Preface xi are Cohen-Macaulay to the Dehn-Sommerville equations for a simple polytope and counting lattice points in multiples of a polytope via the Ehrhart polynomial. Chapter 10 studies toric surfaces, where we add a few twists to this classical subject. After using Hirzebruch-Jung continued fractions to compute the minimal resolution of a toric surface singularity, we discuss the toric meaning of ordinary continued fractions. We then describe unexpected connections with Grobner ¨ fans and the McKay correspondence. Finally, we use the Riemann-Roch theorem on a smooth complete toric surface to explain the mysterious appearance of the number 12 in Chapter 8 when counting lattice points in reflexive polygons. Chapter 11 proves the existence of toric resolutions of singularities for toric varieties of all dimensions. This is more complicated than for surfaces because of the existence of toric flips and flops. We consider simple normal crossing, crepant, log, and embedded resolutions and study how Rees algebras and multiplier ideals can be applied in the resolution problem. We also discuss toric singularities and show that a fan Σ is simplicial if and only if XΣ has at worst finite quotient singular- ities and hence is rationally smooth. We also explain what canonical and terminal singularities mean in the toric context. Chapters 12 and 13 describe the singular and equivariant cohomology of a complete simplicial toric variety XΣ and prove the Hirzebruch-Riemann-Roch and equivariant Riemann-Roch theorems when XΣ is smooth. We compute the funda- mental group of XΣ and study the moment map, with a brief mention of topological models of toric varieties and connections with symplectic geometry. We describe the Chow ring and intersection cohomology of a complete simplicial toric variety. After proving Riemann-Roch, we give applications to the volume polynomial and lattice point enumeration in polytopes. Chapters 14 and 15 explore the rich connections that link geometric invariant theory, the secondary fan, the nef and moving cones, Gale duality, triangulations, wall crossings, flips, extremal contractions, and the toric minimal model program. Appendices. The book ends with three appendices: • Appendix A: The History of Toric Varieties. • Appendix B: Computational Methods. • Appendix C: Spectral Sequences. Appendix A surveys the history of toric geometry since its origins in the early 1970s. It is fun to see how the concepts and terminology evolved. Appendix B discusses some of the software packages for toric geometry and gives examples to illustrate what they can do. Appendix C gives a brief introduction to spectral sequences and describes the spectral sequences used in Chapters 9 and 12. Prerequisites. We assume that the reader is familiar with the material covered in basic graduate courses in algebra and topology, and to a somewhat lesser degree,

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