GRADUATE STUDIES IN MATHEMATICS 188

Introduction to Algebraic Geometry

Steven Dale Cutkosky

GRADUATE STUDIES IN MATHEMATICS 188

Introduction to Algebraic Geometry

Steven Dale Cutkosky

10.1090/gsm/188

EDITORIAL COMMITTEE

Dan AbramovichDaniel S. Freed (Chair)

Gigliola StaffilaniJeff A. Viaclovsky

2010 Mathematics Subject Classification. Primary 14-01.

For additional information and updates on this book, visitwww.ams.org/bookpages/gsm-188

Library of Congress Cataloging-in-Publication Data

Names: Cutkosky, Steven Dale, author.Title: Introduction to algebraic geometry / Steven Dale Cutkosky.Other titles: Algebraic geometryDescription: Providence, Rhode Island : American Mathematical Society, [2018] | Series: Gradu-

ate studies in mathematics ; volume 188 | Includes bibliographical references and index.Identifiers: LCCN 2017045552 | ISBN 9781470435189 (alk. paper)Subjects: LCSH: Geometry, Algebraic. | AMS: Algebraic geometry – Instructional exposition

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10 9 8 7 6 5 4 3 2 1 23 22 21 20 19 18

To Hema, Ashok, and Maya

Contents

Preface xi

Chapter 1. A Crash Course in Commutative Algebra 1

§1.1. Basic algebra 1

§1.2. Field extensions 6

§1.3. Modules 8

§1.4. Localization 9

§1.5. Noetherian rings and factorization 10

§1.6. Primary decomposition 13

§1.7. Integral extensions 16

§1.8. Dimension 19

§1.9. Depth 20

§1.10. Normal rings and regular rings 22

Chapter 2. Affine Varieties 27

§2.1. Affine space and algebraic sets 27

§2.2. Regular functions and regular maps of affine algebraic sets 33

§2.3. Finite maps 40

§2.4. Dimension of algebraic sets 42

§2.5. Regular functions and regular maps of quasi-affine varieties 48

§2.6. Rational maps of affine varieties 58

Chapter 3. Projective Varieties 63

§3.1. Standard graded algebras 63

v

vi Contents

§3.2. Projective varieties 67

§3.3. Grassmann varieties 73

§3.4. Regular functions and regular maps of quasi-projectivevarieties 74

Chapter 4. Regular and Rational Maps of Quasi-projective Varieties 87

§4.1. Criteria for regular maps 87

§4.2. Linear isomorphisms of projective space 90

§4.3. The Veronese embedding 91

§4.4. Rational maps of quasi-projective varieties 93

§4.5. Projection from a linear subspace 95

Chapter 5. Products 99

§5.1. Tensor products 99

§5.2. Products of varieties 101

§5.3. The Segre embedding 105

§5.4. Graphs of regular and rational maps 106

Chapter 6. The Blow-up of an Ideal 111

§6.1. The blow-up of an ideal in an affine variety 111

§6.2. The blow-up of an ideal in a projective variety 120

Chapter 7. Finite Maps of Quasi-projective Varieties 127

§7.1. Affine and finite maps 127

§7.2. Finite maps 131

§7.3. Construction of the normalization 135

Chapter 8. Dimension of Quasi-projective Algebraic Sets 139

§8.1. Properties of dimension 139

§8.2. The theorem on dimension of fibers 141

Chapter 9. Zariski’s Main Theorem 147

Chapter 10. Nonsingularity 153

§10.1. Regular parameters 153

§10.2. Local equations 155

§10.3. The tangent space 156

§10.4. Nonsingularity and the singular locus 159

§10.5. Applications to rational maps 165

Contents vii

§10.6. Factorization of birational regular maps of nonsingularsurfaces 168

§10.7. Projective embedding of nonsingular varieties 170

§10.8. Complex manifolds 175

Chapter 11. Sheaves 181

§11.1. Limits 181

§11.2. Presheaves and sheaves 185

§11.3. Some sheaves associated to modules 196

§11.4. Quasi-coherent and coherent sheaves 200

§11.5. Constructions of sheaves from sheaves of modules 204

§11.6. Some theorems about coherent sheaves 209

Chapter 12. Applications to Regular and Rational Maps 221

§12.1. Blow-ups of ideal sheaves 221

§12.2. Resolution of singularities 225

§12.3. Valuations in algebraic geometry 228

§12.4. Factorization of birational maps 232

§12.5. Monomialization of maps 236

Chapter 13. Divisors 239

§13.1. Divisors and the class group 240

