Miranda R. — Graduate studies in mathematics (vol.5). Algebraic curves and Riemann surfaces :: Электронная библиотека попечительского совета мехмата МГУ

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Miranda R. — Graduate studies in mathematics (vol.5). Algebraic curves and Riemann surfaces

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Название: Graduate studies in mathematics (vol.5). Algebraic curves and Riemann surfaces

Автор: Miranda R.

Аннотация:

This text is an introduction to the theory of algebraic curves defined over the complex numbers. It begins with the definitions and first properties of Riemann surfaces, with special
attention paid to the Riemann sphere, complex tori, hyperelliptic curves, smooth plane curves, and projective curves. The heart of the book is the treatment of divisors and rational functions, culminating in the theorems of Riemann-Roch and Abel and the analysis of the canonical map.
Sheaves, cohomology, the Zariski topology, line bundles, and the Picard group are developed after these main theorems are proved and applied, as a bridge from the classical material to the modern language of algebraic geometry.

Язык:

Рубрика: Математика/Алгебра/Алгебраическая геометрия/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1995

Количество страниц: 390

Добавлена в каталог: 23.03.2005

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Предметный указатель

Pole of a meromorphic function is isolated      29
Presheaf      269
Presheaf, constant      271
Principal divisor      130
Principal part      181
Projection      98
Projective algebraic set      95
Projective curve      36
Projective line      8
Projective line has no deformations      367
Projective line, automorphism      43
Projective line, holomorphic map to, defined by monodromy      91
Projective line, isomorphic to the Riemann Sphere      40
Projective line, meromorphic function      24 32
Projective linePole of a meromorphic n-fold differential      238
Projective n-space      16 94
Projective n-space, dual      166
Projective n-space, holomorphic embedding in      36
Projective n-space, holomorphic map to      153
Projective plane      13
Projective plane curve      14
Projectively normal      231
Projectivization of a vector space      94 147 166
Properly discontinuous action      83
Pullback of a 1-form      115
Pullback of a divisor      134
Pullback of a path      254
Ramification divisor      134
Ramification point      45\
Rational function field      see “Function field”
Rational normal curve      165 216
Rational normal curve has no inflection points      245
Rational normal curve is the canonical image of a hyperelliptic curve      204
Rational normal curve, achieves the Castelnuovo bound      229
Rational section of a line bundle      342
Rational section of a line bundle, divisor of      343
Rational section of a line bundle, order of      343
Refinement of a triangulation      51
Refinement of a triangulation, common refinements exist      51
Refinement of a triangulation, elementary      51
Refinement of an open covering      293
Refining map      293
Regular function      309
Regular, one-form      310
Regular, section of a line bundle      337
Regular, tangent vector field      341 356
Removable singularity      23
Reparametrization of a path      118
Residue      121
Residue map      188
Residue of a trace      253
Residue theorem      123 186 200
Residue Theorem, algebraic proof      253
Resolving a node      69
Restriction maps      116 269
Reversal of a path      118
Riemann bilinear relations      262
Riemann sphere      4
Riemann Sphere as a rational normal curve      165
Riemann Sphere has trivial Jacobian      248
Riemann Sphere is an algebraic curve      170
Riemann Sphere is the only curve of genus zero      197
Riemann Sphere, computation of L(D)      149
Riemann Sphere, finite groups acting on      80
Riemann Sphere, function field      177
Riemann Sphere, holomorphic maps to      41
Riemann Sphere, isomorphic to $\mathbb{P}^1$       40
Riemann Sphere, linear equivalence on      140
Riemann Sphere, meromorphic function on      24 30
Riemann Sphere, meromorphic one-form      111
Riemann surface      4
Riemann — Roch problem      186
Riemann — Roch problem, motivating Cech cohomology      290
Riemann — Roch theorem      186 192
Riemann — Roch Theorem, Geometric Form      208
Riemann — Roch Theorem, implies algebraicity      195
SCROLL      209
Secant line      99
Second countable      4
Section of a line bundle      337
Section of a presheaf      269
Section of the canonical bundle      339
Section of the tangent bundle      341
Section, global      269
Section, rational, of a line bundle      342
Separates points      169
Separates tangents      169
Serre duality      188
Sheaf      272
Sheaf of $\mathcal{C}^{\infty}$ functions      270
Sheaf of $\mathcal{O}$ -modules      323
Sheaf of forms      271
Sheaf of harmonic functions      271
Sheaf of holomorphic functions      270
Sheaf of meromorphic functions      271
Sheaf of meromorphic functions with poles bounded by D      271
Sheaf of nonzero holomorphic functions      271

