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Bibliographic Details
Main Authors:	Cox, David A. 1948- (Author), Little, John B. 1956- (Author), Schenck, Hal 1963- (Author)
Format:	Book
Language:	German
Published:	Providence, Rhode Island American Mathematical Society [2011]
Series:	Graduate studies in mathematics Volume 124
Subjects:	Torische Varietät
Links:	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 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
Physical Description:	XXIV, 841 Seiten Diagramme
ISBN:	9780821848197 0821848194

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DE-BY-UBR_location	UB Lesesaal Mathematik UB Handapparat Mathematik Prof. Künnemann UB Handapparat Mathematik Prof. Gubler
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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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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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