Toric varieties:
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | German |
| Published: |
Providence, Rhode Island
American Mathematical Society
[2011]
|
| Series: | Graduate studies in mathematics
Volume 124 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | XXIV, 841 Seiten Diagramme |
| ISBN: | 9780821848197 0821848194 |
Staff View
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| 245 | 1 | 0 | |a Toric varieties |c David A. Cox ; John B. Little ; Henry K. Schenck |
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c [2011] | |
| 300 | |a XXIV, 841 Seiten |b Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
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| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate studies in mathematics |v Volume 124 | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
Preface
јх
Notation xv
Part
I.
Basic
Theory of
Torte
Varieties
1
Chapter
1.
Affine Toric
Varieties
3
§1.0.
Background:
Affine
Varieties
3
§1.1.
Introduction to
Affine
Toric Varieties
10
§ 1.2.
Cones and
Affine
Toric Varieties
23
§1.3.
Properties of
Affine
Toric Varieties
35
Appendix: Tensor Products of Coordinate Rings
48
Chapter
2.
Projective
Toric Varieties
49
§2.0.
Background:
Projective
Varieties
49
§2.1.
Lattice Points and
Projective
Toric Varieties
54
§2.2.
Lattice Points and Poly topes
62
§2.3.
Polytopes and
Projective
Toric Varieties
75
§2.4.
Properties of
Projective
Toric Varieties
86
Chapter
3.
Normal Toric Varieties
93
§3.0.
Background: Abstract Varieties
93
§3.1.
Fans and Normal Toric Varieties
105
§3.2.
The Orbit-Cone Correspondence
П4
§3.3.
Toric Morphisms
125
§3.4.
Complete and Proper I39
vi
Contents
Appendix: Nonnormal Toric
Varieties
150
Chapter
4. Divisors
on
Toric
Varieties
155
§4.0.
Background: Valuations, Divisors and Sheaves
155
§4.1.
Weil Divisors on Toric Varieties
170
§4.2.
Cartier
Divisors on Toric Varieties
176
§4.3.
The Sheaf of a Torus-Invariant Divisor
189
Chapter
5.
Homogeneous Coordinates on Toric Varieties
195
§5.0.
Background: Quotients in Algebraic Geometry
195
§5.1.
Quotient Constructions of Toric Varieties
205
§5.2.
The Total Coordinate Ring
219
§5.3.
Sheaves on Toric Varieties
226
§5.4.
Homogenization and Polytopes
232
Chapter
6.
Line Bundles on Toric Varieties
245
§6.0.
Background: Sheaves and Line Bundles
245
§6.1.
Ample and
Basepoint
Free Divisors on Complete Toric Varieties
262
§6.2.
Polytopes and
Projective
Toric Varieties
277
§6.3.
The
Nef
and Mori Cones
286
§6.4.
The Simplicial Case
298
Appendix: Quasicoherent Sheaves on Toric Varieties
309
Chapter
7.
Projective
Toric Morphisms
313
§7.0.
Background: Quasiprojective Varieties and
Projective
Morphisms
313
§7.1.
Polyhedra and Toric Varieties
318
§7.2.
Projective
Morphisms and Toric Varieties
328
§7.3.
Projective
Bundles and Toric Varieties
335
Appendix: More on
Projective
Morphisms
345
Chapter
8.
The Canonical Divisor of a Toric Variety
347
§8.0.
Background: Reflexive Sheaves and Differential Forms
347
§8.1.
One-Forms on Toric Varieties
358
§8.2.
Differential Forms on Toric Varieties
365
§8.3.
Fano
Toric Varieties
379
Chapter
9.
Sheaf Cohomology of Toric Varieties
387
§9.0.
Background: Sheaf Cohomology
387
§9.1.
Cohomology of Toric Divisors
398
Contents vjj
§9.2.
Vanishing
Theorems
I 409
§9.3.
Vanishing
Theorems
II
420
§9.4. Lattice Polytopes and Differential
Forms
43О
§9.5.
Local Cohomology and the
Total
Coordinate Ring
443
Part
II. Topics in
Toric
Geometry
457
Chapter
10. Toric
Surfaces
459
§10.1.
Singularities of
Toric
Surfaces and Their Resolutions
459
§ 10.2.
Continued Fractions and Toric Surfaces
467
§10.3. Gröbner
Fans and McKay Correspondences
485
§10.4.
Smooth Toric Surfaces
495
§10.5.
Riemann-Roch and Lattice Polygons
502
Chapter
11.
Toric Resolutions and Toric Singularities
513
§11.1.
Resolution of Singularities
5 ] 3
§11.2.
Other Types of Resolutions
525
§11.3.
Rees
Algebras and Multiplier Ideals
534
§11.4.
Toric Singularities
546
Chapter
12.
The Topology of Toric Varieties
561
§12.1.
The Fundamental Group
561
§12.2.
The Moment Map
568
§12.3.
