Beltrametti_titelei 14.7.2009 16:02 Uhr Seite 1 Beltrametti_titelei 14.7.2009 16:02 Uhr Seite 2 EMS Textbooks in Mathematics EMS Textbooks in Mathematics isa series of books aimed at students or professional mathemati- cians seeking an introduction into a particular field. The individual volumes are intended not only to provide relevant techniques, results, and applications, but also to afford insight into the motiva- tions and ideas behind the theory. Suitably designed exercises help to master the subject and prepare the reader for the study of more advanced and specialized literature. Jørn Justesen and Tom Høholdt, A Course In Error-Correcting Codes Markus Stroppel, Locally Compact Groups Peter Kunkel and Volker Mehrmann, Differential-Algebraic Equations Dorothee D. Haroske and Hans Triebel, Distributions, Sobolev Spaces, Elliptic Equations Thomas Timmermann, An Invitation to Quantum Groups and Duality Oleg Bogopolski, Introduction to Group Theory Marek Jarnicki und Peter Pflug,First Steps in Several Complex Variables: Reinhardt Domains Tammo tom Dieck, Algebraic Topology Beltrametti_titelei 14.7.2009 16:02 Uhr Seite 3 Mauro C. Beltrametti Ettore Carletti Dionisio Gallarati Giacomo Monti Bragadin Lectures on Curves, Surfaces and Projective Varieties A Classical View of Algebraic Geometry Translated from the Italian by Francis Sullivan Beltrametti_titelei 14.7.2009 16:02 Uhr Seite 4 Authors: Translator: Mauro C. Beltrametti Francis Sullivan Ettore Carletti Dipartimento di Matematica Pura ed Applicata Dionisio Gallarati Università di Padova Giacomo Monti Bragadin Via Trieste, 63 35121 Padova Dipartimento di Matematica Italy Università degli Studi di Genova Via Dodecaneso, 35 16146 Genova Italy Originally published in Italian by Bollati Boringhieri, Torino, under the title “Letture su curve, superficie e varietà proiettive speciali – Un’introduzione alla geometria algebrica”. © 2003 Bollati Boringhieri editore s.r.l., Torino, corso Vittorio Emanuele II, 86 2000 Mathematical Subject Classification (primary; secondary): 14-01; 14E05, 14E07, 14H50, 14J26, 14J70, 14M99, 14N05 Key words: Algebraic geometry, projective varieties, curves, surfaces, special varieties ISBN 978-3-03719-064-7 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2009 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum FLI C4 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the author’s TEX files: I. Zimmermann, Freiburg Printed in Germany 9 8 7 6 5 4 3 2 1 TomydaughterBarbara MauroC.Beltrametti Inmemoryofmyfather EttoreCarletti TomysonMario DionisioGallarati TomychildrenAlessandroandMargherita GiacomoMontiBragadin Preface Chéselavocetuasaràmolesta nelprimogusto,vitalnutrimento lasceràpoi,quandosaràdigesta. (Dante,Paradiso,XVII,130–132) Thisbookconsistsoflecturesonclassicalalgebraicgeometry,thatis,themethods andresultscreatedbythegreatgeometersofthelatenineteenthandearlytwentieth centuries. Thisbookisaimedatstudentsofthelasttwoyearsofanundergraduateprogram inmathematics: itcontainssomeratheradvancedtopicsthatcouldformmaterialfor specializedcoursesandwhicharesuitableforthefinaltwoyearsofundergraduate study,aswellasinterestingtopicsforaseniorthesis. Thebookwillbewelcomedby teachersandstudentsofalgebraicgeometrywhoareseekingaclearandpanoramic pathleadingfromthebasicfactsaboutlinearsubspaces,conicsandquadrics,learned in courses on linear algebra and advanced calculus to a systematic discussion of classicalalgebraicvarietiesandthetoolsneededtostudythem. The topics chosen throw light on the intuitive concepts that were the starting pointformuchcontemporaryresearch,andshouldtherefore,inouropinion,make uppartoftheculturalbaggageofanyyoungstudentintendingtoworkinalgebraic geometry. Our hope is that this text, which can be a first step in recovering an importantandfascinatingpatrimonyofmathematicalideas,willstimulateinsome readersthedesiretolookintotheoriginalworksofthegreatgeometersofthepast, andperhapseventofindthereinmotivationforsignificantnewresearch. Anotherreasonwhichinducedustowritethisbookistheobservationthatmany young researchers, though able to obtain significant results by using the sophis- ticated techniques presently available, can also encounter notable difficulty when faced with questions for which classical methods are particularly indicated. This book combines the more classical and intuitive approach with the more formally