Sheaf of not identically zero meromorphic functions      271
Sheaf of rational 1-forms      310
Sheaf of rational functions      310
Sheaf of regular functions      309
Sheaf of regular one-forms      310
Sheaf, algebraic      309
Sheaf, axiom      272
Sheaf, constant      273
Sheaf, direct sum      277
Sheaf, homomorphism      278
Sheaf, invertible      324
Sheaf, isomorphism      287
Sheaf, locally constant      273
Sheaf, map      278
Sheaf, map, 1-1      282
Sheaf, map, induces a cochain map      291
Sheaf, map, induces a cohomology map      293 296
Sheaf, map, kernel      282
Sheaf, map, onto      282
Sheaf, map, short exact sequence      284
Sheaf, restriction to an open subset      276
Sheaf, skyscraper      273
Sheaf, stalk      277
Sheaf, totally discontinuous      273
Short exact sequence      181
Short exact sequence of sheaves      284
Simple plane curve singularities      72
Simple tangency      219
Simply connected      84
Singularity      23
Skyscraper sheaf      273
Small loop around a point      88
Small path enclosing a point      118
Smooth complete intersection curve      17
Smooth projective curve      36
Smooth projective plane curve      16
Span of a divisor      208
Special divisor      198
Stabilizer      75
Stalk      277
Standard identified polygon      247
Stoke’s Theorem      123
Subchart      1
Support of a divisor      129
Support of a function      302
Support of a line bundle chart      331
Tacnode      72
Tangent bundle      337 341 367
Tangent hyperplane      217
Tangent line      100 241 266
Tangent sheaf      329
Tangent vector field      341
Tautological line bundle      334
Tensor product of invertible sheaves      327
Theta-function      34 50
Totally discontinous sheaf      273
Trace of a 1-form      253
Trace of a function      251
Transform of a $\mathcal{C}^{\infty}$ 1-form      109
Transform of a 2-form      111
Transform of a holomorphic 1-form      105
Transform of a meromorphic 1-form      106
Transform of a meromorphic n-fold differential      236
Transition function between charts      2
Transition function for line bundle charts      332
Transition function for line bundle charts for the canonical bundle      337
Transition function for line bundle charts for the tangent bundle      337
Transition function for line bundle charts, cocycle conditions      335
Transition function for line bundle charts, defining a line bundle      336
Transverse hyperplane      218 219
Transverse line      266
Triangulation      50
Trigonal algebraic curve      209
Triple point      72
Trivialization of an invertible sheaf      324
Twisted cubic curve      17 100 102 145 165 216
Two-form on a Riemann surface      111
Two-form, $\mathcal{C}^{\infty}$       110
Two-form, defined with an atlas      111
Two-form, integral      122
Two-form, pullback      115
Two-form, sheaf      271
Two-form, transform of      111
Universal cover      85
Universal cover of a compact Riemann surface with $g \geq 2$       83
Universal cover of a torus      62
Veronese map      165 204
Very ample divisor      163 195
Web      147
Wedge product      113
Weierstrass points      242
Zariski topology      95 311
Zero Mean Theorem      318
Zero of a meromorphic 1-form      107
Zero of a meromorphic function      26
Zero of a meromorphic function is isolated      29
Zero of a meromorphic n-fold differential      238

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