§13.2. The sheaf associated to a divisor 242

§13.3. Divisors associated to forms 249

§13.4. Calculation of some class groups 249

§13.5. The class group of a curve 254

§13.6. Divisors, rational maps, and linear systems 259

§13.7. Criteria for closed embeddings 264

§13.8. Invertible sheaves 269

§13.9. Transition functions 271

Chapter 14. Differential Forms and the Canonical Divisor 279

§14.1. Derivations and Kahler differentials 279

§14.2. Differentials on varieties 283

§14.3. n-forms and canonical divisors 286

Chapter 15. Schemes 289

§15.1. Subschemes of varieties, schemes, and Cartier divisors 289

§15.2. Blow-ups of ideals and associated graded rings of ideals 293

viii Contents

§15.3. Abstract algebraic varieties 295

§15.4. Varieties over nonclosed fields 296

§15.5. General schemes 296

Chapter 16. The Degree of a Projective Variety 299

Chapter 17. Cohomology 307

§17.1. Complexes 307

§17.2. Sheaf cohomology 308

§17.3. Cech cohomology 310

§17.4. Applications 312

§17.5. Higher direct images of sheaves 320

§17.6. Local cohomology and regularity 325

Chapter 18. Curves 333

§18.1. The Riemann-Roch inequality 334

§18.2. Serre duality 335

§18.3. The Riemann-Roch theorem 340

§18.4. The Riemann-Roch problem on varieties 343

§18.5. The Hurwitz theorem 345

§18.6. Inseparable maps of curves 348

§18.7. Elliptic curves 351

§18.8. Complex curves 358

§18.9. Abelian varieties and Jacobians of curves 360

Chapter 19. An Introduction to Intersection Theory 365

§19.1. Definition, properties, and some examples of intersectionnumbers 366

§19.2. Applications to degree and multiplicity 375

Chapter 20. Surfaces 379

§20.1. The Riemann-Roch theorem and the Hodge indextheorem on a surface 379

§20.2. Contractions and linear systems 383

Chapter 21. Ramification and Etale Maps 391

§21.1. Norms and Traces 392

§21.2. Integral extensions 393

§21.3. Discriminants and ramification 398

Contents ix

§21.4. Ramification of regular maps of varieties 406

§21.5. Completion 408

§21.6. Zariski’s main theorem and Zariski’s subspace theorem 413

§21.7. Galois theory of varieties 421

§21.8. Derivations and Kahler differentials redux 424

§21.9. Etale maps and uniformizing parameters 426

§21.10. Purity of the branch locus and the Abhyankar-Jungtheorem 433

§21.11. Galois theory of local rings 438

§21.12. A proof of the Abhyankar-Jung theorem 441

Chapter 22. Bertini’s Theorems and General Fibers of Maps 451

§22.1. Geometric integrality 452

§22.2. Nonsingularity of the general fiber 454

§22.3. Bertini’s second theorem 457

§22.4. Bertini’s first theorem 458

Bibliography 469

Index 477

Preface

This book is an introductory course in algebraic geometry, proving most ofthe fundamental classical results of algebraic geometry.

Algebraic geometry combines the intuition of geometry with the preci-sion of algebra. Starting with geometric concepts, we introduce machineryas necessary to model important ideas from algebraic geometry and to provefundamental results. Emphasis is put on developing facility with connect-ing geometric and algebraic concepts. Examples are constructed or citedillustrating the scope of these results. The theory in this book is developedin increasing sophistication, giving (and refining) definitions as required toaccommodate new geometric ideas.

We work as much as possible with quasi-projective varieties over analgebraically closed field of arbitrary characteristic. This allows us to inter-pret varieties through their function fields. This approach and the use ofmethods of algebraic number theory in algebraic geometry have been cen-tral to algebraic geometry at least since the time of Dedekind and Weber(Theorie der algebraischen Functionen einer Veranderlichen [46], translatedin [47]). By interpreting the geometric concept of varieties through theirregular functions, we are able to use the techniques of commutative algebra.

