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Full Description
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.
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
