24 Chapter 1. Affine Toric Varieties Cone(e1,−e1) ⊆ R2 is the x-axis, while Cone(e1,−e1,e2) is the closed upper half- plane {(x,y) ∈ R2 | y ≥ 0}. As we will see below, these last two examples are not strongly convex. We can also create cones using polytopes, which are defined as follows. Definition 1.2.2. A polytope in NR is a set of the form P = Conv(S) = u∈S λuu | λu ≥ 0, u∈S λu = 1 ⊆ NR, where S ⊆ NR is finite. We say that P is the convex hull of S. Polytopes include all polygons in R2 and bounded polyhedra in R3. As we will see in later chapters, polytopes play a prominent role in the theory of toric varieties. Here, however, we simply observe that a polytope P ⊆ NR gives a polyhedral cone C(P) ⊆ NR ×R, called the cone of P and defined by C(P) = {λ· (u,1) ∈ NR ×R | u ∈ P, λ ≥ 0}. If P = Conv(S), then we can also describe this as C(P) = Cone(S ×{1}). Figure 3 shows what this looks when P is a pentagon in the plane. P Figure 3. The cone C(P) of a pentagon P ⊆ R2 The dimension dim σ of a polyhedral cone σ is the dimension of the smallest subspace W = Span(σ) of NR containing σ. We call Span(σ) the span of σ. Dual Cones and Faces. As usual, the pairing between MR and NR is denoted , . Definition 1.2.3. Given a polyhedral cone σ ⊆ NR, its dual cone is defined by σ∨ = {m ∈ MR | m,u≥ 0 for all u ∈ σ}. Duality has the following important properties. Proposition 1.2.4. Let σ ⊆ NR be a polyhedral cone. Then σ∨ is a polyhedral cone in MR and (σ∨)∨ = σ.
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