Differences between the theory in characteristic 0 and positive charac-teristic are emphasized in this book. We extend our view to schemes, al-lowing rings with nilpotents, to study fibers of regular maps and to developintersection theory. We discuss the cases of nonclosed ground fields andnonseparated schemes and some of the extra considerations which appear inthese situations. A list of exercises is given at the end of many sections andchapters.

xi

xii Preface

The classic textbooks Basic Algebraic Geometry [136] by Shafarevich,Introduction to Algebraic Geometry [116] by Mumford, and AlgebraicGeometry [73] by Hartshorne, as well as the works of Zariski, Abhyankar,Serre, and Grothendieck, have been major influences on this book.

The necessary commutative algebra is introduced and reviewed, begin-ning with Chapter 1, “A Crash Course in Commutative Algebra”. We statedefinitions and theorems, explain concepts, and give examples from com-mutative algebra for everything that we will need, proving some results andgiving a few examples, but mostly giving references to books on commuta-tive algebra for proofs. As such, this book is intended to be self-contained,although a reader may be curious about the proofs for some cited resultsin commutative algebra and will want to either derive them or look up thereferences. We give references to several books, mostly depending on whichbook has the exact statement we require.

A reader should be familiar with the material through Section 1.6 onprimary decomposition before beginning Chapter 2 on affine varieties. De-pending on the background of students, the material in Chapter 1 can beskipped, quickly reviewed at the beginning of a course, or be used as anoutline of a semester-long course in commutative algebra before beginningthe study of geometry in Chapter 2.

Chapters 2–10 give a one-semester introduction to algebraic geometry,through affine and projective varieties. The sections on integral extensions,dimension, depth, and normal and regular local rings in Chapter 1 can bereferred to as necessary as these concepts are encountered within a geometriccontext. Chapters 11–20 provide a second-semester course, which includessheaves, schemes, cohomology, divisors, intersection theory, and the appli-cation of these concepts to curves and surfaces.

Chapter 21 (on ramification and etale maps) and Chapter 22 (onBertini’s theorems and general fibers of maps) could be the subject of atopics course for a third-semester course. The distinctions between char-acteristic 0 and positive characteristic are especially explored in these lastchapters. These two chapters could be read any time after the completionof Chapter 14.

I thank the students of my classes, especially Razieh Ahmadian,Navaneeth Chenicheri Chathath, Suprajo Das, Arpan Dutta, Melissa Emory,Kyle Maddox, Smita Praharaj, Thomas Polstra, Soumya Sanyal, Pham AnVinh, and particularly Roberto Nunez, for their feedback and helpful com-ments on preliminary versions of this book. I thank Maya Cutkosky for helpwith the figures.

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Index

(L1 · · · Lt), 369

(L1 · · · Lt;F)V , 368

(L1 · · · Lt ·W ), 369

(φ), 286

(φ)X , 286

(f), 241

(f)0, 241

(f)∞, 242

A-torsion, 208

B(I), 113, 121

B(I), 221–223C∗(U,F), 310

D(B/A), 399

D(F ), 71, 103

D(f), 48

D ∩ U , 242

D ≡ 0, 382

D1 ≥ D2, 240

D1 ∼ D2, 242

DL/K(a1, . . . , an), 398

F : Xp → X, 349

F a, 71

Fr(x), 349

G(L,K), 7

G(Y/X), 421

Gi(S/R), 440

Gs(S/R), 438

Hi(A∗), 307

Hi(X,F ), 308, 309

HiY (X,F), 330

HiI(M), 326

H0Sing(U,G), 358

I(S/R), 435

I(Y ), 29, 33, 69, 70, 103

I(Z), 292

Isat, 66

IX(Y ), 33, 70

J(S/R), 435

Ks, 438

KX , 286

M(n), 63

M∨, 243

M(F ), 64

M(p), 64

N(X), 382

NL/K , 392

P (A), 243

PGX , 189

PM (z), 301

PY , 301

Riφ∗F , 320

R(d), 64

S(I;R), 56

S(W ), 103

S(X), 70

S(X × Pn), 105

S(Y ), 105

S(Pm × Pn), 102

S(Pn), 68

T (A), 24

TA(M), 208

Tp(X), 157

Vν , 228

WF , 103

X × Y , 101

X(r), 360

477

478 Index

XF , 71XZ , 290Xf , 48Xp, 290

Y H , 422Z(A), 102Z(I), 68Z(J), 70Z(T ), 28, 33Z(U), 68Z(f), 28Z1 ∩ Z2, 290ZX(T ), 33ZX(U), 70Zred, 289[L : K]i, 7[L : K]s, 7Am × An, 101An, 27C{x1, . . . , xn}, 176Δ, 176ΔX , 110ΔPn , 106ΓY (X,F), 330Γν , 228Γφ, 106, 107N, 1Ωn

