xviii Notation Varieties V(I) affine or projective variety of an ideal (1,2) Vf subset of an affine variety V where f = 0 (1) S Zariski closure of S in a variety (1,3) Tp(X) Zariski tangent space of a variety at a point (1,3) dim X, dimp X dimension of a variety and dimension at a point (1,3) Spec(R) affine variety of coordinate ring R (1) Proj(S) projective variety of graded ring S (7) X ×Y product of varieties (1,3) X ×S Y fiber product of varieties (3) X affine cone of a projective variety X (2) Toric Varieties YA , XA affine and projective toric variety of A ⊆ M (1,2) Uσ = Uσ,N affine toric variety of a cone σ ⊆ NR (1) XΣ = XΣ,N toric variety of a fan Σ in NR (3) XP projective toric variety of a lattice polytope or polyhedron (2.7) φ lattice homomorphism of a toric morphism φ : XΣ 1 → XΣ 2 (1,3) φR real extension of φ (1) γσ distinguished point of Uσ (3) O(σ) torus orbit corresponding to σ ∈ Σ (3) V(σ) = O(σ) closure of orbit of σ ∈ Σ, toric variety of Star(σ) (3) UP affine toric variety of recession cone of a polyhedron (7) UΣ affine toric variety of a fan with convex support (7) Specific Varieties Cn, Pn affine and projective n-dimensional space (1,2) P(q0,...,qn) weighted projective space (2) Cd, Cd rational normal cone and curve (1,2) Bl0(Cn) blowup of Cn at the origin (3) BlV(σ)(XΣ) blowup of XΣ along V(σ), toric variety of Σ∗(σ) (3) Hr Hirzebruch surface (3) Sa,b rational normal scroll (3) Total Coordinate Ring S total coordinate ring of XΣ (5) xρ variable in S corresponding to ρ ∈ Σ(1) (5) Sβ graded piece of S in degree β ∈ Cl(XΣ) (5) deg(xα) degree in Cl(XΣ) of a monomial in S (5)

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