Singular Cohomology of Toric Varieties
577
§12.4.
The Cohomology Ring
592
§ 12.5.
The Chow Ring and Intersection Cohomology
612
Chapter
13.
Toric Hirzebrach-Riemann-Roch
623
§13.1.
Chern Characters, Todd Classes, and
HRR
624
§13.2.
Brion s Equalities
632
§13.3.
Toric Equivariant Riemann-Roch
641
§13.4.
The Volume Polynomial
654
§13.5.
The Khovanskii-Pukhlikov Theorem
663
Appendix: Generalized Gysin Maps
672
Chapter
14.
Toric GIT and the Secondary Fan
677
§14.1.
Introduction to Toric GIT
677
§14.2.
Toric GIT and Polyhedra
685
§14.3.
Toric GIT and Gale Duality
699
viii Contents
§ 14.4.
The Secondary Fan
712
Chapter
15.
Geometry of the Secondary Fan
725
§15.1.
The
Nef
and Moving Cones
725
§15.2.
Gale Duality and
Triangulations
734
§15.3.
Crossing a Wall
747
§15.4.
Extremal Contractions and Flips
762
§15.5.
The Toric Minimal Model Program
772
Appendix A. The History of Toric Varieties
787
§
A.
1.
The First Ten Years
787
§A.2. The Story Since
1980 794
Appendix B. Computational Methods
797
§B.l. The Rational Quartic
798
§B.2. Polyhedral Computations
799
§B.3. Normalization and Normaliz
803
§B.4. Sheaf Cohomology and Resolutions
805
§B
.5.
Sheaf Cohomology on the Hirzebruch Surface
Ж2
806
ŞB.6.
Resolving Singularities
808
§B.7. Intersection Theory and Hirzebruch-Riemann-Roch
809
§B.8. Anticanonical Embedding of
a Fano
Toric Variety
810
Appendix C. Spectral Sequences
811
§C.
1.
Definitions and Basic Properties
811
§C2. Spectral Sequences Appearing in the Text
814
Bibliography
817
Index
831
Toric varieties form a
beautiful and accessible
part of modern algebraic
geometry. This book
covers the standard
topics in toric geom¬
etry; a novel feature is
that each of the first
nine chapters contains
an introductory section on the necessary background material in algebraic geom¬
etry. Other topics covered include quotient constructions, vanishing theorems,
equivariant cohomology, GIT quotients, the secondary fan, and the minimal model
program for toric varieties. The subject lends itself to rich examples reflected in
the 1
34
illustrations included in the text. The book also explores connections
with commutative algebra and polyhedral geometry, treating both polytopes and
their unbounded cousins, polyhedra.There are appendices on the history of toric
varieties and the computational tools available to investigate
nontrivial
examples
in toric geometry.
Readers of this book should be familiar with the material covered in basic grad¬
uate courses in algebra and topology, and to a somewhat lesser degree, complex
analysis. In addition, the authors assume that the reader has had some previous
experience with algebraic geometry at an advanced undergraduate level. The book
will be a useful reference for graduate students and researchers who are interested
in algebraic geometry, polyhedral geometry, and toric varieties.
ISBN
978-0-8218-4819-7
9 780821 848197
GSM/1
24
For additional information
_^_ and updates on this book, visit
www.ams.org/bookpages/gsm-1
24
AXIS on the Web
|
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| indexdate | 2024-07-09T23:22:44Z |
| institution | BVB |
| isbn | 9780821848197 0821848194 |
| language | German |
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| spelling | Cox, David A. 1948- Verfasser (DE-588)137410832 aut Toric varieties David A. Cox ; John B. Little ; Henry K. Schenck Providence, Rhode Island American Mathematical Society [2011] XXIV, 841 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics Volume 124 Torische Varietät (DE-588)4786945-8 gnd rswk-swf Torische Varietät (DE-588)4786945-8 s DE-604 Little, John B. 1956- Verfasser (DE-588)137410859 aut Schenck, Hal 1963- Verfasser (DE-588)14378160X aut Graduate studies in mathematics Volume 124 (DE-604)BV009739289 124 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022519082&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022519082&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Cox, David A. 1948- Little, John B. 1956- Schenck, Hal 1963- Toric varieties Graduate studies in mathematics Torische Varietät (DE-588)4786945-8 gnd |
| subject_GND | (DE-588)4786945-8 |
| title | Toric varieties |
| title_auth | Toric varieties |
| title_exact_search | Toric varieties |
| title_full | Toric varieties David A. Cox ; John B. Little ; Henry K. Schenck |
| title_fullStr | Toric varieties David A. Cox ; John B. Little ; Henry K. Schenck |
| title_full_unstemmed | Toric varieties David A. Cox ; John B. Little ; Henry K. Schenck |
| title_short | Toric varieties |
| title_sort | toric varieties |
| topic | Torische Varietät (DE-588)4786945-8 gnd |
| topic_facet | Torische Varietät |
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