By Garrett P.
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Note also that the dual of a smooth (resp. simplicial) cone is again smooth (resp. simplicial). Later in the section we will give examples of simplicial cones that are not smooth. Semigroup Algebras and Affine Toric Varieties. Given a rational polyhedral cone ✻❏✦ ❄ ❅ , the lattice points ✷✜✸ ❘ ✼ ✻ ✾✲❀ ✦✑❀ form a semigroup. A key fact is that this semigroup is finitely generated. 17 (Gordan’s Lemma). hence is an affine semigroup. ✷ ✸ ❘ ✻ ✼ ✾❏❀ is finitely generated and ❂ ■❃✙ ✢ ✹ ✠ ✽ ✠ Proof. Since ✻ ✼ is rational polyhedral, ✻ ✼ ❘ for a finite set ✰ ✦ ❀ .
1) ✠✠ ✠ ✱ ✭✍✁✔✓✔✓✔✓✍✁ ✱ ✧ generate ✻◆✼ , then it is ❯ ✾ ❖ ☛✓ 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. 5. A face of a cone of the polyhedral cone ✻ is ❍ ❘ some ✱ ✳ ✻ ✼ . Using ✱ ❘ ☛ shows that ✻ is a face of itself. Faces called proper faces. for are The faces of a polyhedral cone have the following obvious properties. 6. Let ✻ ❘ ❂ ■❃✙ ✢ ✹❁▲❍✽ be a polyhedral cone. Then: (a) Every face of ✻ is a polyhedral cone.
The dots are the lattice ❄ ❘ the white ones are ✆✔✾ ❄ . ←ρ ↑ ray generator Figure 6. 15. A strongly convex rational polyhedral cone is generated by the ray generators of its edges. ☛ It is customary to call the ray generators of the edges the minimal generators of a strongly convex rational polyhedral cone. Figures 1 and 2 show -dimensional strongly convex rational polyhedral cones and their ray generators. Chapter 1. 12. Here are some especially important strongly convex cones. 16. Let ✻❏✦ (a) (b) ✻ ❄ ❅ be a strongly convex rational polyhedral cone.
A rationality principle by Garrett P.