An introduction to invariants and moduli:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English Japanese |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge Univ. Press
2003
|
| Ausgabe: | 1. publ. |
| Schriftenreihe: | Cambridge studies in advanced mathematics
81 |
| Schlagworte: | |
| Online-Zugang: | Publisher description Table of contents Inhaltsverzeichnis |
| Beschreibung: | Aus dem Japan. übers. |
| Beschreibung: | XX, 503 S. graph. Darst. |
| ISBN: | 0521809061 9780521809061 |
Internformat
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| 100 | 1 | |a Mukai, Shigeru |e Verfasser |4 aut | |
| 240 | 1 | 0 | |a Mojurai riron |
| 245 | 1 | 0 | |a An introduction to invariants and moduli |c Shigeru Mukai |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2003 | |
| 300 | |a XX, 503 S. |b graph. Darst. | ||
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| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Cambridge studies in advanced mathematics |v 81 | |
| 500 | |a Aus dem Japan. übers. | ||
| 650 | 4 | |a Invariants | |
| 650 | 4 | |a Modules, Théorie des | |
| 650 | 4 | |a Invariants | |
| 650 | 4 | |a Moduli theory | |
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Datensatz im Suchindex
| _version_ | 1804129288617721856 |
|---|---|
| adam_text | Contents
Preface page xi
Acknowledgements xiii
Introduction xv
(a) What is a moduli space? xv
(b) Algebraic varieties and quotients of algebraic
varieties xvi
(c) Moduli of bundles on a curve xix
1 Invariants and moduli 1
1.1 A parameter space for plane conies 1
1.2 Invariants of groups 9
(a) Hilbert series 9
(b) Molien s formula 13
(c) Polyhedral groups 15
1.3 Classical binary invariants 19
(a) Resultants and discriminants 19
(b) Binary quartics 26
1.4 Plane curves 32
(a) Affine plane curves 32
(b) Projective plane curves 35
1.5 Period parallelograms and cubic curves 41
(a) Invariants of a lattice 41
(b) The Weierstrass p function 44
(c) The p function and cubic curves 47
2 Rings and polynomials 51
2.1 Hilbert s Basis Theorem 51
2.2 Unique factorisation rings 55
2.3 Finitely generated rings 58
v
vi Contents
2.4 Valuation rings 61
(a) Power series rings 61
(b) Valuation rings 63
2.5 A diversion: rings of invariants which are not finitely
generated 68
(a) Graded rings 69
(b) Nagata s trick 70
(c) An application of Liouville s Theorem 73
3 Algebraic varieties 77
3.1 Affine varieties 78
(a) Affine space 78
(b) The spectrum 81
(c) Some important notions 86
Morphims 86
Products 87
General spectra and nilpotents 88
Dominant morphisms 89
Open immersions 90
Local properties 91
3.2 Algebraic varieties 91
(a) Gluing affine varieties 91
(b) Projective varieties 95
3.3 Functors and algebraic groups 98
(a) A variety as a functor from algebras to sets 98
(b) Algebraic groups 100
3.4 Completeness and toric varieties 103
(a) Complete varieties 103
(b) Toric varieties 107
(c) Approximation of valuations 111
4 Algebraic groups and rings of invariants 116
4.1 Representations of algebraic groups 117
4.2 Algebraic groups and their Lie spaces 122
(a) Local distributions 122
(b) The distribution algebra 124
(c) The Casimir operator 128
4.3 Hilbert s Theorem 130
(a) Linear reductivity 130
(b) Finite generation 135
4.4 The Cayley Sylvester Counting Theorem 137
(a) SL(2) 137
(b) The dimension formula for 5L(2) 140
Contents vii
(c) A digression: Weyl measure 142
(d) The Cay ley Sylvester Formula 143
(e) Some computational examples 148
4.5 Geometric reductivity of SL(2) 152
5 The construction of quotient varieties 158
5.1 Affine quotients 159
(a) Separation of orbits 159
(b) Surjectivity of the affine quotient map 163
(c) Stability 165
5.2 Classical invariants and the moduli of smooth
hypersurfaces in P 167
(a) Classical invariants and discriminants 167
(b) Stability of smooth hypersurfaces 171
