Cox rings:
Gespeichert in:
| Hauptverfasser: | , , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge
Cambridge University Press
2015
|
| Schriftenreihe: | Cambridge studies in advanced mathematics
144 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | viii, 530 Seiten Illustrationen |
| ISBN: | 9781107024625 |
Internformat
MARC
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| 300 | |a viii, 530 Seiten |b Illustrationen | ||
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Datensatz im Suchindex
| _version_ | 1804152560371630080 |
|---|---|
| adam_text | Titel: Cox rings
Autor: Arzhantsev, Ivan
Jahr: 2015
e 1 6 7 7 10 13 13 16 20 24 24 27 30 32 35 35 39 41 44 47 47 50 52 56 56 59 63 65 69 Contents Introduction 1 Basic concepts 1.1 Graded algebras 1.1.1 Monoid graded algebras 1.1.2 Veronese subalgebras 1.2 Gradings and quasitorus actions 1.2.1 Quasitori 1.2.2 Affine quasitorus actions 1.2.3 Good quotients 1.3 Divisorial algebras 1.3.1 Sheaves of divisorial algebras 1.3.2 The relative spectrum 1.3.3 Unique factorization in the global ring 1.3.4 Geometry of the relative spectrum 1.4 Cox sheaves and Cox rings 1.4.1 Free divisor class group 1.4.2 Torsion in the divisor class group 1.4.3 Well-definedness 1.4.4 Examples 1.5 Algebraic properties of the Cox ring 1.5.1 Integrity and normality 1.5.2 Localization and units 1.5.3 Divisibility properties 1.6 Geometric realization of the Cox sheaf 1.6.1 Characteristic spaces 1.6.2 Divisor classes and isotropy groups 1.6.3 Total coordinate space and irrelevant ideal 1.6.4 Characteristic spaces via GIT Exercises to Chapter 1 v
vi Contents 2 Tone varieties and Gale duality 75 2.1 Tone varieties 75 2.1.1 Toric varieties and fans 75 2.1.2 Some toric geometry 78 2.1.3 The Cox ring of a toric variety 82 2.1.4 Geometry of Cox’s construction 86 2.2 Linear Gale duality 89 2.2.1 Fans and bunches of cones _ 89 2.2.2 The GKZ decomposition 93 2.2.3 Proof of Theorem 2.2.1.14 96 2.2.4 Proof of Theorems 2.2.2.2, 2.2.2.3, and 2.2.2.6 100 2.3 Good toric quotients 103 2.3.1 Characterization of good toric quotients 103 2.3.2 Combinatorics of good toric quotients 107 2.4 Toric varieties and bunches of cones 110 2.4.1 Toric varieties and lattice bunches 110 2.4.2 Toric geometry via bunches 113 Exercises to Chapter 2 117 3 Cox rings and combinatorics 123 3.1 GIT for affine quasitorus actions 124 3.1.1 Orbit cones 124 3.1.2 Semistable quotients 128 3.1.3 A 2 -quotients 133 3.1.4 Quotients of H -factorial affine varieties 137 3.2 Bunched rings 142 3.2.1 Bunched rings and their varieties 142 3.2.2 Proofs to Section 3.2.1 147 3.2.3 Example: Flag varieties 151 3.2.4 Example: Quotients of quadrics 155 3.2.5 The canonical toric embedding 161 3.3 Geometry via defining data 167 3.3.1 Stratification and local properties 167 3.3.2 Base loci and cones of divisors 172 3.3.3 Complete intersections 178 3.3.4 Mori dream spaces 181 3.4 Varieties with a torus action of complexity 1 186 3.4.1 Detecting factorial gradings 186 3.4.2 Factorially graded rings of complexity 1 191 3.4.3 T -varieties of complexity 1 via bunched rings 197 3.4.4 Geometry of T -varieties of complexity 1 202 Exercises to Chapter 3 207
Contents vii 4 Selected topics 215 4.1 Tone ambient modifications 216 4.1.1 The Cox ring of an embedded variety 216 4.1.2 Algebraic modification 220 4.1.3 Toric ambient modifications 226 4.1.4 Computing examples 232 4.2 Lifting automorphisms 236 4.2.1 Quotient presentations 236 4.2.2 Linearization of line bundles 241 4.2.3 Lifting group actions 246 4.2.4 Automorphisms of Mori dream spaces 251 4.3 Finite generation 257 4.3.1 General criteria 257 4.3.2 Finite generation via multiplication map 262 4.3.3 Finite generation after Hu and Keel 268 4.3.4 Cox-Nagata rings 274 4.4 Varieties with torus action 279 4.4.1 The Cox ring of a variety with torus action 279 4.4.2 //-factorial quasiaffine varieties 284 4.4.3 Proof of Theorems 4.4.1.2, 4.4.1.3, and 4.4.1.6 291 4.5 Almost homogeneous varieties 297 4.5.1 Homogeneous spaces 297 4.5.2 Small embeddings 303 4.5.3 Examples of small embeddings 308 4.5.4 Spherical varieties 314 4.5.5 Wonderful varieties and algebraic monoids 319 Exercises to Chapter 4 326 5 Surfaces 334 5.1 Mori dream surfaces 334 5.1.1 Basic surface geometry 334 5.1.2 Nef and semiample cones 339 5.1.3 Rational surfaces 345 5.1.4 Extremal rational elliptic surfaces 350 5.1.5 K3 surfaces 356 5.1.6 Enriques surfaces 361 5.2 Smooth del Pezzo surfaces 366 5.2.1 Preliminaries 366 5.2.2 Generators of the Cox ring 369 5.2.3 The ideal of relations 374 5.2.4 Del Pezzo surfaces and flag varieties 381
