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Book DetailsGraduate Studies in MathematicsVolume: 124; 2011; 841 ppMSC: Primary 14;
Toric varieties form a beautiful and accessible part of modern algebraic geometry. This book covers the standard topics in toric geometry; a novel feature is that each of the first nine chapters contains an introductory section on the necessary background material in algebraic geometry. 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 134 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 graduate 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.ReadershipGraduate students and research mathematicians interested in algebraic geometry, polyhedral geometry, and toric varieties.

Table of Contents

Part I. Basic theory of toric varieties

Chapter 1. Affine toric varieties

Chapter 2. Projective toric varieties

Chapter 3. Normal toric varieties

Chapter 4. Divisors on toric varieties

Chapter 5. Homogeneous coordinates on toric varieties

Chapter 6. Line bundles on toric varieties

Chapter 7. Projective toric morphisms

Chapter 8. The canonical divisor of a toric variety

Chapter 9. Sheaf cohomology of toric varieties

Topics in toric geometry

Chapter 10. Toric surfaces

Chapter 11. Toric resolutions and toric singularities

Chapter 12. The topology of toric varieties

Chapter 13. Toric HirzebruchRiemannRoch

Chapter 14. Toric GIT and the secondary fan

Chapter 15. Geometry of the secondary fan

Appendix A. The history of toric varieties

Appendix B. Computational methods

Appendix C. Spectral sequences


Additional Material

Reviews

The book under review is an excellent modern introduction to the subject. It covers both classical results and a large number of topics previously available only in the research literature. The presentation is very explicit, and the material is illustrated by many examples, figures, and exercises. ... The book combines many advantages of an introductory course, a textbook, a monograph, and an encyclopaedia. It is strongly recommended to a wide range of readers from beginners in algebraic geometry to experts in the area.
Ivan V. Arzhantsev, Mathematical Reviews 
This masterfully written book will become a standard text on toric varieties, serving both students and researchers. The book's leisurely pace and wealth of background material makes it perfect for graduate courses on toric varieties or for selfstudy. Researchers will discover gems throughout the book and will find it to be a valuable resource.
Sheldon Katz


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 Request Exam/Desk Copy
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Toric varieties form a beautiful and accessible part of modern algebraic geometry. This book covers the standard topics in toric geometry; a novel feature is that each of the first nine chapters contains an introductory section on the necessary background material in algebraic geometry. 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 134 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 graduate 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.
Graduate students and research mathematicians interested in algebraic geometry, polyhedral geometry, and toric varieties.

Part I. Basic theory of toric varieties

Chapter 1. Affine toric varieties

Chapter 2. Projective toric varieties

Chapter 3. Normal toric varieties

Chapter 4. Divisors on toric varieties

Chapter 5. Homogeneous coordinates on toric varieties

Chapter 6. Line bundles on toric varieties

Chapter 7. Projective toric morphisms

Chapter 8. The canonical divisor of a toric variety

Chapter 9. Sheaf cohomology of toric varieties

Topics in toric geometry

Chapter 10. Toric surfaces

Chapter 11. Toric resolutions and toric singularities

Chapter 12. The topology of toric varieties

Chapter 13. Toric HirzebruchRiemannRoch

Chapter 14. Toric GIT and the secondary fan

Chapter 15. Geometry of the secondary fan

Appendix A. The history of toric varieties

Appendix B. Computational methods

Appendix C. Spectral sequences

The book under review is an excellent modern introduction to the subject. It covers both classical results and a large number of topics previously available only in the research literature. The presentation is very explicit, and the material is illustrated by many examples, figures, and exercises. ... The book combines many advantages of an introductory course, a textbook, a monograph, and an encyclopaedia. It is strongly recommended to a wide range of readers from beginners in algebraic geometry to experts in the area.
Ivan V. Arzhantsev, Mathematical Reviews 
This masterfully written book will become a standard text on toric varieties, serving both students and researchers. The book's leisurely pace and wealth of background material makes it perfect for graduate courses on toric varieties or for selfstudy. Researchers will discover gems throughout the book and will find it to be a valuable resource.
Sheldon Katz