Cox Rings PDF
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CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 144 EditorialBoard B. BOLLOBA´S, W. FULTON, A. KATOK, F. KIRWAN, P. SARNAK, B. SIMON, B. TOTARO COX RINGS Coxringsaresignificantglobalinvariantsofalgebraicvarieties,naturallygeneralizing homogeneouscoordinateringsofprojectivespaces.Thisbookprovidesalargelyself- contained introduction to Cox rings, with a particular focus on concrete aspects of thetheory.Besidestherigorouspresentationofthebasicconcepts,othercentraltopics includethecaseoffinitelygeneratedCoxringsanditsrelationtotoricgeometry;various classesofvarietieswithgroupactions;thesurfacecase;andapplicationsinarithmetic problems,inparticularManin’sconjecture.Theintroductorychaptersrequireonlybasic knowledgeofalgebraicgeometry.Themoreadvancedchaptersalsotouchonalgebraic groups,surfacetheory,andarithmeticgeometry. Eachchapterendswithexercisesandproblems.Thesecomprisemini-tutorialsand examplescomplementingthetext,guidedexercisesfortopicsnotdiscussedinthetext, and,finally,severalopenproblemsofvaryingdifficulty. IvanArzhantsevreceivedhisdoctoraldegreein1998fromLomonosovMoscowState Universityandisaprofessorinitsdepartmentofhigheralgebra.Hisresearchareasare algebraicgeometry,algebraicgroups,andinvarianttheory. UlrichDerenthalreceivedhisdoctoraldegreein2006fromUniversita¨tGo¨ttingen.He isaprofessorofmathematicsatLeibnizUniversita¨tHannover.Hisresearchinterests includearithmeticgeometryandnumbertheory. Ju¨rgen Hausen received his doctoral degree in 1995 from Universita¨t Konstanz. He is a professor of mathematics at Eberhard Karls Universita¨t Tu¨bingen. His field of research is algebraic geometry, in particular algebraic transformation groups, torus actions,geometricinvarianttheory,andcombinatorialmethods. Antonio Laface received his doctoral degree in 2000 from Universita` degli Studi di Milano. He is an associate professor of mathematics at Universidad de Concepcio´n. His field of research is algebraic geometry, more precisely linear systems, algebraic surfaces,andtheirCoxrings. CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS EDITORIAL BOARD B.Bolloba´s,W.Fulton,A.Katok,F.Kirwan,P.Sarnak,B.Simon,B.Totaro AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridgeUniversity Press.Foracompleteserieslistingvisit:www.cambridge.org/mathematics. Alreadypublished 104 A.Ambrosetti&A.MalchiodiNonlinearanalysisandsemilinearellipticproblems 105 T.Tao&V.H.VuAdditivecombinatorics 106 E.B.DaviesLinearoperatorsandtheirspectra 107 K.KodairaComplexanalysis 108 T.Ceccherini-Silberstein,F.Scarabotti,&F.TolliHarmonicanalysisonfinitegroups 109 H.GeigesAnintroductiontocontacttopology 110 J.FarautAnalysisonLiegroups:Anintroduction 111 E.ParkComplextopologicalK-theory 112 D.W.StroockPartialdifferentialequationsforprobabilists 113 A.Kirillov,JrAnintroductiontoLiegroupsandLiealgebras 114 F.Gesztesyetal.Solitonequationsandtheiralgebro-geometricsolutions,II 115 E.deFaria&W.deMeloMathematicaltoolsforone-dimensionaldynamics 116 D.ApplebaumLe´vyprocessesandstochasticcalculus(2ndEdition) 117 