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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 surfacesАвтор: 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
Язык: Рубрика: Математика /Алгебра /Алгебраическая геометрия /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 1995Количество страниц: 390Добавлена в каталог: 23.03.2005Операции: Положить на полку |
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157 234 27 5 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 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, 5 Atlas, equivalence of 4 Atlas, line bundle 333 336 339 352-355 Atlas, n-dimensional complex 6 Automorphism 40 168 84 Automorphism of 43 Automorphism of 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, 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 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 8 Chart on 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 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 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 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
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