Notation xix x ˆ monomial ρ/σ(1) xρ for σ ∈ Σ (5) B(Σ) irrelevant ideal of S, generated by the x ˆ (5) x m Laurent monomial ρ xρm,uρ , m ∈ M (5) x m,D homogenization of χm, m ∈ PD ∩ M (5) xF facet variable of a facet F ≺ P (5) x m,P P-monomial associated to m ∈ P ∩ M (5) x v,P vertex monomial associated to vertex v ∈ P ∩ M (5) M graded S-module (5) M(α) shift of M by α ∈ Cl(XΣ) (5) Quotient Construction X/G good geometric quotient (5) X//G good categorical quotient (5) Z(Σ) exceptional set in quotient construction, equals V(B(Σ)) (5) G group in quotient construction, equals HomZ(Cl(XΣ),C∗) (5) Divisors OX,D local ring of a variety at a prime divisor (4) νD discrete valuation of a prime divisor D (4) div( f ) principal divisor of a rational function (4) D ∼ E linear equivalence of divisors (4) D ≥ 0 effective divisor (4) Div0(X) group of principal divisors on X (4) Div(X) group of Weil divisors on X (4) CDiv(X) group of Cartier divisors on X (4) Cl(X) divisor class group of a normal variety X (4) Pic(X) Picard group of a normal variety X (4) Pic(X)R Pic(X) ⊗Z R (6) Supp(D) support of a divisor (4) D| U restriction of a divisor to an open set (4) {(Ui, fi)} local data of a Cartier divisor on X (4) |D| complete linear system of D (6) D , D “round down” and “round up” of a Q-divisor (9) Torus-Invariant Divisors Dρ = O(ρ) torus-invariant prime divisor on XΣ of ray ρ ∈ Σ(1) (4) DF torus-invariant prime divisor on XP of facet F ≺ P (4) DivT N (XΣ) group of torus-invariant Weil divisors on XΣ (4) CDivT N (XΣ) group of torus-invariant Cartier divisors on XΣ (4) {mσ}σ∈Σ Cartier data of a torus-invariant Cartier divisor on XΣ (4)
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