Classical algebraic geometry: a modern view (with correction)

726 Pages·2012·2.953 MB·English

by  Dolgachev I.V.

Classical Algebraic Geometry: a modern
view
IGORV.DOLGACHEV

Preface
Themainpurposeofthepresenttreatiseistogiveanaccountofsomeofthe
topicsinalgebraicgeometrywhichwhilehavingoccupiedthemindsofmany
mathematicians in previous generations have fallen out of fashion in modern
times.Ofteninthehistoryofmathematicsnewideasandtechniquesmakethe
work of previous generations of researchers obsolete, especially this applies
tothefoundationsofthesubjectandthefundamentalgeneraltheoreticalfacts
used heavily in research. Even the greatest achievements of the past genera-
tions which can be found for example in the work of F. Severi on algebraic
cycles or in the work of O. Zariski’s in the theory of algebraic surfaces have
beengreatlygeneralizedandclarifiedsothattheynowremainonlyofhistor-
icalinterest.Incontrast,thefactthatanonsingularcubicsurfacehas27lines
orthataplanequartichas28bitangentsissomethingthatcannotbeimproved
upon and continues to fascinate modern geometers. One of the goals of this
presentworkisthentosavefromobliviontheworkofmanymathematicians
whodiscoveredtheseclassictenetsandmanyotherbeautifulresults.
Inwritingthisbookthegreatestchallengetheauthorhasfacedwasdistilling
the material down to what should be covered. The number of concrete facts,
examples of special varieties and beautiful geometric constructions that have
accumulatedduringtheclassicalperiodofdevelopmentofalgebraicgeometry
is enormous and what the reader is going to find in the book is really only a
tip of the iceberg; a work that is sort of a taste sampler of classical algebraic
geometry. It avoids most of the material found in other modern books on the
subject, such as, for example, [10] where one can find many of the classical
resultsonalgebraiccurves.Instead,ittriestoassembleor,inotherwords,to
createacompendiumofmaterialthateithercannotbefound,istoodispersedto
befoundeasily,orissimplynottreatedadequatelybycontemporaryresearch
papers.Ontheotherhand,whilemostofthematerialtreatedinthebookexists
inclassicaltreatisesinalgebraicgeometry,theirsomewhatarchaicterminology
iv Preface
andwhatisbynowcompletelyforgottenbackgroundknowledgemakesthese
booksusefultobutahandfulofexpertsintheclassicalliterature.Lastly,one
must admit that the personal taste of the author also has much sway in the
choiceofmaterial.
Thereadershouldbewarnedthatthebookisbynomeansanintroduction
toalgebraicgeometry.Althoughsomeoftheexpositioncanbefollowedwith
only a minimum background in algebraic geometry, for example, based on
Shafarevich’sbook[577],itoftenreliesoncurrentcohomologicaltechniques,
such as those found in Hartshorne’s book [311]. The idea was to reconstruct
aresultbyusingmoderntechniquesbutnotnecessarilyitsoriginalproof.For
one,theingeniousgeometricconstructionsinthoseproofswereoftenbeyond
the authors abilities to follow them completely. Understandably, the price of
thiswasoftentoreplaceabeautifulgeometricargumentwithadullcohomo-
logical one. For those looking for a less demanding sample of some of the
topicscoveredinthebooktherecentbeautifulbook[39]maybeofgreatuse.
Noattempthasbeenmadetogiveacompletebibliography.Togiveanidea
of such an enormous task one could mention that the report on the status of
topics in algebraic geometry submitted to the National Research Council in
Washington in 1928 [582] contains more than 500 items of bibliography by
130 different authors only in the subject of planar Cremona transformations
(covered in one of the chapters of the present book.) Another example is the
bibliography on cubic surfaces compiled by J. E. Hill [326] in 1896 which
alone contains 205 titles. Meyer’s article [425] cites around 130 papers pub-
lished1896-1928.ThetitlesearchinMathSciNetrevealsmorethan200papers
refereedsince1940,manyofthempublishedonlyinthelasttwentyyears.How
saditiswhenoneconsiderstheimpossibilityofsavingfromoblivionsomany
names of researchers of the past years who have contributed so much to our
subject.
A word about exercises: some of them are easy and follow from the defi-
nitions, some of them are hard and are meant to provide additional facts not
coveredinthemaintext.Inthiscaseweindicatethesourcesforthestatements
andsolutions.
I am very grateful to many people for their comments and corrections to
manypreviousversionsofthemanuscript.IamespeciallythankfultoSergey
Tikhomirovwhosehelpinmathematicaleditingofthebookwasessentialfor
getting rid of many mistakes in the previous versions. For all the errors still
foundinthebooktheauthorbearssoleresponsibility.
