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

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5). If instead we ❘ ✞ intersect with ✻ , we obtain and ❖  ✾ ❋✹ ❘ ✻ ✞ ✽ ❘ ❍ ❖  ✾ ❋✹ ❘ ✻ ✞✍✽ ❘ ✹ ❍ ❘ ❍ ✽ ✾ ❋✹ ❘ ✻ ✞✍✽ ❘ ❘ ❍ ✁ follows. 13 we call a separating hyperplane. and ❀ be dual✰ lattices with associated vector ✏ ✯ ✆ . For ✆ we usually use the lattice ✰ . 14. A polyhedral cone ✻ ✦ ❄✟❅ is rational if ✻ ❘ for ▲ some finite set ✦ ❄ . Rational Polyhedral Cones. Let ❄ ✏ ✯ ✆ and ❀❏❅ ❘ spaces ❄ ❅ ❘ ❄ The cones appearing in Figures 1, 2 and 5 are rational. We note without proof that faces and duals of rational polyhedral cones are rational.

Consider the affine semigroup ✦ generated by and , so ✷ ✁✑✏✒✁ ✒✁✔✓✔✓✔✓✖✕ . 5). 4. Hence ✞ ✞ ☛ ✟❋✪ ✷ ✲ ❘ ✟✫✪ ✁ ✁ ✁ ☛ ✲ ✁ ✟❆✪ ✬ ✁ ✲ ☞ ❑ ✬ ☛ ❘ ✍ ✸ is the curve ✬ ☛ ❘ ✞ . and the affine toric variety ▲ ✁ Equivalence of Constructions. We can now state the main result of this section, which asserts that our various approaches to affine toric varieties all give the same class of objects. 16. Let (a) ❀ (b) (c) (d) ❀ be an affine variety. 3. ❀ ❘ ✍☎✸ for a finite set ✩ in a lattice. is an affine variety defined by a toric ideal.

A) Find generators for the toric ideal ✏✢✭ ✕✑✤ ●✩✬❫✄ ❇◗❍ ❞ ✁ ✷ ❞ ✷ ❞ ✷ ❞ ✠ ✷ ❞ ❏ . ✹ ♦ ✮ (b) Show that ③❲④⑥⑤✲●▼✭✳✲ . 8 holds over ❇ . ✁ ✄ ✂ (c) Show that ✏✩✭✵✴ ❞ ✠✠ ✖ ❞ ✰✁ ❞ ✷ ❞ ✠ ✖ ❞ ✹ ❞ ♦ . 8. Instead of working over , we will work over an algebraically closed field ✯✮ parametrized by characteristic ✭ . Consider the affine toric variety of Chapter 1. , It follows that is a set-theoretic complete intersection. The paper [3] shows that if we replace with an algebrically closed field of characteristic ✭ , then the above parametrization is never a set-theoretic complete intersection.