Algebraic Geometry: Notes on a Course

332 Pages·2022·3.959 MB·English

by  Michael Artin

G
S GRADUATE STUDIES 222
M
IN MATHEMATICS
222
Algebraic
Geometry
This book is an introduction to the geometry of complex algebraic varieties. It is
intended for students who have learned algebra, analysis, and topology, as taught
in standard undergraduate courses. So it is a suitable text for a beginning graduate Notes on a Course
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course or an advanced undergraduate course.
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The book begins with a study of plane algebraic curves, then introduces affine e
and projective varieties, going on to dimension and construcibility. ​-modules b
O r
(quasicoherent sheaves) are defined without reference to sheaf theory, and their a Michael Artin
cohomology is defined axiomatically. The Riemann-Roch Theorem for curves is ic
proved using projection to the projective line.
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Some of the points that aren’t always treated in beginning courses are Hensel’s e
o
Lemma, Chevalley’s Finiteness Theorem, and the Birkhoff-Grothendieck Theorem.
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The book contains extensive discussions of finite group actions, lines in P3​, and
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double planes, and it ends with applications of the Riemann-Roch Theorem.
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For additional information
and updates on this book, visit
www.ams.org/bookpages/gsm-222
GSM/222.E
www.ams.org

3-Color Cover: Pantone 632 C (light blue), Pantone 302 C (blue), Pantone 123 C (yellow) 330 pages • softcover • trim size 7 x 10 • spine width 11/16"
10.1090/gsm/222
Algebraic
Geometry
Notes on a Course

222
GRADUATE STUDIES
IN MATHEMATICS
Algebraic
Geometry
Notes on a Course
Michael Artin
EDITORIAL COMMITTEE
Matthew Baker
Marco Gualtieri
Gigliola Staffilani (Chair)
Jeff A. Viaclovsky
Rachel Ward
2020 Mathematics Subject Classification. Primary 14-01.
For additional informationand updates on this book, visit
www.ams.org/bookpages/gsm-222
Library of Congress Cataloging-in-Publication Data
Names: Artin,Michael,author.
Title: Algebraicgeometry: notesonacourse/MichaelArtin.
Description: Providence,RhodeIsland: AmericanMathematicalSociety,2022. |Series: Graduate
studiesinmathematics,1065-7339;222|Includesbibliographicalreferencesandindex.
Identifiers: LCCN 2022012175 | ISBN 9781470468484 (hardcover : acid-free paper) | ISBN
9781470471118(paperback: acid-freepaper)|ISBN9781470471101(ebook)
Subjects: LCSH: Geometry, Algebraic. | AMS: Algebraic geometry – Instructional exposition
(textbooks,tutorialpapers,etc.).
Classification: LCCQA564.A7452022|DDC516.3/5–dc23/eng20220614
LCrecordavailableathttps://lccn.loc.gov/2022012175
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Contents
Preface ix
ANotefortheStudent x
Chapter1. PlaneCurves 1
§1.1. TheAffinePlane 1
§1.2. TheProjectivePlane 4
§1.3. PlaneProjectiveCurves 8
§1.4. TangentLines 14
§1.5. TheDualCurve 19
§1.6. ResultantsandDiscriminants 25
§1.7. NodesandCusps 30
§1.8. Hensel’sLemma 39
§1.9. Bézout’sTheorem 44
§1.10. ThePlückerFormulas 50
Exercises 51
Chapter2. AffineAlgebraicGeometry 55
§2.1. TheZariskiTopology 55
§2.2. SomeAffineVarieties 62
§2.3. Hilbert’sNullstellensatz 63
§2.4. TheSpectrum 67
§2.5. MorphismsofAffineVarieties 71
§2.6. Localization 76
§2.7. FiniteGroupActions 80
Exercises 85
v
vi Contents
Chapter3. ProjectiveAlgebraicGeometry 89
§3.1. ProjectiveVarieties 90
§3.2. HomogeneousIdeals 93
§3.3. ProductVarieties 97
§3.4. RationalFunctions 101
§3.5. Morphisms 104
§3.6. AffineVarieties 111
§3.7. LinesinThree-Space 113
Exercises 120
Chapter4. IntegralMorphisms 125
§4.1. TheNakayamaLemma 125
§4.2. IntegralExtensions 127
§4.3. Normalization 130
§4.4. GeometryofIntegralMorphisms 135
§4.5. Dimension 138
§4.6. Chevalley’sFinitenessTheorem 143
§4.7. DoublePlanes 149
Exercises 155
Chapter5. StructureofVarietiesintheZariskiTopology 159
§5.1. LocalRings 159
§5.2. SmoothCurves 164
§5.3. ConstructibleSets 170
§5.4. ClosedSets 173
§5.5. ProjectiveVarietiesAreProper 175
§5.6. FibreDimension 178
Exercises 179
Chapter6. Modules 183
§6.1. TheStructureSheaf 183
§6.2. 𝒪-Modules 184
§6.3. TheSheafProperty 187
§6.4. MoreModules 190
§6.5. DirectImage 198
§6.6. Support 202
Contents vii
§6.7. Twisting 203
§6.8. ExtendingaModule: Proof 209
Exercises 213
Chapter7. Cohomology 217
§7.1. Cohomology 217
§7.2. Complexes 219
§7.3. CharacteristicProperties 222
§7.4. ExistenceofCohomology 223
§7.5. CohomologyoftheTwistingModules 230
§7.6. CohomologyofHypersurfaces 232
§7.7. ThreeTheoremsaboutCohomology 234
§7.8. Bézout’sTheorem 239
§7.9. UniquenessoftheCoboundaryMaps 241
Exercises 243
Chapter8. TheRiemann-RochTheoremforCurves 247
§8.1. Divisors 247
§8.2. TheRiemann-RochTheoremI 253
§8.3. TheBirkhoff-GrothendieckTheorem 256
§8.4. Differentials 258
§8.5. BranchedCoverings 262
§8.6. TraceofaDifferential 264
§8.7. TheRiemann-RochTheoremII 271
§8.8. UsingRiemann-Roch 273
§8.9. WhatIsNext 283
Exercises 284
Chapter9. Background 289
§9.1. RingsandModules 289
§9.2. TheImplicitFunctionTheorem 302
§9.3. TranscendenceDegree 303
Glossary 305
IndexofNotation 311
Bibliography 313
Index 315