T.SzamuelyGaloisgroupsandfundamentalgroups 118 G.W.Anderson,A.Guionnet,&O.ZeitouniAnintroductiontorandommatrices 119 C.Perez-Garcia&W.H.SchikhofLocallyconvexspacesovernon-Archimedeanvalued fields 120 P.K.Friz&N.B.VictoirMultidimensionalstochasticprocessesasroughpaths 121 T.Ceccherini-Silberstein,F.Scarabotti,&F.TolliRepresentationtheoryofthesymmetric groups 122 S.Kalikow&R.McCutcheonAnoutlineofergodictheory 123 G.F.Lawler&V.LimicRandomwalk:Amodernintroduction 124 K.Lux&H.PahlingsRepresentationsofgroups 125 K.S.Kedlayap-adicdifferentialequations 126 R.Beals&R.WongSpecialfunctions 127 E.deFaria&W.deMeloMathematicalaspectsofquantumfieldtheory 128 A.TerrasZetafunctionsofgraphs 129 D.Goldfeld&J.HundleyAutomorphicrepresentationsandL-functionsforthegeneral lineargroup,I 130 D.Goldfeld&J.HundleyAutomorphicrepresentationsandL-functionsforthegeneral lineargroup,II 131 D.A.CravenThetheoryoffusionsystems 132 J.Va¨a¨na¨nenModelsandgames 133 G.Malle&D.TestermanLinearalgebraicgroupsandfinitegroupsofLietype 134 P.LiGeometricanalysis 135 F.MaggiSetsoffiniteperimeterandgeometricvariationalproblems 136 M.Brodmann&R.Y.SharpLocalcohomology(2ndEdition) 137 C.Muscalu&W.SchlagClassicalandmultilinearharmonicanalysis,I 138 C.Muscalu&W.SchlagClassicalandmultilinearharmonicanalysis,II 139 B.HelfferSpectraltheoryanditsapplications 140 R.Pemantle&M.C.WilsonAnalyticcombinatoricsinseveralvariables 141 B.Branner&N.FagellaQuasiconformalsurgeryinholomorphicdynamics 142 R.M.DudleyUniformcentrallimittheorems(2ndEdition) 143 T.LeinsterBasiccategorytheory Cox Rings IVAN ARZHANTSEV MoscowStateUniversity ULRICH DERENTHAL LeibnizUniversita¨tHannover JU¨ RGEN HAUSEN EberhardKarlsUniversita¨tTu¨bingen ANTONIO LAFACE UniversidaddeConcepcio´n 32AvenueoftheAmericas,NewYork,NY10013-2473,USA CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learning,andresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781107024625 (cid:2)C IvanArzhantsev,UlrichDerenthal,Ju¨rgenHausen,andAntonioLaface2015 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2015 PrintedintheUnitedStatesofAmerica AcatalogrecordforthispublicationisavailablefromtheBritishLibrary. LibraryofCongressCataloginginPublicationData Arzhantsev,I.V.(IvanVladimirovich),1972– [Kol’tsaKoksa.English] Coxrings/IvanArzhantsev,DepartmentofAlgebra,FacultyofMechanicsand Mathematics,Moscow[andthreeothers]. pages cm.–(Cambridgestudiesinadvancedmathematics) Includesbibliographicalreferencesandindex. ISBN978-1-107-02462-5(hardback) 1.Algebraicvarieties. 2.Rings(Algebra) I.Title. QA564.A7913 2015 516.3(cid:3)53–dc23 2014005540 ISBN978-1-107-02462-5Hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofURLsfor externalorthird-partyInternetWebsitesreferredtointhispublicationanddoesnotguarantee thatanycontentonsuchWebsitesis,orwillremain,accurateorappropriate. Contents Introduction page1 1 Basicconcepts 6 1.1 Gradedalgebras 7 1.1.1 Monoidgradedalgebras 7 1.1.2 Veronesesubalgebras 10 1.2 Gradingsandquasitorusactions 13 1.2.1 Quasitori 13 1.2.2 Affinequasitorusactions 16 1.2.3 Goodquotients 20 1.3 Divisorialalgebras 24 1.3.1 Sheavesofdivisorialalgebras 