Classical Algebraic Geometry: A Modern View PDF
Preview Classical Algebraic Geometry: A Modern View
ClassicalAlgebraicGeometry Algebraicgeometryhasbenefitedenormouslyfromthepowerfulgeneralmachinery developedinthelatterhalfofthetwentiethcentury.Thecosthasbeenthatmuchofthe researchofpreviousgenerationsisinalanguageunintelligibletomodernworkers,in particular,therichlegacyofclassicalalgebraicgeometry,suchasplanealgebraiccurves oflowdegree,specialalgebraicsurfaces,thetafunctions,Cremonatransformations,the theoryofapolarityandthegeometryoflinesinprojectivespaces. Theauthor’scontemporaryapproachmakesthislegacyaccessibletomodernalgebraic geometersandtootherswhoareinterestedinapplyingclassicalresults.Thevast bibliographyofover600referencesiscomplementedbyhistoricalnotesandexercises thatextendorexemplifyresultsgiveninthebook. igor v. dolgachevisProfessorEmeritusintheDepartmentofMathematicsatthe UniversityofMichigan. Classical Algebraic Geometry A Modern View IGOR V. DOLGACHEV UniversityofMichigan cambridge university press Cambridge,NewYork,Melbourne,Madrid,CapeTown, Singapore,Sa˜oPaulo,Delhi,Tokyo,MexicoCity CambridgeUniversityPress TheEdinburghBuilding,CambridgeCB28RU,UK PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork www.cambridge.org Informationonthistitle:www.cambridge.org/9781107017658 (cid:2)c IgorV.Dolgachev2012 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2012 PrintedandBoundintheUnitedKingdombytheMPGBooksGroup AcatalogrecordforthispublicationisavailablefromtheBritishLibrary LibraryofCongressCataloging-in-PublicationData Dolgachev,I.(IgorV.) Classicalalgebraicgeometry:amodernview/IgorV.Dolgachev. p. cm. ISBN978-1-107-01765-8(hardback) 1. Geometry,Algebraic. I. Title. QA564.D638 2012 516.3(cid:3)5–dc23 2012018486 ISBN 978-1-107-01765-8Hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceor accuracyofURLsforexternalorthird-partyinternetwebsitesreferredto inthispublication,anddoesnotguaranteethatanycontentonsuch websitesis,orwillremain,accurateorappropriate. Dedicatedto Natasha,DenisandAndrey Contents Preface pagexi 1 Polarity 1 1.1 Polarhypersurfaces 1 1.2 Thedualhypersurface 28 1.3 Polars-hedra 35 1.4 Dualhomogeneousforms 48 1.5 Firstexamples 60 Exercises 64 Historicalnotes 66 2 Conicsandquadricsurfaces 69 2.1 Self-polartriangles 69 2.2 Ponceletrelation 81 2.3 Quadricsurfaces 91 Exercises 108 Historicalnotes 111 3 Planecubics 114 3.1 Equations 114 3.2 Polarsofaplanecubic 124 3.3 Projectivegenerationofcubiccurves 133 3.4 Invarianttheoryofplanecubics 136 Exercises 141 Historicalnotes 143 4 Determinantalequations 146 4.1 Planecurves 146 4.2 Determinantalequationsforhypersurfaces 160 Exercises 184 Historicalnotes 186 5 Thetacharacteristics 188 5.1 Oddandeventhetacharacteristics 188 vii viii Contents 5.2 Hyperellipticcurves 192 5.3 Thetafunctions 197 5.4 Oddthetacharacteristics 204 5.5 Scorzacorrespondence 212 Exercises 224 Historicalnotes 224 6 Planequartics 226 6.1 Bitangents 226 6.2 Determinantequationsofaplanequartic 235 6.3 Eventhetacharacteristics 243 6.4 Invarianttheoryofplanequartics 265 6.5 Automorphismsofplanequarticcurves 266 Exercises 276 Historicalnotes 278 7 Cremonatransformations 280 7.1 Homaloidallinearsystems 280 7.2 Firstexamples 294 7.3 PlanarCremonatransformations 303 7.4 Elementarytransformations 320 7.5 Noether’sFactorizationTheorem 329 Exercises 342 Historicalnotes 344 8 delPezzosurfaces 347 8.1 Firstproperties 347 8.2 TheE -lattice 358 N 8.3 Anticanonicalmodels 379 8.4 delPezzosurfacesofdegree≥6 386 8.5 delPezzosurfacesofdegree5 389 8.6 QuarticdelPezzosurfaces 396 8.7 delPezzosurfacesofdegree2 405 8.8 delPezzosurfacesofdegree1 411 Exercises 422 Historicalnotes 423 9 Cubicsurfaces 426 9.1 Linesonanonsingularcubicsurface 426 9.2 Singularities 443 9.3 Determinantalequations 449 9.4 Representationsassumsofcubes 459 9.5 Automorphismsofcubicsurfaces 483 Exercises 502 Historicalnotes 504 Contents ix 10 Geometryoflines 508 10.1 Grassmanniansoflines 508 10.2 Linearlinecomplexes 517 10.3 Quadraticlinecomplexes 531 10.4 Ruledsurfaces 557 Exercises 588 Historicalnotes 589 References 593 Symbolindex 620 Subjectindex 623
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