rigorousandmodernapproach,andsocontributestofillingagapintheliterature. Thisbook,whichweconsidernewandcertainlydifferentfromtextspublished inthelastfiftyyears,isthetextwewouldhavelikedonourdeskwhenwebegan our studies; it is our hope that it will serve as a useful introduction toAlgebraic Geometryalongclassicallines. Theidealuseforthistextcouldwellbetoprovideasolidpreliminarycourseto bemasteredbeforeapproachingmoreadvancedandabstractbooks. Thuswelaya firmclassicalfoundationforunderstandingmodernexpositionssuchasHartshorne [50],Mumford[68],Liu[65],oralsoDolgachev’sforthcomingtreatise[34]. Our viii Preface text can also be considered as a more modern version of Walker’s classic book [113],butgreatlyenrichedwithrespecttothelatterbythediscussionofimportant classesofhigherdimensionalvarietiesasmentionedabove. Prerequisites. We suppose that the reader knows the foundational elements of ProjectiveGeometry,andthegeometryofprojectivespaceanditssubspaces. These aretopicsordinarilyencounteredinthefirsttwoyearsofundergraduateprograms in mathematics. The basic references for these topics are the classic treatment of Cremona [31] and the texts of Berger [13] and Hodge and Pedoe [52,Vols. 1, 2]. Theintroductorytext[10]bytheauthorsofthepresentvolumeisalsouseful. For theconvenienceofthereaderinthepurelyintroductoryChapter1wehavegivena concisereviewofthosefactsthatwillbemostfrequentlyusedinthesequel. Moreover to understand the book, in addition to a few elementary facts from Analysis, the reader should also be familiar with the basic structures ofAlgebra (groups,rings,polynomialrings,ideals,primeandmaximalideals,integraldomains andfields,thecharacteristicofaring),aswellasextensionsoffields(algebraicand transcendentalelements,minimalpolynomials,algebraicallyclosedfields)asfound inthetextsof[35]or[75]. Possible“Itineraries”. Thebookcontainsseveral“itineraries”thatcouldsuggest orconstitutetopicsfordifferentadvancedundergraduatecoursesinmathematics, andalsoforgraduatelevelcourses. Herearesomemorepreciseindications,which alsoofferaviewofthetopicstreatedhere. (cid:2) Chapters 2 and 3 can be the introduction to any course in algebraic geometry. They contain the essential notions regarding algebraic and projective sets: the Hilbert Nullstellensatz, morphisms and rational maps, dimension, simple points andsingularpointsofanalgebraicset,tangentspacesandtangentcones,theorder ofaprojectivevariety. Ifonethenaddsthebriefcommentsoneliminationtheory inChapter4,onehasenoughmaterialforasemestercourse. (cid:2) Chapter5isdedicatedtohypersurfacesinPnwithparticularattentiontoalgebraic planecurvesandsurfacesinP3. Itassumesonlytherudimentsofthegeometryof projectivespaceandathoroughfamiliaritywithprojectivecoordinates. Thetopics coveredinthischapter,suitablyamplifiedandaccompaniedbytheexercisesgiven inSections5.7,5.8,caninthemselvesformtheprogramforacoursethatprobably requires more than a semester, especially if one adds the first two paragraphs of Chapter 9 which are dedicated to quadratic transformations between planes and theirmostimportantapplications,forexampletheproofoftheexistenceofaplane modelwithonlyordinarysingularitiesforanyalgebraiccurve. (cid:2) Chapter 6, which deals with linear systems of algebraic hypersurfaces in Pn, containstopicsnecessaryforthesubsequentchapters. Veronesevarietiesandmap- Preface ix pings are introduced, as well as the notion of the blowing up of Pn with center a subvarietyofcodimension(cid:3)2. (cid:2) Theprogramforaspecializedonesemestercourseforadvancedundergraduates could be furnished by Chapter 7 and the first two sections of Chapter 9, which are dedicated respectively to algebraic curves in Pn (with particular attention to rationalcurvesandthecurvesonaquadricinP3)andtoquadratictransformations between planes. The genus of a curve is introduced, and its birational nature is placedinevidence.AddingtheremainingresultsdiscussedinChapter9,whichhas someoriginalitywithrespecttotheexistingliteratureonCremonatransformations, wouldgiverisetoafullyearcourse. (cid:2) Chapter 8 is the natural continuation and completion of Chapter 7, and also makes use of some results from the first two sections of Chapter 9. It deals with thetheoryoflinearseriesonanalgebraiccurve,includinganextensivediscussion ontheRiemann–Rochtheorem,andanapproachtotheclassificationofalgebraic curvesinPnintermsofpropertiesofthecanonicalseriesandthecanonicalcurve. ThischapterwaslargelyinspiredbySeveri’sclassictext[100],wherethesocalled “quickmethod”forstudyingthegeometryofalgebraiccurvesisexpounded. The