24 1.3.2 Therelativespectrum 27 1.3.3 Uniquefactorizationintheglobalring 30 1.3.4 Geometryoftherelativespectrum 32 1.4 CoxsheavesandCoxrings 35 1.4.1 Freedivisorclassgroup 35 1.4.2 Torsioninthedivisorclassgroup 39 1.4.3 Well-definedness 41 1.4.4 Examples 44 1.5 AlgebraicpropertiesoftheCoxring 47 1.5.1 Integrityandnormality 47 1.5.2 Localizationandunits 50 1.5.3 Divisibilityproperties 52 1.6 GeometricrealizationoftheCoxsheaf 56 1.6.1 Characteristicspaces 56 1.6.2 Divisorclassesandisotropygroups 59 1.6.3 Totalcoordinatespaceandirrelevantideal 63 1.6.4 CharacteristicspacesviaGIT 65 ExercisestoChapter1 69 v vi Contents 2 ToricvarietiesandGaleduality 75 2.1 Toricvarieties 75 2.1.1 Toricvarietiesandfans 75 2.1.2 Sometoricgeometry 78 2.1.3 TheCoxringofatoricvariety 82 2.1.4 GeometryofCox’sconstruction 86 2.2 LinearGaleduality 89 2.2.1 Fansandbunchesofcones 89 2.2.2 TheGKZdecomposition 93 2.2.3 ProofofTheorem2.2.1.14 96 2.2.4 ProofofTheorems2.2.2.2,2.2.2.3,and2.2.2.6 100 2.3 Goodtoricquotients 103 2.3.1 Characterizationofgoodtoricquotients 103 2.3.2 Combinatoricsofgoodtoricquotients 107 2.4 Toricvarietiesandbunchesofcones 110 2.4.1 Toricvarietiesandlatticebunches 110 2.4.2 Toricgeometryviabunches 113 ExercisestoChapter2 117 3 Coxringsandcombinatorics 123 3.1 GITforaffinequasitorusactions 124 3.1.1 Orbitcones 124 3.1.2 Semistablequotients 128 3.1.3 A -quotients 133 2 3.1.4 QuotientsofH-factorialaffinevarieties 137 3.2 Bunchedrings 142 3.2.1 Bunchedringsandtheirvarieties 142 3.2.2 ProofstoSection3.2.1 147 3.2.3 Example:Flagvarieties 151 3.2.4 Example:Quotientsofquadrics 155 3.2.5 Thecanonicaltoricembedding 161 3.3 Geometryviadefiningdata 167 3.3.1 Stratificationandlocalproperties 167 3.3.2 Baselociandconesofdivisors 172 3.3.3 Completeintersections 178 3.3.4 Moridreamspaces 181 3.4 Varietieswithatorusactionofcomplexity1 186 3.4.1 Detectingfactorialgradings 186 3.4.2 Factoriallygradedringsofcomplexity1 191 3.4.3 T-varietiesofcomplexity1viabunchedrings 197 3.4.4 GeometryofT-varietiesofcomplexity1 202 ExercisestoChapter3 207 Contents vii 4 Selectedtopics 215 4.1 Toricambientmodifications 216 4.1.1 TheCoxringofanembeddedvariety 216 4.1.2 Algebraicmodification 220 4.1.3 Toricambientmodifications 226 4.1.4 Computingexamples 232 4.2 Liftingautomorphisms 236 4.2.1 Quotientpresentations 236 4.2.2 Linearizationoflinebundles 241 4.2.3 Liftinggroupactions 246 4.2.4 AutomorphismsofMoridreamspaces 251 4.3 Finitegeneration 257 4.3.1 Generalcriteria 257 4.3.2 Finitegenerationviamultiplicationmap 262 4.3.3 FinitegenerationafterHuandKeel 268 4.3.4 Cox–Nagatarings 274 4.4 Varietieswithtorusaction 279 4.4.1 TheCoxringofavarietywithtorusaction 279 4.4.2 H-factorialquasiaffinevarieties 284 4.4.3 ProofofTheorems4.4.1.2,4.4.1.3,and4.4.1.6 291 4.5 Almosthomogeneousvarieties 297 4.5.1 Homogeneousspaces 297 4.5.2 Smallembeddings 303 4.5.3 Examplesofsmallembeddings 308 4.5.4 Sphericalvarieties 314 4.5.5 Wonderfulvarietiesandalgebraicmonoids 319 ExercisestoChapter4 326 5 Surfaces 334 5.1 Moridreamsurfaces 334 5.1.1 Basicsurfacegeometry 334 5.1.2 Nefandsemiamplecones 339 5.1.3 