This new-in-paperback edition provides an introduction to algebraic and arithmetic geometry, starting with the theory of schemes, followed by applications to arithmetic surfaces and to the theory of reduction of algebraic curves. Clear explanations of both theory and applications, and almost 600 exercises are included in the text. 1 Some topics in commutative algebra 1(25) 1.1 Tensor products 1(5) 1.1.1 Tensor product of modules 1(3) 1.1.2 Right-exactness of the tensor product 4(1) 1.1.3 Tensor product of algebras 5(1) 1.2 Flatness 6(9) 1.2.1 Left-exactness: flatness 6(3) 1.2.2 Local nature of flatness 9(3) 1.2.3 Faithful flatness 12(3) 1.3 Formal completion 15(11) 1.3.1 Inverse limits and completions 15(5) 1.3.2 The Artin Rees lemma and applications 20(2) 1.3.3 The case of Noetherian local rings 22(4) 2 General properties of schemes 26(52) 2.1 Spectrum of a ring 26(7) 2.1.1 Zariski topology 26(3) 2.1.2 Algebraic sets 29(4) 2.2 Ringed topological spaces 33(8) 2.2.1 Sheaves 33(4) 2.2.2 Ringed topological spaces 37(4) 2.3 Schemes 41(18) 2.3.1 Definition of schemes and examples 42(3) 2.3.2 Morphisms of schemes 45(5) 2.3.3 Projective schemes 50(5) 2.3.4 Noetherian schemes, algebraic varieties 55(4) 2.4 Reduced schemes and integral schemes 59(8) 2.4.1 Reduced schemes 59(2) 2.4.2 Irreducible components 61(3) 2.4.3 Integral schemes 64(3) 2.5 Dimension 67(11) 2.5.1 Dimension of schemes 68(2) 2.5.2 The case of Noetherian schemes 70(3) 2.5.3 Dimension of algebraic varieties 73(5) 3 Morphisms and base change 78(37) 3.1 The technique of base change 78(9) 3.1.1 Fibered product 78(3) 3.1.2 Base change 81(6) 3.2 Applications to algebraic varieties 87(12) 3.2.1 Morphisms of finite type 87(2) 3.2.2 Algebraic varieties and extension of the base field 89(3) 3.2.3 Points with values in an extension of the base field 92(2) 3.2.4 Frobenius 94(5) 3.3 Some global properties of morphisms 99(16) 3.3.1 Separated morphisms 99(4) 3.3.2 Proper morphisms 103(4) 3.3.3 Projective morphisms 107(8) 4 Some local properties 115(42) 4.1 Normal schemes 115(11) 4.1.1 Normal schemes and extensions of regular functions 115(4) 4.1.2 Normalization 119(7) 4.2 Regular schemes 126(9) 4.2.1 Tangent space to a scheme 126(2) 4.2.2 Regular schemes and the Jacobian criterion 128(7) 4.3 Flat morphisms and smooth morphisms 135(14) 4.3.1 Flat morphisms 136(3) 4.3.2 Etale morphisms 139(2) 4.3.3 Smooth morphisms 141(8) 4.4 Zariski's 'Main Theorem' and applications 149(8) 5 Coherent sheaves and Cech cohomology 157(53) 5.1 Coherent sheaves on a scheme 157(21) 5.1.1 Sheaves of modules 157(2) 5.1.2 Quasi-coherent sheaves on an affine scheme 159(2) 5.1.3 Coherent sheaves 161(3) 5.1.4 Quasi-coherent sheaves on a projective scheme 164(14) 5.2 Cech cohomology 178(17) 5.2.1 Differential modules and cohomology with values in a sheaf 178(7) 5.2.2 Cech cohomology on a separated scheme 185(3) 5.2.3 Higher direct image and flat base change 188(7) 5.3 Cohomology of projective schemes 195(15) 5.3.1 Direct image theorem 195(3) 5.3.2 Connectedness principle 198(3) 5.3.3 Cohomology of the fibers 201(9) 6 Sheaves of differentials 210(42) 6.1 K er differentials 210(10) 6.1.1 Modules of relative differential forms 210(5) 6.1.2 Sheaves of relative differentials (of degree 1) 215(5) 6.2 Differential study of smooth morphisms 220(7) 