§1.2. Cones and Affine Toric Varieties 23 1.1.15. Suppose that φ : M M is a group isomorphism. Fix a finite set A M and let B = φ(A ). Prove that the toric varieties YA and YB are equivariantly isomorphic (meaning that the isomorphism respects the torus action). §1.2. Cones and Affine Toric Varieties We begin with a brief discussion of rational polyhedral cones and then explain how they relate to affine toric varieties. Convex Polyhedral Cones. To start, we fix a pair of dual vector spaces MR and NR. The geometry of convex polyhedral cones in MR and NR is a lovely topic that plays an important role in toric geometry. For space reasons, most of the results we need will be stated without proof. We refer the reader to [105] for more details and [219, App. A.1] for careful statements. See also [51, 128, 242]. Definition 1.2.1. A convex polyhedral cone in NR is a set of the form σ = Cone(S) = u∈S λuu | λu 0 NR, where S NR is finite. We say that σ is generated by S. Also set Cone(∅) = {0}. A convex polyhedral cone σ is in fact convex, meaning that x,y σ implies λx + (1 λ)y σ for all 0 λ 1, and is a cone, meaning that x σ implies λx σ for all λ 0. Since we will only consider convex cones, the cones satisfying Definition 1.2.1 will be called simply “polyhedral cones”. Examples of polyhedral cones include the first quadrant in R2 or first octant in R3. For another example, the cone Cone(e1,e2,e1 + e3,e2 + e3) R3 is pictured in Figure 2. It is also possible to have cones that contain entire lines. For example, z y x Figure 2. Cone in R3 generated by e1,e2,e1 + e3,e2 + e3
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