Reid M. — Undergraduate algebraic geometry :: Электронная библиотека попечительского совета мехмата МГУ

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Reid M. — Undergraduate algebraic geometry

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Название: Undergraduate algebraic geometry

Автор: Reid M.

Аннотация:

Algebraic geometry is, essentially, the study of the solution of equations and occupies a central position in pure mathematics. With the minimum of prerequisites, Dr. Reid introduces the reader to the basic concepts of algebraic geometry, including: plane conics, cubics and the group law, affine and projective varieties, and nonsingularity and dimension. He stresses the connections the subject has with commutative algebra as well as its relation to topology, differential geometry, and number theory. The book contains numerous examples and exercises illustrating the theory.

Язык:

Рубрика: Математика/Алгебра/Алгебраическая геометрия/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1988

Количество страниц: 131

Добавлена в каталог: 12.03.2005

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Предметный указатель

a.c.c. (ascending chain condition)      48—49 53 55 63
Abstract variety      4 79—80 117—118
Affine change of coordinates      12 24
Affine coordinates      14 38 43 111
Affine covering of projective variety      83
Affine curve      39 45 79
Affine piece of projective variety      13 38 79 83—84 92
Affine space $\mathbb{A}^n_k$       50 53 60 64 66 69 77 79 89 94 100 113 121
Affine variety      4 48 50 70 74 78
Aftine cone over projective variety      81 82
Aftine scheme      118
Algebraic (sub-) set      50—55 64 66 72 78 81—86
Algebraically closed field      52 54 55 64 71 77 115—116 118 119 120
Algebraically independent      59 89 97 100
Asymptotic line      9 12 14 60 111
Bezout’s Theorem      17—18 33 35—36
Birational equivalence      87—89 99 100—101 107
Birational maps      87—89 91 92 99 100—101
Blow-up      100—101
Categories of geometry      2—4 46
Category theory      4 114 118 121
Characteristic p      4 14 16 24 28 61—62 123
Classification of varieties      43—47 115
Complete variety      117
Complex analytic geometry      3 36 43—47 95 116
Complex function theory      6 45—47 112 116
conic      9—20 25—33 37—38 45 85 93
Coordinate ring k[V]      66—72 73 74 75 118 121
Cubic curve      1 2 7 27—42 43—44 75—77 79 92 115
Cubic surface      102—111 114 115
Cuspidal cubic      27 41 68 74 103 111
Denominator of a rational function      4 68 72 76—77 78
Dense open set      36 51 67 71 72 73 88 95 97 99
DIMENSION      2 57 59 60 62 64 97 99 102 123—124
Diophantine problems      1 9—10 24 28 41—42 45—47 123
Discrete valuation ring (d.v.r.)      122 123
Discriminant      22 23 106
Domain of definition dom f      71—73 77 78 83—84 85 87 91
Dominant      73—74 87
Elimination theory      25 57 64 104—105
Empty set $\emptyset$       45 52 53 55 73 82
Equivalence of categories $V \mapsto k[V]$       69 118 120
Finite algebra      4 57—58 59 60 61 64
Finitely generated algebra      4 49 54 57—59 71 118 122
Finitely generated ideal      48 49 50 81
FORM      16—17 22 25 30 99
Function field k(V)      62—63 71 73 74 78 83 85 87 88 89 97 99 112 121 122
Generic point      119 120—121 122
Genus of a curve      43—47 115
Group law on cubic      33—36 39—41 46 76
Homogeneous ideal      80—81 84
Homogeneous polynomial (= form)      16—17 22 25 30 80—81 99—100
Homogeneous V-I correspondences      81—82
Hypersurface      50 56—57 62—63 64 88—89 94—95 99 101
Inflexion      34 38 41 103 111
Infniite descent      29 42
Intersection of plane curves      17 33 35—36 64
Intersection of two conies      20—25 115
Intersection of two quadrics      91—92 115
Irreducible algebraic set      33 52—53 55 57 63 67 71 78 82 84 92 95
Irreducible hypersurface      56—57 64
Isomorphism      4 68 70 74—75 77 78 79 85 87 90 92 93 99
Jacobson ring      118
Jokes (not for exam)      51 55 69 91 114
Linear projection      10 60 65 68 86 92 107
Linear system of plane curves      18—20 30—33
Local ring $\mathcal{O}_{V,P}$       71 83 116 122
Localisation $A[S^{-1}]$       49 63 122
Maximal ideal       54 55 64 118 119 120
Military funding      13 112 113
Moduli      46 47 114 121 123
Morphism (= regular map)      4 36 74 76 77 80 85 90 93 107 111

