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Kunz E. — Introduction to Plane Algebraic Curves
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Название: Introduction to Plane Algebraic CurvesАвтор: 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.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 2007Количество страниц: 293Добавлена в каталог: 10.05.2008Операции: Положить на полку |
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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 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 11 Divisor (class) group on 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 33 Field of rational functions on 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 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, 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 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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