Contents
1 Polarity page1
1.1 Polarhypersurfaces 1
1.1.1 Thepolarpairing 1
1.1.2 Firstpolars 7
1.1.3 Polarquadrics 13
1.1.4 TheHessianhypersurface 15
1.1.5 Parabolicpoints 18
1.1.6 TheSteinerianhypersurface 21
1.1.7 TheJacobianhypersurface 25
1.2 Thedualhypersurface 32
1.2.1 Thepolarmap 32
1.2.2 Dualvarieties 33
1.2.3 Plu¨ckerformulas 37
1.3 Polars-hedra 40
1.3.1 Apolarschemes 40
1.3.2 Sumsofpowers 42
1.3.3 Generalizedpolars-hedra 44
1.3.4 Secantvarietiesandsumsofpowers 45
1.3.5 TheWaringproblems 52
1.4 Dualhomogeneousforms 54
1.4.1 Catalecticantmatrices 54
1.4.2 Dualhomogeneousforms 57
1.4.3 TheWaringrankofahomogeneousform 58
1.4.4 Mukai’sskew-symmetricform 59
1.4.5 Harmonicpolynomials 63
1.5 Firstexamples 68
1.5.1 Binaryforms 68
vi Contents
1.5.2 Quadrics 70
Exercises 73
HistoricalNotes 75
2 Conicsandquadricsurfaces 77
2.1 Self-polartriangles 77
2.1.1 Veronesequarticsurfaces 77
2.1.2 Polarlines 79
2.1.3 Thevarietyofself-polartriangles 81
2.1.4 Conjugatetriangles 85
2.2 Ponceletrelation 91
2.2.1 Darboux’sTheorem 91
2.2.2 Ponceletcurvesandvectorbundles 96
2.2.3 Complexcircles 99
2.3 Quadricsurfaces 102
2.3.1 Polarpropertiesofquadrics 102
2.3.2 Invariantsofapairofquadrics 108
2.3.3 Invariantsofapairofconics 112
2.3.4 TheSalmonconic 117
Exercises 121
HistoricalNotes 125
3 Planecubics 127
3.1 Equations 127
3.1.1 Ellipticcurves 127
3.1.2 TheHesseequation 131
3.1.3 TheHessepencil 133
3.1.4 TheHessegroup 134
3.2 Polarsofaplanecubic 138
3.2.1 TheHessianofacubichypersurface 138
3.2.2 TheHessianofaplanecubic 139
3.2.3 Thedualcurve 143
3.2.4 Polars-gons 144
3.3 Projectivegenerationofcubiccurves 149
3.3.1 Projectivegeneration 149
3.3.2 Projectivegenerationofaplanecubic 151
3.4 Invarianttheoryofplanecubics 152
3.4.1 Mixedconcomitants 152
3.4.2 Clebsch’stransferprinciple 153
3.4.3 Invariantsofplanecubics 155
Exercises 157
Contents vii
HistoricalNotes 160
4 Determinantalequations 162
4.1 Planecurves 162
4.1.1 Theproblem 162
4.1.2 Planecurves 163
4.1.3 Thesymmetriccase 168
4.1.4 Contactcurves 170
4.1.5 Firstexamples 174
4.1.6 Themodulispace 176
4.2 Determinantalequationsforhypersurfaces 178
4.2.1 Determinantalvarieties 178
4.2.2 ArithmeticallyCohen-Macaulaysheaves 182
4.2.3 Symmetricandskew-symmetricaCMsheaves 187
4.2.4 Singularplanecurves 189
4.2.5 Lineardeterminantalrepresentationsofsurfaces 197
4.2.6 Symmetroidsurfaces 201
Exercises 205
HistoricalNotes 207
5 Thetacharacteristics 209
5.1 Oddandeventhetacharacteristics 209
5.1.1 Firstdefinitionsandexamples 209
5.1.2 Quadraticformsoverafieldofcharacteristic2 210
5.2 Hyperellipticcurves 213
5.2.1 Equationsofhyperellipticcurves 213
5.2.2 2-torsionpointsonahyperellipticcurve 214
5.2.3 Thetacharacteristicsonahyperellipticcurve 216
5.2.4 Families of curves with odd or even theta
characteristic 218
5.3 Thetafunctions 219
5.3.1 Jacobianvariety 219
5.3.2 Thetafunctions 222
5.3.3 Hyperellipticcurvesagain 224
5.4 Oddthetacharacteristics 226
5.4.1 Syzygetictriads 226
5.4.2 Steinercomplexes 229
5.4.3 Fundamentalsets 233
5.5 Scorzacorrespondence 236
5.5.1 Correspondencesonanalgebraiccurve 236
5.5.2 Scorzacorrespondence 240
viii Contents
5.5.3 Scorzaquartichypersurfaces 243
5.5.4 Contacthyperplanesofcanonicalcurves 245
Exercises 249
HistoricalNotes 249
6 PlaneQuartics 251
6.1 Bitangents 251
6.1.1 28bitangents 251
6.1.2 Aronholdsets 253
6.1.3 Riemann’sequationsforbitangents 256
