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Notation xxi Intersection Theory deg(D) degree of a divisor on a smooth complete curve (6) D ·C intersection product of Cartier divisor and complete curve (6) D ≡ D , C ≡ C numerically equivalent Cartier divisors and complete curves (6) N1(X), N1(X) (CDiv(X)/≡) ⊗Z R and ({proper 1-cycles on X}/≡) ⊗Z R (6) Nef(X) cone in N1(X) generated by nef divisors (6) Mov(X) moving cone of a variety X in N1(X) (15) Eff(X) pseudoeffective cone of a variety X in N1(X) (15) NE(X) cone in N1(X) generated by complete curves (6) NE(X) Mori cone, equals the closure of NE(X) (6) Differential Forms and Sheaves ΩR/C module of Kahler ¨ differentials of a C-algebra R (8) ΩX, 1 TX cotangent and tangent sheaves of a variety X (8) IY /IY2, NY/X conormal and normal sheaves of Y ⊆ X (8) ΩX, p ΩX p sheaves of p-forms and Zariski p-forms on X (8) KX, ωX canonical divisor and canonical sheaf ΩX, n n = dim X (8) ΩX(logD) 1 sheaf of 1-forms with logarithmic poles on D (8) Sheaf Cohomology H 0 (X,F ) global sections Γ(X,F ) of a sheaf F on X (9) H p (X,F ) pth sheaf cohomology group of a sheaf F on X (9) Rp f∗F higher direct image sheaf (9) Extp OX (G ,F ) Ext groups of sheaves of OX-modules G , F (9) ˇ•(U ,F ) ˇ Cech complex for sheaf cohomology (9) χ(F ) Euler characteristic of F , equals ∑ p (−1)pdim H p (X,F ) (9) Sheaf Cohomology of a Toric Variety H p (XΣ,L )m graded piece of sheaf cohomology of L = OX Σ (D) for m ∈ M (9) VD,m, V supp D,m subsets of |Σ| used to compute H p (XΣ,L )m (9) Local Cohomology H p I (M) pth local cohomology of an R-module M for the ideal I ⊆ R (9) ˇ•(f,M) ˇ Cech complex for local cohomology when I = f (9) Extp(N,M) R Ext groups of R-modules N, M (9) Resolution of Singularities Xsing singular locus of a variety (11) Exc(φ) exceptional locus of a resolution of singularities (11)