xii Preface complex analysis. In addition, we assume that the reader has had some previous experience with algebraic geometry, at the level of any of the following texts: Ideals, Varieties and Algorithms by Cox, Little and O’Shea [69]. Introduction to Algebraic Geometry by Hassett [133]. Elementary Algebraic Geometry by Hulek [151]. Undergraduate Algebraic Geometry by Reid [239]. Computational Algebraic Geometry by Schenck [247]. An Invitation to Algebraic Geometry by Smith, Kahanp¨¨ Kek¨ ¨ ainen and Traves [254]. Chapters 9 and 12 assume knowledge of some basic algebraic topology. The books by Hatcher [135] and Munkres [211] are useful references here. Readers who have studied more sophisticated algebraic geometry texts such as Harris [130], Hartshorne [131], or Shafarevich [246] certainly have the background needed to read our book. For readers with a more modest background, an important prerequisite is a willingness to absorb a lot of algebraic geometry. Background Sections. Since we do not assume a complete knowledge of algebraic geometry, Chapters 1–9 each begin with a background section that introduces the definitions and theorems from algebraic geometry that are needed to understand the chapter. References where proofs can be found are provided. The remaining chapters do not have background sections. For some of those chapters, no further background is necessary, while for others, the material is more sophisticated and the requisite background is given by careful references to the literature. What Is Omitted. We work exclusively with varieties defined over the complex numbers C. The toric variety of a fan can be defined as a scheme over Spec(Z), and many properties of toric varieties hold in this generality (see [82]). Most results in the book are valid over an algebraically closed field (see, for example, [172]). When the field is not algebraically closed, a more sophisticated situation can occur where the torus is not split and the “toric variety” contains not the torus but rather a principal homogeneous space for the torus (see [92] for a treatment of this topic). None of this is in the book because of our focus on C. We also do not consider toric stacks (see [39] for an introduction). Moreover, our viewpoint is primarily algebro-geometric. Thus, while we hint at some of the connections with symplectic geometry and topology in Chapter 12, we do not do justice to this side of the story. Even within the algebraic geometry of toric varieties, there are many topics we have had to omit, though we provide some references that should help readers who want to explore these areas. We have also omitted any discussion of how toric varieties are used in physics and applied mathematics. Some pointers to the literature are given in our discussion of the recent history of toric varieties in §A.2 of Appendix A.
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