xxiv Notation Brion’s Equalities Z[M] integral semigroup algebra of M (13) Z[[M]] formal semigroup module of M, formal sums ∑ m∈M amχm (13) Z[[M]]Sum summable elements in Z[[M]] (13) S( f ) sum of an element f ∈ Z[[M]]Sum (13) χ(L ) ∑ m∈M ∑n i=0 dimHi(X,L )m χm ∈ Z[[M]] (13) Geometric Invariant Theory G character group of algebraic subgroup G ⊆ (C∗)r (14) Lχ sheaf of sections of rank 1 line bundle on Cr for character χ ∈ G (14) (Cr)ss, χ (Cr)s χ semistable and stable points for χ (14) Rχ graded ring ∞ d=0 Γ(Cr,L χd )G (14) Cr//χ G GIT quotient of Cr by G for χ, equals Proj(Rχ) = (Cr)ss//G χ (14) B(χ) irrelevant ideal of χ (14) Z(χ) exceptional set of χ, equals V(B(χ)) (14) Pχ, Pa polyhedra in Rr and MR for χ = χa (14) Fi,χ, Fi,a ith virtual facet of Pχ, Pa (14) The Secondary Fan β, ν lists of r vectors in GR and NR (14) Cβ, Cν cones generated by β and ν (14) ΓΣ,I ∅ , ΓΣ,I ∅ GKZ cones determined by Σ,I∅ (14) B(Σ,I∅) irrelevant ideal determined by Σ,I∅ (14) ΣGKZ secondary fan of Σ (14) MovGKZ moving cone of the secondary fan (15) PGKZ secondary polytope, normal fan is ΣGKZ (15) Toric Minimal Model Program R extremal ray of the Mori cone (15) D ·R intersection product D ·C for [C] ∈ R\{0} (15) f∗D birational transform of a divisor by a birational map (15) J−, J+ index sets determined by a wall relation (15) Σ−, Σ+ fans determined by a wall relation (15) φ−, φ+ toric morphisms determined by a wall relation (15) Miscellaneous μd multiplicative group of dth roots of unity in C (1) [a1,...,as] ordinary continued fraction of a rational number (10) [[b1,...,br]] Hirzebruch-Jung continued fraction of a rational number (10)

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