(c) A moduli space for hypersurfaces in P 174
(d) Nullforms and the projective quotient map 175
6 The projective quotient 181
6.1 Extending the idea of a quotient: from values
to ratios 182
(a) The projective spectrum 186
(b) The Proj quotient 189
(c) The Proj quotient by a GL(n) action of ray type 195
6.2 Linearisation and Proj quotients 197
6.3 Moving quotients 201
(a) Flops 201
(b) Toric varieties as quotient varieties 205
(c) Moment maps 208
7 The numerical criterion and some applications 211
7.1 The numerical criterion 212
(a) 1 parameter subgroups 212
(b) The proof 213
7.2 Examples and applications 219
(a) Stability of projective hypersurfaces 219
(b) Cubic surfaces 224
(c) Finite point sets in projective space 230
8 Grassmannians and vector bundles 234
8.1 Grassmannians as quotient varieties 235
(a) Hilbert series 237
(b) Standard monomials and the ring of invariants 239
(c) Young tableaux and the Pliicker relations 241
(d) Grassmannians as projective varieties 245
(e) A digression: the degree of the Grassmannian 247
viii Contents
8.2 Modules over a ring 251
(a) Localisation 251
(b) Local versus global 254
(c) Free modules 257
(d) Tensor products and flat modules 259
8.3 Locally free modules and flatness 262
(a) Locally free modules 262
(b) Exact sequences and flatness 264
8.4 The Picard group 268
(a) Algebraic number fields 268
(b) Two quadratic examples 271
8.5 Vector bundles 276
(a) Elementary sheaves of modules 277
(b) Line bundles and vector bundles 279
(c) The Grassmann functor 282
(d) The tangent space of the functor 284
9 Curves and their Jacobians 287
9.1 Riemann s inequality for an algebraic curve 288
(a) Prologue: gap values and the genus 290
(b) Divisors and the genus 292
(c) Divisor classes and vanishing index of
speciality 294
9.2 Cohomology spaces and the genus 297
(a) Cousin s problem 297
(b) Finiteness of the genus 301
(c) Line bundles and their cohomology 304
(d) Generation by global sections 307
9.3 Nonsingularity of quotient spaces 309
(a) Differentials and differential modules 309
(b) Nonsingularity 311
(c) Free closed orbits 313
9.4 An algebraic variety with the Picard group as its set
of points 316
(a) Some preliminaries 316
(b) The construction 319
(c) Tangent spaces and smoothness 323
9.5 Duality 327
(a) Dualising line bundles 327
(b) The canonical line bundle 330
(c) De Rham cohomology 332
Contents ix
9.6 The Jacobian as a complex manifold 334
(a) Compact Riemann surfaces 335
(b) The comparison theorem and the Jacobian 337
(c) Abel s Theorem 342
10 Stable vector bundles on curves 348
10.1 Some general theory 349
(a) Subbundles and quotient bundles 350
(b) The Riemann Roch formula 352
(c) Indecomposable bundles and stable bundles 355
(d) Grothendieck s Theorem 359
(e) Extensions of vector bundles 361
10.2 Rank 2 vector bundles 365
(a) Maximal line subbundles 365
(b) Nonstable vector bundles 366
(c) Vector bundles on an elliptic curve 369
10.3 Stable bundles and Pfaffian semiinvariants 371
(a) Skew symmetric matrices and Pfaffians 371
Even skew symmetric matrices 372
Odd skew symmetric matrices 374
Skew symmetric matrices of rank 2 375
(b) Gieseker points 376
(c) Semistability of Gieseker points 379
10.4 An algebraic variety with SUC(2, L) as its set
of points 386
(a) Tangent vectors and smoothness 387
(b) Proof of Theorem 10.1 391
(c) Remarks on higher rank vector bundles 394
11 Moduli functors 398
11.1 The Picard functor 400
(a) Fine moduli and coarse moduli 400
(b) Cohomology modules and direct images 403
(c) Families of line bundles and the Picard functor 407
(d) Poincare line bundles 410
11.2 The moduli functor for vector bundles 413
(a) Rank 2 vector bundles of odd degree 416
(b) Irreducibility and rationality 418
(c) Rank 2 vector bundles of even degree 419
11.3 Examples 422
(a) The Jacobian of a plane quartic 422
