Preface The study of toric varieties is a wonderful part of algebraic geometry. There are elegant theorems and deep connections with polytopes, polyhedra, combinatorics, commutative algebra, symplectic geometry, and topology. Toric varieties also have unexpected applications in areas as diverse as physics, coding theory, algebraic statistics, and geometric modeling. Moreover, as noted by Fulton [105], “toric varieties have provided a remarkably fertile testing ground for general theories.” At the same time, the concreteness of toric varieties provides an excellent context for someone encountering the powerful techniques of modern algebraic geometry for the first time. Our book is an introduction to this rich subject that assumes only a 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 provide background material when needed. In addition, we cover a large number of topics previously available only in the research literature. ix
Previous Page Next Page