Contents viii 5.3 K3 surfaces 385 5.3.1 Abelian coverings 385 5.3.2 Picard numbers 1 and 2 390 5.3.3 Nonsymplectic involutions 397 5.3.4 Cox rings of K3 surfaces 400 5.4 Rational K*-surfaces 405 5.4.1 Defining data and their surfaces 405 5.4.2 Intersection numbers — 411 5.4.3 Resolution of singularities 417 5.4.4 Gorenstein log del Pezzo K*-surfaces 422 Exercises to Chapter 5 431 6 Arithmetic applications 437 6.1 Universal torsors and Cox rings 437 6.1.1 Quasitori and principal homogeneous spaces 437 6.1.2 Universal torsors 442 6.1.3 Cox rings and characteristic spaces 449 6.2 Existence of rational points 454 6.2.1 The Hasse principle and weak approximation 454 6.2.2 Brauer-Manin obstructions 457 6.2.3 Descent and universal torsors 459 6.2.4 Results 461 6.3 Distribution of rational points 464 6.3.1 Heights and Manin’s conjecture 464 6.3.2 Parameterization by universal torsors and Cox rings 467 6.4 Toward Manin’s conjecture for del Pezzo surfaces 469 6.4.1 Classification and results 469 6.4.2 Strategy 474 6.4.3 Parameterization via Cox rings 475 6.4.4 Counting integral points on universal torsors 478 6.4.5 Interpretation of the integral 485 6.5 Manin’s conjecture for a singular cubic surface 489 6.5.1 Statement of the result 489 6.5.2 Geometry and Cox ring 490 6.5.3 Parameterization via Cox rings 491 6.5.4 Counting integral points on universal torsors 493 6.5.5 Interpretation of the integral 497 Exercises to Chapter 6 498 Bibliography 501 Index 517
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| id | DE-604.BV042098219 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T01:12:39Z |
| institution | BVB |
| isbn | 9781107024625 |
| language | English |
| lccn | 014005540 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027538939 |
| oclc_num | 894763268 |
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| owner | DE-188 DE-20 DE-11 DE-83 |
| owner_facet | DE-188 DE-20 DE-11 DE-83 |
| physical | viii, 530 Seiten Illustrationen |
| publishDate | 2015 |
| publishDateSearch | 2015 |
| publishDateSort | 2015 |
| publisher | Cambridge University Press |
| record_format | marc |
| series | Cambridge studies in advanced mathematics |
| series2 | Cambridge studies in advanced mathematics |
| spelling | Arzhantsev, Ivan aut Kolʹt͡sa Koksa <English> Cox rings Ivan Arzhantsev ; Ulrich Derenthal ; Jürgen Hausen ; Antonio Laface Cambridge Cambridge University Press 2015 viii, 530 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Cambridge studies in advanced mathematics 144 Includes bibliographical references and index Ring Mathematik (DE-588)4128084-2 gnd rswk-swf Ring Mathematik (DE-588)4128084-2 s DE-604 Derenthal, Ulrich 1978- (DE-588)132749351 aut Hausen, Jürgen aut LAFACE, ANTONIO aut Cambridge studies in advanced mathematics 144 (DE-604)BV000003678 144 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027538939&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Arzhantsev, Ivan Derenthal, Ulrich 1978- Hausen, Jürgen LAFACE, ANTONIO Cox rings Cambridge studies in advanced mathematics Ring Mathematik (DE-588)4128084-2 gnd |
| subject_GND | (DE-588)4128084-2 |
| title | Cox rings |
| title_alt | Kolʹt͡sa Koksa <English> |
| title_auth | Cox rings |
| title_exact_search | Cox rings |
| title_full | Cox rings Ivan Arzhantsev ; Ulrich Derenthal ; Jürgen Hausen ; Antonio Laface |
| title_fullStr | Cox rings Ivan Arzhantsev ; Ulrich Derenthal ; Jürgen Hausen ; Antonio Laface |
| title_full_unstemmed | Cox rings Ivan Arzhantsev ; Ulrich Derenthal ; Jürgen Hausen ; Antonio Laface |
| title_short | Cox rings |
| title_sort | cox rings |
| topic | Ring Mathematik (DE-588)4128084-2 gnd |
| topic_facet | Ring Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027538939&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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