6.2 Determinantequationsofaplanequartic 261
6.2.1 Quadraticdeterminantalrepresentations 261
6.2.2 Symmetricquadraticdeterminants 265
6.3 Eventhetacharacteristics 270
6.3.1 Contactcubics 270
6.3.2 Cayleyoctads 272
6.3.3 Sevenpointsintheplane 275
6.3.4 TheClebschcovariantquartic 279
6.3.5 ClebschandLu¨rothquartics 283
6.3.6 AFanomodelofVSP(f,6) 292
6.4 Invarianttheoryofplanequartics 294
6.5 Automorphismsofplanequarticcurves 296
6.5.1 Automorphismsoffiniteorder 296
6.5.2 Automorphismgroups 299
6.5.3 TheKleinquartic 302
Exercises 306
HistoricalNotes 308
7 Cremonatransformations 311
7.1 Homaloidallinearsystems 311
7.1.1 Linearsystemsandtheirbaseschemes 311
7.1.2 Resolutionofarationalmap 313
7.1.3 ThegraphofaCremonatransformation 316
7.1.4 F-locusandP-locus 318
7.1.5 Computationofthemultidegree 323
7.2 Firstexamples 327
7.2.1 Quadro-quadratictransformations 327
7.2.2 BilinearCremonatransformations 329
7.2.3 deJonquie`restransformations 334
7.3 PlanarCremonatransformations 337
7.3.1 Exceptionalconfigurations 337
Contents ix
7.3.2 Thebubblespaceofasurface 341
7.3.3 Netsofisologuesandfixedpoints 344
7.3.4 Quadratictransformations 349
7.3.5 SymmetricCremonatransformations 351
7.3.6 deJonquie`restransformationsandhyperellip-
ticcurves 353
7.4 Elementarytransformations 356
7.4.1 Minimalrationalruledsurfaces 356
7.4.2 Elementarytransformations 359
7.4.3 BirationalautomorphismsofP1×P1 361
7.5 Noether’sFactorizationTheorem 366
7.5.1 Characteristicmatrices 366
7.5.2 TheWeylgroups 372
7.5.3 Noether-Fanoinequality 376
7.5.4 Noether’sFactorizationTheorem 377
Exercises 381
HistoricalNotes 383
8 DelPezzosurfaces 386
8.1 Firstproperties 386
8.1.1 SurfacesofdegreedinPd 386
8.1.2 Rationaldoublepoints 390
8.1.3 Ablow-upmodelofadelPezzosurface 392
8.2 TheE -lattice 398
N
8.2.1 Quadraticlattices 398
8.2.2 TheE -lattice 401
N
8.2.3 Roots 402
8.2.4 Fundamentalweights 408
8.2.5 Gossetpolytopes 410
8.2.6 (−1)-curvesondelPezzosurfaces 412
8.2.7 Effectiveroots 415
8.2.8 Cremonaisometries 418
8.3 Anticanonicalmodels 422
8.3.1 Anticanonicallinearsystems 422
8.3.2 Anticanonicalmodel 427
8.4 DelPezzosurfacesofdegree≥6 429
8.4.1 DelPezzosurfacesofdegree7,8,9 429
8.4.2 DelPezzosurfacesofdegree6 430
8.5 DelPezzosurfacesofdegree5 433
8.5.1 Linesandsingularities 433
x Contents
8.5.2 Equations 434
8.5.3 OADPvarieties 436
8.5.4 Automorphismgroup 437
8.6 QuarticdelPezzosurfaces 441
8.6.1 Equations 441
8.6.2 Cyclidquartics 444
8.6.3 Linesandsingularities 446
8.6.4 Automorphisms 448
8.7 DelPezzosurfacesofdegree2 451
8.7.1 Singularities 451
8.7.2 Geiserinvolution 454
8.7.3 Automorphisms of del Pezzo surfaces of
degree2 457
8.8 DelPezzosurfacesofdegree1 458
8.8.1 Singularities 458
8.8.2 Bertiniinvolution 460
8.8.3 Rationalellipticsurfaces 462
8.8.4 Automorphisms of del Pezzo surfaces of
degree1 463
Exercises 470
HistoricalNotes 471
9 Cubicsurfaces 475
9.1 Linesonanonsingularcubicsurface 475
9.1.1 MoreabouttheE -lattice 475
6
9.1.2 Linesandtritangentplanes 482
9.1.3 Schur’squadrics 486
9.1.4 Eckardtpoints 491
9.2 Singularities 494
9.2.1 Non-normalcubicsurfaces 494
9.2.2 Linesandsingularities 495
9.3 Determinantalequations 501
9.3.1 Cayley-Salmonequation 501
9.3.2 Hilbert-BurchTheorem 504
9.3.3 Cubicsymmetroids 508
9.4 Representationsassumsofcubes 512
9.4.1 Sylvester’spentahedron 512
9.4.2 TheHessiansurface 515
9.4.3 Cremona’shexahedralequations 517
9.4.4 TheSegrecubicprimal 520