vi Contents Appendix: Nonnormal Toric Varieties 150 Chapter 4. Divisors on Toric Varieties 155 §4.0. Background: Valuations, Divisors and Sheaves 155 §4.1. Weil Divisors on Toric Varieties 170 §4.2. Cartier Divisors on Toric Varieties 176 §4.3. The Sheaf of a Torus-Invariant Divisor 189 Chapter 5. Homogeneous Coordinates on Toric Varieties 195 §5.0. Background: Quotients in Algebraic Geometry 195 §5.1. Quotient Constructions of Toric Varieties 205 §5.2. The Total Coordinate Ring 219 §5.3. Sheaves on Toric Varieties 226 §5.4. Homogenization and Polytopes 232 Chapter 6. Line Bundles on Toric Varieties 245 §6.0. Background: Sheaves and Line Bundles 245 §6.1. Ample and Basepoint Free Divisors on Complete Toric Varieties 262 §6.2. Polytopes and Projective Toric Varieties 277 §6.3. The Nef and Mori Cones 286 §6.4. The Simplicial Case 298 Appendix: Quasicoherent Sheaves on Toric Varieties 309 Chapter 7. Projective Toric Morphisms 313 §7.0. Background: Quasiprojective Varieties and Projective Morphisms 313 §7.1. Polyhedra and Toric Varieties 318 §7.2. Projective Morphisms and Toric Varieties 328 §7.3. Projective Bundles and Toric Varieties 335 Appendix: More on Projective Morphisms 345 Chapter 8. The Canonical Divisor of a Toric Variety 347 §8.0. Background: Reflexive Sheaves and Differential Forms 347 §8.1. One-Forms on Toric Varieties 358 §8.2. Differential Forms on Toric Varieties 365 §8.3. Fano Toric Varieties 379 Chapter 9. Sheaf Cohomology of Toric Varieties 387 §9.0. Background: Sheaf Cohomology 387 §9.1. Cohomology of Toric Divisors 398
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