X/k, 286

Ωnk(X)/k, 286

ΩB/A, 280ΩX/k, 283Pm × Pn, 102Pnk , 67

Z+, 1Zan, 359Hp(X,F), 312

Hp(U,F), 311χ(F), 319deg x, 63deg, 250deg(D), 254deg(Y ), 301deg(φ), 254, 371, 406dimR, 19dimX, 42, 139�R(M), 9

M , 409

φ : R → S, 411lim← Ai, 184lim→ Ai, 182F(n), 210

F ⊗ G, 204, 206F ⊗OX G, 204, 206Fan, 359G∗, 245IOX , 205IY , 155, 198, 200IZ , 289IY,p, 155Idiv(s)

, 291

L−1, 269OD, 290OU (V ), 53OX -module, 191OX -module homomorphism, 191OX -submodule, 192OX -torsion, 209OX(D), 242OX(U), 49, 75, 187OX(n), 210O∗

X , 271OU,p, 53OW,(p,q), 103OW (U), 103OX,E , 240OX,Y , 229OX,p, 50, 75T (F), 209

p(ai)i , 245

AnnX(σ), 208Aut(X), 351Base(L), 261Cl(X), 242Cl0(X), 258Cokernel(α), 194DerA(B,M), 279Div(F ), 249Div(X), 240Grass(a, b), 73Image(α), 194Kernel(α), 194Map(U,A1), 49, 75Pic(X), 270Pican(X), 359Proj(S), 67, 297Spec(R), 5, 296Supp D, 240bideg(x), 102div(φ), 286div(φ)X , 286div(f), 242div(f)X , 242

Index 479

div(s), 291gcd(D0, . . . , Dn), 259ht(P), 19reg(M), 328

reg(M), 328trdegKL, 6μ(I), 46νE , 241φ : X ��� Y , 58, 93φ∗, 35, 76φ∗ : Cl(Y ) → Cl(X), 252φ∗M, 205φ−1(Z), 290φL, 260φV , 260Vol(L), 345√I, 3√I, 289

TrL/K , 392

I, 120, 198, 200J , 192M , 196N , 199|D|, 260∧nM , 286ai(M), 327d(R′ : R), 443dφp, 157e(R), 376e(Si/R), 396e(ν∗/ν), 397eP , 346f(Si/R), 396f(ν∗/ν), 397f∗G, 207fh, 71f#, 191f∗F , 191g(R′ : R), 443g(X), 334hi(X,F), 319k, an algebraically closed field, 1k-linear Frobenius map, 349k(U), 53, 82k(W ), 103k(X), 49, 74k(p), 50, 75, 283k[X], 33k[Am × An], 101k[An], 27m-regular, 328

mR, the maximal ideal of a local ring, 1mp, 50, 75, 283mX,p, 50, 75n-fold, 45p-basis, 425p-independent, 425r(R′ : R), 443Aut(L/K), 7Aut(k(Y )/k(X)), 421HomOX (F ,G), 201HomOX (F ,G), 204, 206Num(X), 382QF(R), 1QR(A), 291codimX(Y ), 45depthIM , 326res, 34grmR

(R), 23

Abel, 360Abelian variety, 360Abhyankar, xii, 170, 227, 228, 231, 232,

234, 235, 392, 413, 435Abhyankar-Jung theorem, 436Abramovich, 234Abstract prevariety, 295Abstract variety, 295Adjunction, 287, 380Affine map, 127Affine scheme, 296Affine subscheme, 289Affine subvariety, 34Affine variety, 55Algebraic function field, 7Algebraic local ring, 441Algebraic set, biprojective, 103Algebraic set, in An, 28Algebraic set, projective, 68Algebraic set, quasi-affine, 34Algebraic set, quasi-biprojective, 103Ample divisor, 262Ample invertible sheaf, 270Analytic implicit function theorem, 176Artin, 362Auslander, 25, 433

