12.7.FanoFourfoldsofIndex2withp≥2
12.8.ToricFanoThreefolds

References
Index

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