Multiple roots, multiplicities      16—17 34 35 38 40 52 94 102 106
Nodal cubic      27 40 68 78
Noether normalisation      59—62 64
Noetherian property of Zariski topology      53
Noetherian ring      48—49 63
Non-singular      2 33 92 94—95 97 99 101 102 107 111 116 122
Non-singular cubic      see “Cubic curve”
Normal form of cubic      38 41
Nullstellensatz      30 54 72 81—82 118
Number theory      see “Diophantine problems”
Open set      see “Dense open set” or “Standard open set”
Ouasiprojective variety      4 117
Parallelism      11 12 14 15 25 60
Parametrised curve      9—10 15 17—18 24 27—28 31 40 45 47 68 74 77—78 85 86 88 121
Pascal mystic hexagon      36—37
Pencil of conies      20 21—25
Point at infinity      9 12 13 14 16 17 38 39 40 43 60 76 111
Polynomial curve      27 57 64
Polynomial function      2 3 4 51 66—70 72 96
Polynomial map      2 67—70 74 77 78
Prime ideal      52 54—55 60 61 118 120
Prime spectrum Spec A      118 119 122
Primitive Element Theorem      62
Principal ideal domain (PID)      63
Product of varieties      78 89 92
Projective algebraic geometry      117—118
Projective change of coordinates      11—12 41
Projective curve      13 24 44 75
Projective equivalence      15 18
Projective geometry      9 11 79
Projective line $\mathbb{P}^1$       16 43 79 80 85 90
Projective plane $\mathbb{P}^2$       9 11—20 17 25 30—33 38 47 79 86 107
Projective space $\mathbb{P}^3$ , $\mathbb{P}^n$       4 6 60 80 81 85 86 89 91 100 102 108
Projective variety      4 13 79—90 117
Projective variety and non-singularity      99—100
Quadric surface      64 86 90 91 108 109 110
Radical rad $I = \sqrt{I}$       54—55 63 81—82
Rational curve      45 85 91 115 118
Rational function      2 3 4 28 45 68 71—72 76 78 82—83 116
Rational map      4 28 72—74 76—78 84—87 91 107 111 121
Rational normal curve      85 91
Rational variety      45 88 107 115 117 118 123
Real geometry      6—7 115
Regular function      2 4 71—72 77 78 116 122
Regular function on a projective variety      80 82 83 90 92 116
Regular map (= morphism)      2 4 6 71—72 74 77 78 85 90 92 111
Resultant      25 64 104—105
Riemann sphere      43
Riemann surface      43 45 112
Roots of a form in two variables      see “Zero”
Segre embedding      89
Separability      61—62 95 123
singular      2 94 95 97 100 101 102 111
Singular conic      21—22 25 107
Singular cubic      see “Nodal” or “Cuspidal cubic”
Singular cubic surface      111
Singularity      7 28 94 100—101 116
Singularity theory      2 6 100—101 115
Standard affine pieces $V_{(i)}$       13 38 79 83—84 92
Standard open set Vf      55 72 74—75 98
Tangent space       2 33 34 40 41 94—100 103 111 123
topology      see “Zariski topology”
Topology of a curve      43—44 46
Transcendence degree tr       63 88 89 97 100
Transversal of lines      108
Twisted cubic      85 91 114
Unique factorisation domain (UFD)      28 54 63 71 78
V-I correspondence      50—51 52 53 54 55 60 63—64 66—67 81—82 84
Variety      50 57 70 80 88 89 97 99 102 113 116 117—122
Veronese surface      93
Zariski topology      36 50—51 64 67 71 73 75 78 81 83 84 89 92 95 116 118 119 120
Zero of a form      16—17 22 23 25 31 34 38 41 104 106

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