x Preface The Fifteen Chapters. To give you a better idea of what is in the book, we now highlight a few topics from each chapter. Chapters 1, 2 and 3 cover the basic material mentioned in items (a)–(c) above. The toric varieties encountered include: • The affine toric variety YA of a finite set A ⊆ M Zn (Chapter 1). • The affine toric variety Uσ of a polyhedral cone σ ⊆ NR Rn (Chapter 1). • The projective toric variety XA of a finite set A ⊆ M Zn (Chapter 2). • The projective toric variety XP of a lattice polytope P ⊆ MR Rn (Chapter 2). • The abstract toric variety XΣ of a fan Σ in NR Rn (Chapter 3). The general definition of toric variety given in Chapter 3 does not assume that the variety is normal. This differs from other standard texts, which deal exclusively with normal toric varieties. Chapters 1 and 2 include numerous examples of non- normal toric varieties. Nevertheless, the vast majority of toric varieties in the book are normal, and whenever we say “the toric variety XΣ”, we are in the normal case since the toric variety of a fan is normal. When nonnormal toric varieties arise in later chapters, we always warn the reader that normality may fail. Chapter 4 introduces Weil and Cartier divisors on toric varieties. We compute the class group and Picard group of a toric variety and define the sheaf OX Σ (D) associated to a Weil divisor D on a toric variety XΣ. Chapter 5 shows that the classical construction Pn = ( Cn+1 \{0} ) /C∗ can be generalized to any toric variety XΣ. The homogeneous coordinate ring C[x0,...,xn] of Pn also has a toric generalization, called the total coordinate ring of XΣ. Chapter 6 relates Cartier divisors to invertible sheaves on XΣ. We introduce ample, basepoint free, and nef divisors and discuss their relation to convexity. The structure of the nef cone and its dual, the Mori cone, are described in detail, as is the intersection pairing between divisors and curves. Chapter 7 extends the relation between polytopes and projective toric varieties to a relation between polyhedra and projective toric morphisms φ : XΣ → Uσ. We also discuss projective bundles over a toric variety and use these to classify smooth projective toric varieties of Picard number 2. Chapter 8 relates Weil divisors to reflexive sheaves of rank one and defines Zariski p-forms. For p = dim X, this gives the canonical sheaf ωX and canonical divisor KX. In the toric case we describe these explicitly and study the relation be- tween reflexive polytopes and Gorenstein Fano toric varieties, meaning that −KX Σ is ample. We find the 16 reflexive polygons in R2 (up to equivalence) and note the relation |∂P∩ M| + |∂P◦ ∩ N| = 12 for a reflexive polygon P and its dual P◦. Chapter 9 is about sheaf cohomology. We give two methods for computing sheaf cohomology on a toric variety and prove a dizzying array of cohomology vanishing theorems. Applications range from showing that normal toric varieties

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