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

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

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$g^n_d$      157 234
$\mathcal{C}^{\infty}$ function      27
$\mathcal{C}^{\infty}$ real manifold      5
$\mathcal{C}^{\infty}$ structure      5
1-1 sheaf map      282
a-period      257 259 262
Abel — Jacobi map      249 250 255 260 319 356 357
Abel — Jacobi map for a curve of genus one      265
Abel — Jacobi map for a torus      140
Abel — Jacobi map has canonical map as derivative      264
Abel — Jacobi map is 1-1 on X      264 265
Abel — Jacobi map is holomorphic      265 267
Abel — Jacobi map is surjective      263
Abel — Jacobi map on divisors      250
Abel — Jacobi map on divisors of degree d      264
Abel — Jacobi map on points      249
Abel — Jacobi map, independent of the base point      250
Abel’s theorem      247 250 263 266 356
Abel’s Theorem for a torus      140
Abel’s Theorem for curves of genus one      265
Abel’s Theorem for the Riemann Sphere      140
Abel’s Theorem, proof of necessity      255
Abel’s Theorem, proof of sufficiency      257 260
Abel’s Theorem, the high road      319 320
Affine conic      13
Affine plane curve      11 14 15 144
Affine plane curve of degree 2      13
Affine plane curve, 1-forms on      111
Affine plane curve, irreducible      12
Affine plane curve, meromorphic functions on      35 36
Affine plane curve, node of      67
Affine plane curve, projections are holomorphic      22
Affine plane curve, ramification of projection      46
Affine plane curve, smooth part      12
Algebraic curve      169 195 254 309 325
Algebraic curve is projective      196
Algebraic curve of genus 0      197
Algebraic curve of genus 1      197
Algebraic curve of genus 2      198
Algebraic curve of genus 3      206
Algebraic curve of genus 4      207
Algebraic curve of genus 5      209
Algebraic curve, complex torus      170
Algebraic curve, computing the function field      177
Algebraic curve, existence of 1-forms      200 202 203
Algebraic curve, function field has tr. deg. one      174
Algebraic curve, hyperelliptic      170
Algebraic curve, Riemann Sphere      170
Algebraic curve, smooth projective curve      170
Algebraic curve, trigonal      209
Algebraic set      95
Analytic genus      192
Arithmetic genus      192
Atlas      3 341
Atlas on $\mathbb{P}^n$      16
Atlas on a graph      10
Atlas on a smooth affine plane curve      11
Atlas on a torus      9
Atlas on X/G      78
Atlas, $C^{\infty}$      5
Atlas, equivalence of      4
Atlas, line bundle      333 336 339 352-355
Atlas, n-dimensional complex      6
Automorphism      40 168 84
Automorphism of $\mathbb{P}^1$      43
Automorphism of $\mathcal{O}_X$      358
Automorphism of a torus      64 65
Automorphism of a trivial extension      363
Automorphism of the trivial line bundle      358
Automorphism, first-order      366
Automorphism, group is finite      82 243
Automorphism, linear      97
Automorphism, number of fixed points      210
b-period      257 259
b-period, normalized matrix      262
Base point of a linear system      157
Base point of fundamental group      84
Base point, canonical system has none      200
Base point, gap number criterion      234
Base point, removing from a linear system      160
Base-point-free linear system      157
Bezout’s Theorem      143
Biholomorphism      40
Bitangent hyperplane      220
Boundary chain      123 126
Boundary chain, group      126 247
Boundary divisor of a chain      133
Branch divisor      134
Branch point      45
Branch point and monodromy      88 91
Branch point of a quotient map      80
Branch point of hyperelliptic projection      61
Branch point, behaviour of the trace      251
Branched covering      49
Branched covering, number of parameters for      213
Canonical bundle      337
Canonical bundle, sheaf of sections      340
Canonical class      139
Canonical divisor      131
Canonical divisor from a section of the canonical bundle      357
Canonical divisor has degree 2g - 2      133 139 191
Canonical divisor on a torus      137
Canonical divisor, $H^1$ has dimension one      191
Canonical divisor, pullback formula      135
Canonical map      203
Canonical map for a curve of genus 3      206
Canonical map for a curve of genus 4      207
Canonical map for a curve of genus 5      209
Canonical map for a hyperelliptic curve      204
Canonical map for a nonhyperelliptic curve      203
Canonical map is the derivative of Abel — Jacobi      264
Castelnuovo curves      230
Castelnuovo’s bound      229
Cech coboundary      292
Cech cochain      291
Cech cocycle      292
Cech cohomology      304 313
Cech cohomology for $C^{\infty}$ sheaves      302
Cech cohomology for constant sheaves      304
Cech cohomology for skyscraper sheaves      303
Cech cohomology with respect to a cover      292
Cech cohomology, long exact sequence      298 299
Cech cohomology, Zariski topology      313
Center of projection      98
Chain      120
Chart on $\mathbb{P}^1$      8
Chart on $\mathbb{P}^n$      16
Chart on a covering      89
Chart on a quotient      78
Chart on a smooth affine plane curve      11
Chart on a torus      9
Chart on the Riemann Sphere      3
Chart, centered at a point      1
Chart, complex      1
Chart, complex n-dimensional      6
Chart, defining a Riemann surface      8
Chart, domain      1
Chart, hole      66
Chart, line bundle      331
Chart, real      5
