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Kunz E. — Introduction to Plane Algebraic Curves
Kunz E. — Introduction to Plane Algebraic Curves

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Название: Introduction to Plane Algebraic Curves

Автор: Kunz E.

Аннотация:

This work is an introduction to commutative ring theory and algebraic plane curves, requiring of the student only a basic knowledge of algebra, with all of the algebraic facts collected into several appendices that can be easily referred to, as needed. Kunz's proven conception of teaching topics in commutative algebra together with their applications to algebraic geometry makes this book significantly different from others on plane algebraic curves. The exposition focuses on the purely algebraic aspects of plane curve theory, leaving the topological and analytical viewpoints in the background, with only casual references to these subjects and suggestions for further reading.

Most important to this text:
* Emphasizes and utilizes the theory of filtered algebras, their graduated rings and Rees algebras, to deduce basic facts about the intersection theory of plane curves
* Presents residue theory in the affine plane and its applications to intersection theory

* Methods of proof for the Riemann-Roch theorem conform to the presentation of curve theory, formulated in the language of filtrations and associated graded rings
* Examples, exercises, figures and suggestions for further study round out this fairly self-contained textbook


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2007

Количество страниц: 293

Добавлена в каталог: 10.05.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$X_n$-general      270
Abstract Riemann surface      57
Addition on elliptic curve      90
Additivity of intersection multiplicity      43 170
Adic filtration      200
Adjoint curve      184
Affine coordinate ring      9
Algebraic function field (of one variable)      36
Algebroid curve      169
Angle between curves      120
Artin — Rees lemma      222
Ascending filtration      199
Associated graded algebra      200
Associated graded ideal      205
Astroid      173
Asymptote      121
Base change      237
Bezout’s Theorem      26 42
Birationally equivalent      36
Bounded below grading      194
Branch      162
Branch, integral, regular      163
Cancellation rule for residues      112
Canonical divisor (class)      153
Canonical module      245
Canonical multiplication map      241
Canonical trace      245
Cardioid      6
Cauchy, product      261
Cauchy, sequence      261
Cayley — Bacharach theorem      47
Center of a projection      74
Central projection      73
Centroid      49 123
Chain of prime ideals      223
Chain, rule for traces      110
Chinese remainder theorem      218 265
Cissoid of Diocles      5
Classification of double points      166
Classification of elliptic curves      96
Classification of irreducible curves (birational)      36
Classification of singular cubics      68
Comaximal ideals      217
Compatibility of conductors with completion      179
Complementary module      149 280
complete      262
Completion      268
Component, homogeneous      191
Component, irreducible (of a curve)      10 19
Conchoid of Nichomedes      6
Conductor      175
Conductor divisor (degree)      176
Conductor of a numerical semigroup      183
Conductor of an affine curve (a singularity)      176
Conic sections      4
Convergent series (sum)      261
Converse of Pascal’s theorem      49
Coordinate ring of a divisor      11
Coordinate ring of the intersection of two curves      24 25
Coordinate ring, affine      9
Coordinate ring, projective      23
Cremona transformation      74
Criterion for irreducibility      164
Criterion for regularity      56
Criterion, Jacobian      54
Cubics      5
Curvature      129
Curve, adjoint      184
Curve, affine (plane)      3
Curve, elliptic      89
Curve, irreducible      9 18
Curve, nonsingular (regular, smooth)      53
Curve, projective      14
Curve, reduced      20
Cusp      166
Cycle on $\mathbb{P}^2$      42
Decomposable tensor      237
Decomposition into homogeneous components      191
Decomposition into irreducible components      10 19
Dedekind complementary module      149 280
Dedekind different theorem      156
Dedekind’s formula for conductor and complementary modules      177
Defined over      3 14
Degree of a curve      9 14
Degree of a cycle      42
Degree of a divisor      11 20
Degree of a zero-dimensional scheme      48
Degree, form      201
Degree, formula for valuations      232
Degree-filtration      199
Dehomogenization      17
Denominator, ideal      232
Denominator, set      209
Dense subset of a curve      34
Derivation      243
Diagonal of the enveloping algebra      241
Different divisor (exponent)      156
Differential      242
Differential form      109
Dimension of a linear system      67
Dimension of a ring      223
Discrete valuation ring      224
Divisor      11 20 60 131
Divisor (class) group on $\mathbb{A}^2$      11
Divisor (class) group on $\mathbb{P}^2$      20
Divisor (class) group on an abstract Riemann surface      60
Divisor, canonical      153
Divisor, effective      11 20
Divisor, principal      60
Divisor, special      141
Domain of definition of a rational map      73
Double point      166
Dual basis      245
Dual basis with respect to a trace      248
Dualizing module      245
Effective divisor      11 20
ellipse      4
Elliptic curve      5 89
Embedding dimension      222
Enveloping algebra      240
Epicycloid      80
Equality of fractions      212
Equation of integral dependence      228
Euler — Poincar?e characteristic      141
Euler’s formula      192
F-divisor      133
Fermat curves      5