Rationalsurfaces 345 5.1.4 Extremalrationalellipticsurfaces 350 5.1.5 K3surfaces 356 5.1.6 Enriquessurfaces 361 5.2 SmoothdelPezzosurfaces 366 5.2.1 Preliminaries 366 5.2.2 GeneratorsoftheCoxring 369 5.2.3 Theidealofrelations 374 5.2.4 DelPezzosurfacesandflagvarieties 381 viii Contents 5.3 K3surfaces 385 5.3.1 Abeliancoverings 385 5.3.2 Picardnumbers1and2 390 5.3.3 Nonsymplecticinvolutions 397 5.3.4 CoxringsofK3surfaces 400 5.4 RationalK∗-surfaces 405 5.4.1 Definingdataandtheirsurfaces 405 5.4.2 Intersectionnumbers 411 5.4.3 Resolutionofsingularities 417 5.4.4 GorensteinlogdelPezzoK∗-surfaces 422 ExercisestoChapter5 431 6 Arithmeticapplications 437 6.1 UniversaltorsorsandCoxrings 437 6.1.1 Quasitoriandprincipalhomogeneousspaces 437 6.1.2 Universaltorsors 442 6.1.3 Coxringsandcharacteristicspaces 449 6.2 Existenceofrationalpoints 454 6.2.1 TheHasseprincipleandweakapproximation 454 6.2.2 Brauer–Maninobstructions 457 6.2.3 Descentanduniversaltorsors 459 6.2.4 Results 461 6.3 Distributionofrationalpoints 464 6.3.1 HeightsandManin’sconjecture 464 6.3.2 ParameterizationbyuniversaltorsorsandCoxrings 467 6.4 TowardManin’sconjecturefordelPezzosurfaces 469 6.4.1 Classificationandresults 469 6.4.2 Strategy 474 6.4.3 ParameterizationviaCoxrings 475 6.4.4 Countingintegralpointsonuniversaltorsors 478 6.4.5 Interpretationoftheintegral 485 6.5 Manin’sconjectureforasingularcubicsurface 489 6.5.1 Statementoftheresult 489 6.5.2 GeometryandCoxring 490 6.5.3 ParameterizationviaCoxrings 491 6.5.4 Countingintegralpointsonuniversaltorsors 493 6.5.5 Interpretationoftheintegral 497 ExercisestoChapter6 498 Bibliography 501 Index 517 Introduction ThebasicprinciplesregardingCoxringsbecomevisiblealreadyintheclassical example of the projective space Pn over a field K, which we assume to be algebraicallyclosedandofcharacteristiczero.TheelementsofPnarethelines (cid:2)⊆Kn+1 through the origin 0∈Kn+1. Such a line (cid:2) is concretely specified by its homogeneous coordinates [z ,...,z ], where (z ,...,z ) is any point 0 n 0 n on(cid:2),differentfromtheorigin.Hence,thisdescriptioncomeswithanintrinsic ambiguity. More formally speaking, that means that we should regard the projectivespaceasaquotientbyagroupaction Kn+1\{0} ⊆ Kn+1 z [z] /K∗ Pn where K∗ acts on Kn+1 via scalar multiplication. This presentation of the projectivespacePnasthequotientofitscharacteristicspaceKn+1\{0}bythe actionofthecharacteristictorusK∗ isthegeometricwayofthinkingofCox rings.Inalgebraicterms,theactionofK∗onKn+1isencodedbytheassociated decompositionofthepolynomialringintohomogeneousparts (cid:2) K[T ,...,T ]= K[T ,...,T ] , 0 n 0 n k k≥0 where K[T ,...,T ] is the vector space of homogeneous polynomials f of 0 n k degreek,whichmeansthatf(tz)=tkf(z)holdsforalltandz.Thepolynomial ringtogetherwiththisclassicalgradingistheCoxringoftheprojectivespace. NotethattoconstructPnasaK∗-quotient,wehavetoremovetheorigin,which isthevanishinglocusoftheirrelevantideal,fromthetotalcoordinatespace Kn+1. The Cox ring can be seen in terms of algebraic geometry intrinsically 1 .
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