Closed 1-form      114
Closed chain      126
Closed path      118
Coboundary operator      291
Cocycle conditions      335
Collinear      96
Collinearity point      266
Compact Riemann surface      6
Compatibility of complex charts      2 6
Compatibility of line bundle charts      332
Compatibility of real charts      5
Complete linear system      147
Complete linear system as a fiber of Abel — Jacobi      264
Complete linear system is a projective space      147
Complex manifold      6
Complex plane      4
Complex structure      4
Complex structure, n-dimensional      6
Complex torus      9
Complex torus has divisible group law      12
Complex torus is an algebraic curve      170
Complex torus, Abel’s Theorem      140
Complex torus, automorphisms      64
Complex torus, canonical divisor is principal      137
Complex torus, elliptic normal curve      165
Complex torus, every curve of genus one is      198 265
Complex torus, function field      177
Complex torus, g-dimensional      263
Complex torus, isomorphic to $\mathbb{C}/(\mathbb{Z} + \mathbb{Z}\tau)      44
Complex torus, meromorphic functions on      25 35 42 50
Concatenation of paths      119
conic      57 58
Conic is isomorphic to $\mathbb{P}^1$      58
Conic is planar      216
Conic, affine      13
Connecting homomorphism      297
Constant presheaf      271
Constant sheaf      273
Continuous action      75
Covering map      48
Covering space      84
Covering space, cyclic unbranched      360
Cubic      57
Cubic, group law      267
Cusp      72
Cyclic covering      73
Cyclic covering, unbranched      360
De Rham cohomology      305 320
Deformation      364
Deformation, first-order      367
Deformation, locally trivial      365
Deformation, trivial      365
Degree of a canonical divisor is 2g - 2      133 139
Degree of a covering map      86
Degree of a divisor      129
Degree of a holomorphic map      48
Degree of a homogeneous polynomial      14
Degree of a line bundle      345
Degree of a principal divisor is zero      130
Degree of a projective curve      142
Degree of a smooth projective plane curve      16
Degree of the image of a map to $\mathbb{P}^n$      164
Degree, invariant under linear equivalence      138
Degree, minimal      216
Differential form      see “Form”
Differential of a function      113
Differential, n-fold      236
Dimension of a linear system      147
Dimension of a subspace      96
Dimension Theorem      211
Direct limit      295
Direct system      295
Divisor      129
Divisor of a meromorphic 1-form      131
Divisor of a meromorphic function      130
Divisor of a meromorphic n-fold differential      238
Divisor of a rational section of a line bundle      343
Divisor of Laurent tails      179
Divisor of poles      131 132
Divisor of zeroes      131 132
Divisor, boundary divisor of a chain      133
Divisor, branch      134
Divisor, canonical      131
Divisor, complete linear system of      147
Divisor, degree      129
Divisor, fixed      161
Divisor, general      210
Divisor, hyperplane      136 142 159
Divisor, intersection      136
Divisor, inverse image      133
Divisor, invertible sheaf of      271 325
Divisor, linear equivalence      138
Divisor, map      279 319 346
Divisor, partial ordering      136
Divisor, principal      130
Divisor, pullback      134
Divisor, ramification      134
Divisor, skyscraper sheaf      274
Divisor, space of forms with poles bounded by      148
Divisor, space of functions with poles bounded by      146
Divisor, span      208
Divisor, special      198
Divisor, very ample      163 195
Dolbeault cohomology      306
Dolbeault’s Lemma      117 284
Effective action      75
Elliptic function      25 54
Elliptic normal curve      165
Elliptic normal curve, achieves the Castelnuovo bound      229
Embedding      163
Endpoint      118
Equivalence of atlases      5 6
Equivalence of line bundle atlases      333
Essential singularity      23
Euler number      50 70
Euler’s formula      14
Exact 1-form      113
Exact sequence      181
Exact sequence of sheaves      286
Exponential map      281
Extensions of invertible sheaves      362
Fermat curve      54 69 93
Fiber coordinate      332
First homology group      125 126 247
Fixed part of a linear system      161
Flex point      219 233 241 266
Flex point, nine on a cubic      266
Flexed tangent line      266
FORM      see “One-form two-form”
Free linear system      157
Function field      171 177 309
Function field has tr. deg. one      174
Function field, separating points and tangents      169
Fundamental group      84 125
Fundamental group of $\mathbb{P}^1$ minus n points      91
Fundamental group of a curve of genus g      94
Fundamental group of a punctured disc      86
Fundamental group of a torus      86
Fundamental group, abelianization is homology      125
GAGA theorem      316
Gap number      233
General divisor      210
General Position Lemma      225
Genus of a plane curve      70
Genus, analytic      192
Genus, arithmetic      192
Genus, topological      6 192
Global sections      269
Glueing, line bundles      335
Glueing, locally trivial objects      360
Glueing, Riemann surfaces      59
Graph of a function      10
Grassmann variety      212
Group action      75
Group law on a cubic      267
Harmonic function      27 114
Hausdorff      4
hessian      232
Higher-order cusp      72
Higher-order tacnode      72
1 2 3
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