(b) The affine Jacobian of a spectral curve 424
x Contents
(c) The Jacobian of a curve of genus 1 425
(d) Vector bundles on a spectral curve 431
(e) Vector bundles on a curve of genus 2 433
12 Intersection numbers and the Verlinde formula 437
12.1 Sums of inverse powers of trigonometric functions 439
(a) Sine sums 439
(b) Variations 441
(c) Tangent numbers and secant numbers 444
12.2 Riemann Roch theorems 447
(a) Some preliminaries 448
(b) Hirzebruch Riemann Roch 450
(c) Grothendieck Riemann Roch for curves 453
(d) Riemann Roch with involution 455
12.3 The standard line bundle and the Mumford relations 457
(a) The standard line bundle 457
(b) The Newstead classes 461
(c) The Mumford relations 463
12.4 From the Mumford relations to the Verlinde formula 465
(a) Warming up: secant rings 466
(b) The proof of formulae (12.2) and (12.4) 470
12.5 An excursion: the Verlinde formula for
quasiparabolic bundles 476
(a) Quasiparabolic vector bundles 476
(b) A proof of (12.6) using Riemann Roch and the
Mumford relations 480
(c) Birational geometry 483
Bibliography 487
Index 495
|
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| id | DE-604.BV014438776 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:02:46Z |
| institution | BVB |
| isbn | 0521809061 9780521809061 |
| language | English Japanese |
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| spelling | Mukai, Shigeru Verfasser aut Mojurai riron An introduction to invariants and moduli Shigeru Mukai 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2003 XX, 503 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Cambridge studies in advanced mathematics 81 Aus dem Japan. übers. Invariants Modules, Théorie des Moduli theory Invariante (DE-588)4128781-2 gnd rswk-swf Algebraische Mannigfaltigkeit (DE-588)4128509-8 gnd rswk-swf Modultheorie (DE-588)4170336-4 gnd rswk-swf Modultheorie (DE-588)4170336-4 s Invariante (DE-588)4128781-2 s Algebraische Mannigfaltigkeit (DE-588)4128509-8 s DE-604 Cambridge studies in advanced mathematics 81 (DE-604)BV000003678 81 http://www.loc.gov/catdir/description/cam022/2002023422.html Publisher description http://www.loc.gov/catdir/toc/cam024/2002023422.html Table of contents HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009869322&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mukai, Shigeru An introduction to invariants and moduli Cambridge studies in advanced mathematics Invariants Modules, Théorie des Moduli theory Invariante (DE-588)4128781-2 gnd Algebraische Mannigfaltigkeit (DE-588)4128509-8 gnd Modultheorie (DE-588)4170336-4 gnd |
| subject_GND | (DE-588)4128781-2 (DE-588)4128509-8 (DE-588)4170336-4 |
| title | An introduction to invariants and moduli |
| title_alt | Mojurai riron |
| title_auth | An introduction to invariants and moduli |
| title_exact_search | An introduction to invariants and moduli |
| title_full | An introduction to invariants and moduli Shigeru Mukai |
| title_fullStr | An introduction to invariants and moduli Shigeru Mukai |
| title_full_unstemmed | An introduction to invariants and moduli Shigeru Mukai |
| title_short | An introduction to invariants and moduli |
| title_sort | an introduction to invariants and moduli |
| topic | Invariants Modules, Théorie des Moduli theory Invariante (DE-588)4128781-2 gnd Algebraische Mannigfaltigkeit (DE-588)4128509-8 gnd Modultheorie (DE-588)4170336-4 gnd |
| topic_facet | Invariants Modules, Théorie des Moduli theory Invariante Algebraische Mannigfaltigkeit Modultheorie |
| url | http://www.loc.gov/catdir/description/cam022/2002023422.html http://www.loc.gov/catdir/toc/cam024/2002023422.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009869322&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000003678 |
| work_keys_str_mv | AT mukaishigeru mojurairiron AT mukaishigeru anintroductiontoinvariantsandmoduli |