Toric Varieties (Graduate Studies in Mathematics) (Graduate Studies in Mathematics, 124)

The study of toric varieties is a wonderful part of algebraic geometry. There areelegant theorems and deep connections with polytopes, polyhedra, combinatorics,commutative algebra, symplectic geometry, and topology. Toric varieties also haveunexpected applications in areas as diverse as physics, coding theory, algebraicstatistics, and geometric modeling. Moreover, as noted by Fulton [105], “toricvarieties have provided a remarkably fertile testing ground for general theories.”At the same time, the concreteness of toric varieties provides an excellent contextfor someone encountering the powerful techniques of modern algebraic geometryfor the first time. Our book is an introduction to this rich subject that assumes onlya modest background yet leads to the frontier of this active area of research.Brief Summary. The text covers standard material on toric varieties, including:(a) Convex polyhedral cones, polytopes, and fans.(b) Affine, projective, and abstract toric varieties.(c) Complete toric varieties and proper toric morphisms.(d) Weil and Cartier divisors on toric varieties.(e) Cohomology of sheaves on toric varieties.(f) The classical theory of toric surfaces.(g) The topology of toric varieties.(h) Intersection theory on toric varieties.These topics are discussed in earlier texts on the subject, such as [93], [105] and[219]. One difference is that we provide more details, with numerous examples,figures, and exercises to illustrate the concepts being discussed. We also providebackground material when needed. In addition, we cover a large number of topicspreviously available only in the research literature. 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. Dedications Contents Preface Notation Part I. Basic Theory of Toric Varieties 1. Affine Toric Varieties 2. Projective Toric Varieties 3. Normal Toric Varieties 4. Divisors on Toric Varieties 5. Homogeneous Coordinates on Toric Varieties 6. Line Bundles on Toric Varieties 7. Projective Toric Morphisms 8. The Canonical Divisor of a Toric Variety 9. Sheaf Cohomology of Toric Varieties Part II. Topics in Toric Geometry 10. Toric Surfaces 11. Toric Resolutions and Toric Singularities 12. The Topology of Toric Varieties 13. Toric Hirzebruch-Riemann-Roch 14. Toric GIT and the Secondary Fan 15. Geometry of the Secondary Fan Appendix A. The History of Toric Varieties Appendix B. Computational Methods Appendix C. Spectral Sequences Bibliography Index This title 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