contentofthischapterwouldgiverisetoaonesemestercourse. (cid:2) Chapter10canfurnishmaterialforaonesemestercourseforstudentswhoalready haveagoodmasteryofthegeometryofhyperspaces([52,Vol.1, ChapterV]),of plane projective curves (Chapter 5, Section 5.7) and of Cremona transformations between planes (Chapter 9, Sections 9.1, 9.3). Thus this chapter is well adapted foranupper-undergraduatelevelcourseinmathematicsoralsoforgraduatelevel courses. Nevertheless,themethodsusedareratherelementary.Amongthetopicsto whichthemostspaceisdedicated,wementiontherationalnormalruledsurfaces,the Veronesesurface,andtheSteinersurface. Someofthesurfacesalreadydescribed in the last section of Chapter 5 are here rediscovered and seen in a new light. They are studied together with other surfaces that occupy an important place in algebraic-projectivegeometry. (cid:2) Veronese varieties, Segre varieties, and Grassmann varieties are discussed in Chapter 6, Section 6.7, Chapter 11 and Chapter 12 respectively. They constitute examples of special varieties that every student of geometry should know. These topics too could be part of an advanced course or graduate course. Among other things,theymightwellsuggesttopicsforresearchprojectsoraseniorthesis. (cid:2) Thenumerousexercisesofthetextareinpartdistributedthroughoutthevarious chapters, and in part collected in Chapter 13. They can be quite useful to young graduates who are preparing for admission to a doctoral program or a position as researchassistant,oralsotohighschoolteacherspreparingtoqualifyforpromotion. Theeasierexercisesaremerelystated;others,almostalwaysnewtothistext,offer x Preface variouslevelsofdifficulty. Mostofthemareaccompaniedbyacompletesolution, butinsomecasesthemethodofsolutionismerelysuggested. Sources. Inadditiontothealreadycitedtext[10]towhichthepresentvolumemay beseenasthenaturalsuccessor,classicalreferencesandsourcesofinspirationfor partofthematerialcontainedherearethebooksbyBertini[14],[15],Castelnuovo [25], Comessatti[27], EnriquesandChisini[36], FanoandTerracini[38], Hodge andPedoe[52],B.Segre[81],SempleandRoth[92],andC.Segre’smemoir[83]. We have also been influenced by more modern texts like Shafarevich [103] and Harris [48], and, with particular reference to the topics regarding algebraic sets, rational regular functions, and rational maps developed in the second chapter, by Reid’stext[74]. Besides the texts mentioned above, in our opinion the very nice introductory textsofMusili[69]andKunz[60]aswellasKempf’smoreadvancedbook[59]merit specialmention. Wealsocallthereader’sattentiontothecharming“bibliographie commentée”inDieudonné’stext[33]whichoffersapanoramicviewofthebasic andadvancedtextsandthefundamentalarticleswhichhaveconstitutedthehistory and development ofAlgebraic Geometry, from the origins of Greek mathematics up to the late 1960s. The bibliography is rendered even more valuable, topic by topic,throughinterestingcommentsandhistoricalnotesillustratingan“excursus” thatstartsfromHeath’sinterpretationofcertainalgebraicmethodsinDiophantus andarrivesatMumford’sconstructionofthespaceofmoduliforcurvesofagiven genus. Changes and improvements with respect to the Italian version. The present textofferssomesubstantialchangesandimprovementswithrespecttotheoriginal Italian version [9]. Among the major changes are an entirely new chapter, Chap- ter 8, devoted to linear series on algebraic curves, a major revision to Chapter 2, the new Section 4.3, giving greater detail regarding intersection multiplicities in Chapter4. Moreoveranumberofnewexerciseshavebeenaddedthroughoutthe book,including,inparticular,anewfinalsectionofChapter13. AmongtheminorchangesthereisanewfinalparagraphinChapter10dealing withbirationalCremonatransformationsbetweenprojectivespacesofdimension3. Therearealsonumerouscorrectionsofminortypographicalandmathematicaler- rors. We thank the many colleagues and students who have had occasion to read partsoftheItalianversionofthebook,thuscontributingtothecorrectionoferrors and improving the exposition of the material. In particular, we wish to thank our friendsandcolleaguesL.Ba˘descu,E.Catalisano,A.DelPadrone,A.Geramita,P. Ionescu,R.Pardinifortheircomments. WewouldliketothankI.Dolgachevwho firstencouragedustoconsiderthepossibilityofatranslationoftheoriginalversion ofthebook. WewouldalsoliketothankourfriendandcolleagueA.Languascofor hisinvaluableassistanceinresolvingvariousproblemsinvolvingtheuseofLATEX.