Table Of ContentBeltrametti_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.
