By Cox D., Little J., Schenck H.

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C) Consider lex monomial order with ❞ ✔✗✖✯❞ ✁ ✖✙✘✚✘✚✘✛✖✯❞✗✖ . Let ✱✪✝✜✕✑✤ ❣ ✖ ✬ be homogeneous of degree ✟ and let ✢ be the remainder of ✱ on division by the generators of ✏ . Prove that ✢ can be written ✞✢ ✭✖✕ ✔ ✤ ❞ ✔ ✷ ❞ ✁ ✬ ✖✳✕ ✁ ✤ ❞ ✁ ✷ ❞ ✹ ✬ ✖ ✘✚✘✚✘ ✖✳✕ ✖ ✍ ✁ ✤ ❞ ✖ ✍ ✁ ✷ ❞ ✖ ✬ (a) Prove that ✖✙✝ ✕ ❞ ✕ ✖ ✖ ✁ ✆ ✖ ✁ ✖ ✞★ ✆ ✍ ✘ ✷✼✰✻✰✼✰✻✷✸✏✼✘ ✍ ✷♥✘ ✬✮✭❈✠ to show that ✢■✭❱✠ . (d) Use part (c) and ✢❲✤❋✏ ✷✸✏ (e) Use parts (b), (c) and (d) to prove that ✏✩✣ ✭ ✕✑✤ ❣ ✡ ✖ ✬ . Also explain why the generators of ✏ are a Gr¨obner basis for the above lex order.

An edge of ✻ is a face of dimension 1. In Figure 4 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. σ supporting hyperplane Figure 4. A cone ✁ ✂ ✄ with shaded facets and a hyperplane supporting an edge Here are some properties of facets. Chapter 1. Affine Toric Varieties 26 ✰ ✦ ❄✿❅ ✁ ✆ be a polyhedral cone. 8. Let ✻ (a) then ❯ ✻ ❘ ❖ ✆ ✾ ❯ ✾ ❖ ✟☛ ❂ ✻ ✼ ❘ ■❃✙ ✢ ✹ ✱ ✭✍✁✔✓✔✓✔✓☞✁ ✱ ✧ ✽ ✓ (b) Every proper face ❍ of ✻ is the intersection of the facets of ✻ containing ❍ .

7]. ✷ Affine Semigroups. A semigroup is a set with an associative binary operation and an identity element. To be an affine semigroup, we further require that ✷ ✚ The binary operation on is commutative. We will write the operation as ✩ ✦ ✷ gives and the identity element as ☛ . Thus a finite set ✠✶✩ ❘ ✁ ✫ ✝ ✸ ✯ ✱ ✂ ✯ ✳ ✠✄✂ ✦ ✷ ✓ The semigroup is finitely generated, meaning that there is a finite set ✠★✩ ❘ ✷ . such that ✩ ✦ ✷ ❀✰ . ✰ ✠ ✦ . More ✠ generally, The simplest example of an affine semigroup is given a ✩ ✩ The semigroup can be embedded in a lattice lattice ❀ and a finite set ✦ ❀ , we get the affine semigroup isomorphism, all affine semigroups are of this form.