6.2.1 Smoothness criteria 220(3) 6.2.2 Local structure and lifting of sections 223(4) 6.3 Local complete intersection 227(9) 6.3.1 Regular immersions 228(4) 6.3.2 Local complete intersections 232(4) 6.4 Duality theory 236(16) 6.4.1 Determinant 236(2) 6.4.2 Canonical sheaf 238(5) 6.4.3 Grothendieck duality 243(9) 7 Divisors and applications to curves 252(65) 7.1 Cartier divisors 252(15) 7.1.1 Meromorphic functions 252(4) 7.1.2 Cartier divisors 256(4) 7.1.3 Inverse image of Cartier divisors 260(7) 7.2 Weil divisors 267(8) 7.2.1 Cycles of codimension 1 267(5) 7.2.2 Van der Waerden's purity theorem 272(3) 7.3 Riemann Roch theorem 275(9) 7.3.1 Degree of a divisor 275(3) 7.3.2 Riemann Roch for projective curves 278(6) 7.4 Algebraic curves 284(19) 7.4.1 Classification of curves of small genus 284(5) 7.4.2 Hurwitz formula 289(3) 7.4.3 Hyperelliptic curves 292(5) 7.4.4 Group schemes and Picard varieties 297(6) 7.5 Singular curves, structure of Pic (X) 303(14) 8 Birational geometry of surfaces 317(58) 8.1 Blowing-ups 317(15) 8.1.1 Definition and elementary properties 318(5) 8.1.2 Universal property of blowing-up 323(3) 8.1.3 Blowing-ups and birational morphisms 326(4) 8.1.4 Normalization of curves by blowing-up points 330(2) 8.2 Excellent schemes 332(15) 8.2.1 Universally catenary schemes and the dimension formula 332(3) 8.2.2 Cohen Macaulay rings 335(6) 8.2.3 Excellent schemes 341(6) 8.3 Fibered surfaces 347(28) 8.3.1 Properties of the fibers 347(6) 8.3.2 Valuations and birational classes of fibered surfaces 353(3) 8.3.3 Contraction 356(5) 8.3.4 Desingularization 361(14) 9 Regular surfaces 375(79) 9.1 Intersection theory on a regular surface 376(18) 9.1.1 Local intersection 376(5) 9.1.2 Intersection on a fibered surface 381(7) 9.1.3 intersection with a horizontal divison adjunction formula 388(6) 9.2 Intersection and morphisms 394(17) 9.2.1 Factorization theorem 394(3) 9.2.2 Projection formula 397(4) 9.2.3 Birational morphisms and Picard groups 401(3) 9.2.4 Embedded resolutions 404(7) 9.3 Minimal surfaces 411(18) 9.3.1 Exceptional divisors and Castelnuovo's criterion 412(6) 9.3.2 Relatively minimal surfaces 418(3) 9.3.3 Existence of the minimal regular model 421(3) 9.3.4 Minimal desingularization and minimal embedded resolution 424(5) 9.4 Applications to contraction; canonical model 429(25) 9.4.1 Artin's contractability criterion 430(4) 9.4.2 Determination of the tangent spaces 434(4) 9.4.3 Canonical models 438(4) 9.4.4 Weierstrass models and regular models of elliptic curves 442(12) 10 Reduction of algebraic curves 454(103) 10.1 Models and reductions 454(29) 10.1.1 Models of algebraic curves 455(7) 10.1.2 Reduction 462(5) 10.1.3 Reduction map 467(4) 10.1.4 Graphs 471(12) 10.2 Reduction of elliptic curves 483(22) 10.2.1 Reduction of the minimal regular model 484(5) 10.2.2 N n models of elliptic curves 489(9) 10.2.3 Potential semi-stable reduction 498(7) 10.3 Stable reduction of algebraic curves 505(27) 10.3.1 Stable curves 505(6) 10.3.2 Stable reduction 511(10) 10.3.3 Some sufficient conditions for the existence of the stable model 521(11) 10.4 Deligne Mumford theorem 532(25) 10.4.1 Simplifications on the base scheme 533(4) 10.4.2 Proof of Artin Winters 537(6) 10.4.3 Examples of computations of the potential stable reduction 543(14) Bibliography 557(5) Index 562