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§1.2. Cones and Affine Toric Varieties 25 Given m ∈ MR, we define Hm = {u ∈ NR | m,u = 0} ⊆ NR H+ m = {u ∈ NR | m,u≥ 0} ⊆ NR. If m = 0, then Hm is a hyperplane and H+ m is a closed half-space. We define Hm to be a supporting hyperplane of a polyhedral cone σ ⊆ NR if σ ⊆ H+, m and when this happens, we call H+ m a supporting half-space. (This is a mild abuse of terminology since m = 0 is allowed.) Note that Hm is a supporting hyperplane of σ if and only if m ∈ σ∨. Furthermore, if m1,...,ms generate σ∨, then it is straightforward to check that (1.2.1) σ = Hm + 1 ∩···∩ H+ ms . Thus every polyhedral cone is an intersection of finitely many closed half-spaces. We can use supporting hyperplanes and half-spaces to define faces of a cone. Definition 1.2.5. A face of a cone of the polyhedral cone σ is τ = Hm ∩σ for some m ∈ σ∨, written τ σ. Using m = 0 shows that σ is a face of itself, i.e., σ σ. Faces τ = σ are called proper faces, written τ ≺ σ. One can show that the faces of a polyhedral cone have the following properties. Lemma 1.2.6. Let σ = Cone(S) be a polyhedral cone. Then: (a) Every face of σ is a polyhedral cone. (b) An intersection of two faces of σ is again a face of σ. (c) A face of a face of σ is again a face of σ. You will prove the following useful property of faces in Exercise 1.2.1. Lemma 1.2.7. Let τ be a face of a polyhedral cone σ. If v,w ∈ σ and v + w ∈ τ, then v,w ∈ τ. A facet of σ is a face τ of codimension 1, i.e., dim τ = dim σ −1. An edge of σ is a face of dimension 1. In Figure 4 on the next page we illustrate a 3-dimensional cone with shaded facets and a supporting hyperplane (a plane in this case) that cuts out the vertical edge of the cone. Here are some properties of facets. Proposition 1.2.8. Let σ ⊆ NR Rn be a polyhedral cone. Then: (a) If σ = H+ m1 ∩···∩ H+ ms for mi ∈ σ∨, 1 ≤ i ≤ s, then σ∨ = Cone(m1,...,ms). (b) If dim σ = n, then in (a) we can assume that the facets of σ are τi = Hm i ∩ σ. (c) Every proper face τ ≺ σ is the intersection of the facets of σ containing τ. Note how part (b) of the proposition refines (1.2.1) when dim σ = dim NR.