Moduli of curves:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
1998
|
| Series: | Graduate texts in mathematics
187 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 366 S. graph. Darst. |
| ISBN: | 0387984380 0387984291 |
Staff View
MARC
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Record in the Search Index
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| adam_text | Contents
Preface vii
1 Parameter spaces: constructions and examples 1
A Parameters and moduli 1
B Construction of the Hilbert scheme 5
C Tangent space to the Hilbert scheme 12
D Extrinsic pathologies 18
Mumford s example 19
Other examples 24
E Dimension of the Hilbert scheme 26
F Severi varieties 29
G Hurwitz schemes 32
2 Basic facts about moduli spaces of curves 35
A Why do fine moduli spaces of curves not exist? 35
B Moduli spaces we ll be concerned with 40
C Constructions of Mg 43
The Teichimiller approach 43
The Hodge theory approach 44
The geometric invariant theory (G.I.T.) approach ... 46
D Geometric and topological properties 52
Basic properties 52
Local properties 52
Complete subvarieties of Mg 55
Cohomology of Mg: Harer s theorems 58
Cohomology of the universal curve 62
Cohomology of Hilbert schemes 63
Structure of the tautological ring 67
Witten s conjectures and Kontsevich s theorem .... 71
E Moduli spaces of stable maps 76
3 Techniques 81
A Basic facts about nodal and stable curves 81
Dualizing sheaves 82
xii Contents
Automorphisms 85
B Deformation theory 86
Overview 86
Deformations of smooth curves 89
Variations on the basic deformation theory plan ... 92
Universal deformations of stable curves 102
Deformations of maps 105
C Stable reduction 117
Results 117
Examples 120
D Interlude: calculations on the moduli stack 139
Divisor classes on the moduli stack 140
Existence of tautological families 148
E Grothendieck Riemann Roch and Porteous 150
Grothendieck Riemann Roch 150
Chern classes of the Hodge bundle 154
Chern class of the tangent bundle 159
Porteous formula 161
The hyperelliptic locus in M3 162
Relations amongst standard cohomology classes ... 165
Divisor classes on Hilbert schemes 166
F Test curves: the hyperelliptic locus in M$ begun 168
G Admissible covers 175
H The hyperelliptic locus in ZM3 completed 186
4 Construction of Mg 191
A Background on geometric invariant theory 192
The G.I.T. strategy 192
Finite generation of and separation by invariants ... 194
The numerical criterion 199
Stability of plane curves 202
B Stability of Hilbert points of smooth curves 206
The numerical criterion for Hilbert points 206
Gieseker s criterion 211
Stability of smooth curves 216
C Construction of Mg via the Potential Stability Theorem . 220
The plan of the construction and a few corollaries . . 220
The Potential Stability Theorem 224
5 limit Linear Series and Brill Noether theory 240
A Introductory remarks on degenerations 240
B Limits of line bundles 247
C Limits of linear series: motivation and examples 253
D Limit linear series: definitions and applications 263
Limit linear series 263
Contents xiii
Smoothing limit linear series 266
Limits of canonical series and Weierstrass points . . . 269
E Limit linear series on flag curves 274
Inequalities on vanishing sequences 274
The case p = 0 276
Proof of the Gieseker Petri theorem 280
6 Geometry of moduli spaces: selected results 286
A Irreducibility of the moduli space of curves 286
B Diaz theorem 288
The idea: stratifying the moduli space 288
The proof 292
C Moduli of hyperelliptic curves 293
Fiddling around 293
The calculation for an (almost) arbitrary family .... 295
The Picard group_of the hyperelliptic locus 301
D Ample divisors on Mg 303
An inequality for generically Hilbert stable families . 304
Proof of the theorem 305
An inequality for families of pointed curves 308
Ample divisors on Mg 310
E Irreducibility of the Severi varieties 313
Initial reductions 314
Analyzing a degeneration 320
An example 324
Completing the argument 326
F Kodaira dimension of Mg 328
Writing down general curves 328
Basic ideas 330
Pulling back the divisors Drs ^ 335
Divisors on ~Mg that miss j(Mi, W) 336
Divisors on Mg that miss i(Mo,g) 340
Further divisor class calculations 342
Curves defined over Q 342
Bibliography 345
Index 355
|
| any_adam_object | 1 |
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| dewey-full | 516.35 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.35 |
| dewey-search | 516.35 |
| dewey-sort | 3516.35 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T18:22:17Z |
| institution | BVB |
| isbn | 0387984380 0387984291 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008217726 |
| oclc_num | 246207638 |
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| physical | XIII, 366 S. graph. Darst. |
| publishDate | 1998 |
| publishDateSearch | 1998 |
| publishDateSort | 1998 |
| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Harris, Joe 1951- Verfasser (DE-588)112574718 aut Moduli of curves Joe Harris ; Ian Morrison New York [u.a.] Springer 1998 XIII, 366 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 187 Algebraische Kurve - Modulitheorie Curves, Algebraic Moduli theory Algebraische Kurve (DE-588)4001165-3 gnd rswk-swf Modulitheorie (DE-588)4203418-8 gnd rswk-swf Algebraische Kurve (DE-588)4001165-3 s Modulitheorie (DE-588)4203418-8 s DE-604 Morrison, Ian Verfasser aut Graduate texts in mathematics 187 (DE-604)BV000000067 187 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008217726&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Harris, Joe 1951- Morrison, Ian Moduli of curves Graduate texts in mathematics Algebraische Kurve - Modulitheorie Curves, Algebraic Moduli theory Algebraische Kurve (DE-588)4001165-3 gnd Modulitheorie (DE-588)4203418-8 gnd |
| subject_GND | (DE-588)4001165-3 (DE-588)4203418-8 |
| title | Moduli of curves |
| title_auth | Moduli of curves |
| title_exact_search | Moduli of curves |
| title_full | Moduli of curves Joe Harris ; Ian Morrison |
| title_fullStr | Moduli of curves Joe Harris ; Ian Morrison |
| title_full_unstemmed | Moduli of curves Joe Harris ; Ian Morrison |
| title_short | Moduli of curves |
| title_sort | moduli of curves |
| topic | Algebraische Kurve - Modulitheorie Curves, Algebraic Moduli theory Algebraische Kurve (DE-588)4001165-3 gnd Modulitheorie (DE-588)4203418-8 gnd |
| topic_facet | Algebraische Kurve - Modulitheorie Curves, Algebraic Moduli theory Algebraische Kurve Modulitheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008217726&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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