About this Title
David A. Cox, Amherst College, MA, John B. Little, College of the Holy Cross, Worcester, MA and Henry K. Schenck, University of Illinois at Urbana-Champaign, Urbana, IL
Publication: Graduate Studies in Mathematics
Publication Year 2011: Volume 124
ISBNs: 978-0-8218-4819-7 (print); 978-1-4704-1185-5 (online)
MathSciNet review: MR2810322
MSC: Primary 14M25; Secondary 05A15, 05E45, 52B12
This volume is not part of this online collection, but can be purchased through our online bookstore.
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.
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 Hirzebruch-Riemann-Roch
- 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