Base locus, of a linear system, 261Base point free linear system, 261Bayer, 330Bertini, 304, 451Bezout, 375Bezout’s theorem, 375

480 Index

Bhatt, 433Bihomogeneous coordinate ring, 102Birational equivalence, of affine

varieties, 60Birational equivalence, of

quasi-projective varieties, 94Birational map, of affine varieties, 60Birational map, of quasi-projective

varieties, 94Birkar, 387Blow-up, of a subvariety of a projective

variety, 123Blow-up, of a subvariety of an affine

variety, 113Blow-up, of an ideal of a projective

variety, 121Blow-up, of an ideal of an affine variety,

113Blow-up, of an ideal sheaf on a

projective variety, 222Blow-up, of an ideal sheaf on a

quasi-projective variety, 223Blow-up, of an ideal sheaf on an affine

variety, 221Buchsbaum, 25

Canonical divisor, 286Cartier divisor, 245, 273, 291Carvajal-Rojas, 433Cascini, 387Castelnuovo, 328, 383Castelnuovo’s contraction theorem, 383Chain map, 307Chevalier, 411Chinese remainder theorem, 4Christiansen, 235Clifford, 342Clifford’s theorem, 342Closed embedding, of a quasi-affine

variety, 55Closed embedding, of a quasi-projective

variety, 88Closed embedding, of an affine algebraic

set, 38Closed subscheme, 289Codimension, of a quasi-affine algebraic

set, 45Codimension, of a quasi-projective

algebraic set, 141Cohen, 410Coherent OX -module, 200Coherent sheaf, 290

Cohomology, 311, 312Complete abstract variety, 295Complete linear system, 260Complex, 307Complex manifold, 175Constant sheaf, 189Coordinate functions, on an affine

algebraic set, 38Coordinate ring of Pn, 68Coordinate ring, of a biprojective

variety, 103Coordinate ring, of a projective

algebraic set, 70Coordinate ring, of a subvariety of

Am × Pn, 105Cossart, 228Curve, 45Cutkosky, 235, 236, 345

Dedekind, xiDegree, of a divisor on a curve, 254Degree, of a graded module, 301Degree, of a map of curves, 254Degree, of a projective subscheme, 301Degree, of a regular map, 371, 406Degree, of an invertible sheaf on a

curve, 259Del Pezzo, 304Derivation, 279Diagonal, 106, 176Dimension, of a quasi-projective

algebraic set, 139Dimension, of a topological space, 42Direct limit, 181Discrete valuation, 395Discrete valuation ring, 395Discriminant ideal, 398Divisor, 240Divisor class group, 242Divisor of a function, 241Divisor of a section of an invertible

sheaf, 273Divisor of poles of a function, 242Divisor of zeros of a function, 241Divisor, local equation, 245Divisor, of a form, 249Divisor, of a rational differential

n-form, 286Divisor, support, 240Divisorial valuation, 229Dominant map, of affine algebraic sets,

37

Index 481

Dominant rational map, of affinevarieties, 59

Dominant rational map, ofquasi-projective varieties, 94

Dominant regular map, of aquasi-projective variety, 82

Dominate, rings, 1dvr, 395

Effective Cartier divisor, 291Effective divisor, 240Eisenbud, 304, 330Elliptic curve, 341

Etale regular map, 426Euclidean topology, 175Euler, 13Euler characteristic of a sheaf, 319Excision, 331

Fiber cone, 294Finite map, 127Finite map, of affine varieties, 40First theorem of Bertini, 459Fractional ideal, 243Frobenius homomorphism, 349

Gabrielov, 420Galois over X, 422Galois regular map, 422Generated by global sections, 212Genus, of a curve, 334Geometric regularity theorem, 328Geometrically integral ring, 452Geometrically integral scheme, 452Geometrically irreducible ring, 452Geometrically irreducible scheme, 452Geometrically reduced ring, 452Geometrically reduced scheme, 452Germ, 185Godement, 181Going down theorem, 393Going up theorem, 393Goto, 324, 330Graded module, 63Graded ring, 63Grant, 433Graph, of a rational map, 107Graph, of a regular map, 106Grauert, 387Grothendieck, xii, 151, 367, 433