Field of rational functions of an irreducible curve      36
Field of rational functions on $\mathbb{A}^2$      33
Field of rational functions on $\mathbb{P}^2$      31
Filtered algebra      199
Filtration, separated      199
Filtration, trivial      200
Flex      83
Flex tangent      83
Folium of Descartes      5
Formula of B. Segre      128
Formula of Euler      192
Formula of Hurwitz      156
Formula of Jacobi      118
Formula of Reiss      129
Four-leaf rose      6
Frobenius (number) problem      183
Full ring of quotients      212
Function field of an irreducible curve      36
Fundamental theorem of algebra (graded)      192
Fundamental theorem of Max Noether      45
Genus      140 143 147 156
Global regular differentials      140 281
Graded module (submodule)      193
Graded ring      191
Graded version of Nakayama’s lemma      195
Grothendieck residue symbol      109
Harmonic center      49
Hensel’s Lemma      164
Hessian curve (determinant)      84
Hilbert, basis theorem for power series rings      264
Hilbert, function      195
Hilbert, Nullstellensatz      233 234
Hilbert, series      197
Homogeneous component (element)      191
Homogeneous coordinates      13
Homogeneous homomorphism      194
Homogeneous ideal      18 193
Homogeneous linear form (trace)      249
Homogeneous localization      215
Homogenization      15 201
Homogenization of an ideal      205
Humbert’s theorem      122
Hurwitz formula      156
Hyperbola      4
Hyperelliptic function field (curve)      146
Hypocycloid      80
I-adic filtration      200
I-complete      262
Ideal quotient      257
Indeterminate point of a rational function      33
Indeterminate point of a rational map      73
Index of a divisor      141
Inflection point      83
integral      109
Integral at a point      163
Integral closure (dependence)      228
Integrally closed      228
Interpolation problem      44
Intersection, cycle      42
Intersection, multiplicity      41 169
Intersection, scheme      40
Intersection, transversal      41 65
Irreducibility criterion      164
Irreducible, at a point      163
Irreducible, component      10 19
Irreducible, curve      9 18
J-invariant      95
Jacobian criterion      54
Krull dimension      223
Krull intersection theorem      223
Lattice      96
Laurent polynomial      200
Leading form      201
Lemma of Artin — Rees      222
Lemma of Nakayama      221
Lemma of Nakayama, in the graded case      195
Lemniscate      6
Length of a prime ideal chain      223
Limacon of Pascal      60
LIMIT      261
Line at infinity      15
Linear system      67
Linearly equivalent      132 133
Local ring at P      212
Local ring of a point in $\mathbb{P}^2$      32
Local ring of a point on a curve (on an intersection scheme)      40
localization      212
Luroth’s theorem      78
Maclaurin’s theorem      49
Miguel’s theorem      72
Minimal polynomial of a projective curve      14
Minimal polynomial of an affine curve      9
Minimal set of generators      221
Model of an algebraic function field      36
Module of (Kahler) differentials      242
Mordell — Weil theorem      92
Multiplicatively closed      209
Multiplicity of a branch      163
Multiplicity of a point on a curve      51
Multiplicity of asymptotes      121
Multiplicity of tangents      53
Nakayama’s Lemma      221 262
Nakayama’s lemma in the graded case      195
Neil’s parabola      5
Newton — Puiseux series      172
Newton’s theorem on diameters      49 125 126
Node      166
Noether different      114 242
Nonsingular curve      53
Normal crossing      166
Numerical semigroup      155
Olympic emblem      173
Order of a divisor at a point      131
Order of a zero (pole)      59
Order with respect to a filtration      199
Ordinary, double point      166
Ordinary, singularity      170
Orthogonal coordinate transformation      119
Pappus’s theorem      46
parabola      4
Parametric representation      77
Parametric representation of affine curves      79
Parametric representation, analytic (of a branch)      167
Parametric representation, polynomial      80
Pascal’s theorem      46 71
Permutability of completion and residue classes      269
Permutability of quotient with residue class rings      213
Permutability of tensor and direct products      240
Permutability of tensor products and localization      239
Point, $\mu$-fold      41
Point, at finite distance      15
Point, at infinity      15 16
Point, multiple (singular)      53
Point, rational      3 14
Point, regular (simple)      53
Point, strange      87
Polar curve      82
Pole      59 82
Pole of a rational function      59
Pole, divisor      31 132
Pole, order      59
Positively graded      194
Principal divisor      31 60
Problem of Frobenius      183
Product formula for conductors      177
Projective, closure      15
Projective, coordinate transformation      15
Projective, curve (line)      14
Projective, equivalence      95
Projective, plane (n-space)      13
Projective, quadric (nonsingular)      21
Proper flex      83
Pythagorean triple      12
Quadratic transformation      74
Ramification index (point)      156
Rational curve      36
Rational function field      143
Rational function on $\mathbb{P}^2$      31
Rational function on a projective curve      33
Rational function on an affine curve      35
Rational map      73
Rational points of affine curves      3
Rational points of projective curves      14
Reciprocity theorem of Brill — Noether      159
Reduced at a point      163
Reduced curve      20
Reduced polynomial      9
Rees algebra      200
Rees algebra, nonextended      202
Regular curve (point)      53
Regularity criterion      56
Relatively prime ideals      217
Residue (symbol)      109
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