Hacon, 387Harris, 304

Hartshorne, xii, 181Heinzer, 395Herzog, 46, 116Higher direct images of a sheaf, 320Hilbert, 11, 29, 301, 376Hilbert polynomial, of a graded module,

301Hilbert polynomial, of a projective

subscheme, 301Hilbert’s basis theorem, 11Hilbert’s nullstellensatz, 29Hilbert-Samuel polynomial, 376Hironaka, 227, 232, 234, 295Hodge index theorem, 382Homogeneous coordinates, 90Homogeneous coordinates, on Pn, 68Homogeneous ideal, 64Homogeneous ideal of a subscheme, 292Homotopy, 308Huneke, 116Hurwitz, 348

Ideal sheaf, 120, 205Ideal sheaf, of a subvariety, 200Ideal sheaf, on a projective variety, 192Ideal sheaf, on an affine variety, 192Ideal transform, 56Image, of a rational map, 94Indirect limit, 184Inertia field, 440Injective sheaf homomorphism, 194Integral scheme, 290Intersection number, 368Invertible sheaf, 203, 223, 245, 269, 290Irreducible scheme, 290Irreducible topological space, 29Isomorphism, of affine algebraic sets, 38Isomorphism, of quasi-affine varieties,

54Isomorphism, of varieties, 76

Jacobi, 360Jacobian criterion for nonsingularity,

160Jacobian, of a curve, 360, 361Jung, 435

Kahler differentials, 280Kaplansky, 232Karu, 234, 236Kollar, 387Krull, 20

482 Index

Krull’s principal ideal theorem, 20Kunneth, 319Kunneth formula, 319

Lang, 29Lazarsfeld, 345Linear equivalence of divisors, 242Linear subspace, of a projective space,

95Linear system, 260Lipman, 227, 231Local cohomology, 326, 330Local equations, 155Local ring, 1Locally free sheaf, 217Locally principal ideal sheaf, 223Locally ringed space, 191

Map of complexes, 307Matsuki, 234Matsusaka, 452Mayer-Vietoris sequence, 331McKernan, 387Milne, 362Mori, 387Morphism, of affine schemes, 297Morphism, of locally ringed spaces, 191Morphism, of schemes, 297Multiplicity, 376Mumford, xii, 328, 330Mustata, 345

Nagata, 57, 324, 433Nakayama, 8Newton, 437Nishida, 324Noether, 17, 134Noether’s normalization lemma, 17Nondegenerate variety, 299Nonsingular point, of a variety, 160Nonsingular, quasi-projective variety,

160Normal field extension, 394Normal point, 135Normal variety, 135Normalization, in a finite extension, 135Nullhomotopic, 308Numerical equivalence of divisors, 382Numerical polynomial, 300

Oda, 235Open embedding, of quasi-affine

varieties, 55

Open subscheme, 289

Picard group, 270Piltant, 228Presheaf, 185Presheaf, homomorphism, 185Presheaf, isomorphism, 185Prime divisor, 240Primitive element, 401Primitive element theorem, 7Principal divisor, 242Projective algebraic set, 69Projective Noether normalization, 134Projective scheme, 297Projective subscheme, 289Projective variety, 69Proper transform, 123Purity of the branch locus, 433

Quadratic transform, 170Quasi-affine variety, 34Quasi-coherent OX -module, 200Quasi-coherent sheaf, 290Quasi-projective algebraic set, 69Quasi-projective subscheme, 289Quasi-projective variety, 69Quotient of Y by H, 422

Ramification index, of a regular map ofcurves, 346

Ramification locus, in the domain of afinite map, 432, 433

Ramification locus, in the range of afinite map, 407

Ramified extension, of a normaldomain, 400

Ramified extension, of local domains,399

Ramified, map of curves, 346Rashid, 234Rational function field, 7Rational functions, on a biprojective

variety, 103Rational functions, on a projective

variety, 74Rational functions, on a quasi-affine

variety, 53Rational functions, on a

quasi-projective variety, 82Rational functions, on an affine variety,

49Rational map, of affine varieties, 58

Index 483

Rational map, of projective varieties, 93Rational map, of quasi-projective

varieties, 93Rational maps, on a subvariety of

Am × Pn, 105Reduced ramification index, 396Reduced scheme, 290Rees, 57Reflexive rank 1 sheaf, 245Regular functions, on An, 27Regular functions, on a quasi-affine

variety, 49Regular functions, on an affine

algebraic set, 33Regular functions, on an open subset of

a biprojective variety, 103Regular map, of affine algebraic sets, 36Regular map, of an affine variety to An,

35Regular map, of quasi-affine varieties,

54Regular map, of varieties, 76Regular maps, of biprojective varieties,

103Regular parameters, 154Regularity, 328Relative degree, 396Resolution of indeterminancy, 225Resolution of singularities, 225Riemann, 335, 340Riemann-Roch inequality, 335Riemann-Roch problem, 344Riemann-Roch theorem, 340Rigidity lemma, 358Roberts, 324Roch, 335, 340

Sally, 46, 232Samuel, 376Scheme, 289, 297Scheme-theoretic fiber, 290Scheme-theoretic intersection, 290Schwede, 433Second theorem of Bertini, 458Section, nonzero divisor, 291Separable field extension, 452Separable regular map, 346, 452Separably generated field, 7Separated scheme, 297Separated, variety, 82Separating transcendence basis, 7Serre, xii, 301, 319, 335, 339, 468

Serre duality, 319, 339Shafarevich, xii, 168Shannon, 232Sheaf, 186Sheaf cohomology, 308Sheaf, axioms, 186Sheaf, global section, 186Sheaf, homomorphism, 187Sheaf, isomorphism, 187Sheaf, support, 203Sheafification of a graded module on a

projective variety, 199Sheafification of a module on an affine

variety, 196Sheafification of a presheaf, 187Short exact sequence of sheaves, 194Simis, 116Singular cohomology, 358Singular locus, 162Smooth morphism, 456Snapper, 366Snapper polynomial, 366Splitting field, 438Splitting group, 438Srinivas, 345Strict transform, of a projective variety,

123Strict transform, of an affine variety,

116Subscheme, 289Subsheaf, 192Subvariety, 69Surface, 45Surjective sheaf homomorphism, 194Symbolic power, 245System of parameters, 142

Tamely ramified, map of curves, 346Tangent cone, 294Tangent space, extrinsic definition, 156Tangent space, intrinsic definition, 157Torsion sheaf, 366Transcendence basis, 6Transcendence degree, 6Transition functions on an invertible

sheaf, 271Trivialization, of an invertible sheaf, 292Tucker, 433

Ulrich, 116Uniformizing parameters, 285Universal property of blowing up, 223

484 Index

Unramified extension, of a normaldomain, 400

Unramified extension, of local domains,399

Valuation, 228Valuation ring, 228Value group, 228Variety, biprojective, 103Variety, quasi-biprojective, 103Vasconcelos, 116Veronese subring, 64Very ample divisor, 262Very ample invertible sheaf, 270

Walker, 227Watanabe, 324Weber, xiWeierstrass, 176Weierstrass preparation theorem, 176Weil, 362Wildly ramified, map of curves, 346W�lodarczyk, 234

Zariski, xii, 76, 148, 151, 161, 168, 227,229–232, 344, 388, 392, 413, 416,420, 433, 452, 458

Zariski topology, on An, 29Zariski topology, on Pn, 68Zariski topology, on a quasi-projective

algebraic set, 68Zariski topology, on an affine algebraic

set, 34Zariski topology, on the Proj of a

graded ring, 67Zariski topology, on the Spec of a ring,

5Zariski’s connectedness theorem, 151Zariski’s main theorem, 148, 416Zariski’s subspace theorem, 420

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GSM/188

This book presents a readable and accessible introductory course in algebraic geom-etry, with most of the fundamental classical results presented with complete proofs. An emphasis is placed on developing connections between geometric and algebraic aspects of the theory. Differences between the theory in characteristic 0 and posi-tive characteristic are emphasized. The basic tools of classical and modern algebraic geometry are introduced, including varieties, schemes, singularities, sheaves, sheaf cohomology, and intersection theory. Basic classical results on curves and surfaces

With more than 200 exercises, the book is an excellent resource for teaching and learning introductory algebraic geometry.

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