By Cox D., Little J., Schenck H.
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When ✙ satisfies a quadratic equation with integer coefficients, the generators of ✔ are related to continued fractions. For example, when ✍ ✁ ✥✣ and ✤✣ ✙ ✖✜✛ ✢ ✭ is the golden ratio, the minimal generators of ✔ are for , where ✣ is the th Fibonacci number. 15. Suppose that ❪ ✻✱✘ ❤ ✘ is a group isomorphism. Fix a finite set ✛ ✄ ✘ and ✂ ✜ and ✂★✧ are equivariantly isomorphic let ✦ ✭ ❪✤ ✛✖✬ . Prove that the toric varieties (meaning that the isomorphism respects the torus action). 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.
Let ✏✚✵ be a toric ideal and let ✟ ✷✻✰✼✰✻✰✾✷ ✟✚✶ be a basis of the sublattice ✤ ✂ ✄ ✥✧✦ . Define ✂ ✄ ✏✸✷ ✭ ❞ ✆✺✹★ ★ ❞ ✆✩✹✪ ❖ ☛✥✭▲✆✶✷✻✰✻✰✼✰✾✴✷ ✢ ✰ ✂ Prove ✏✰ ✚✵ ☞ ✭ ✏✸✷✼✻ ❞ ✁ ✘✚✘✽✘♣❞ ✦ ✄ ✑ . Hint: Given s❩✴✷ ✯ ✳ ✝ ✱ ✦ with s✟✲★ ✯✪✳✝ ✤ , write s✟✜★ ✯✛✭ ✾ ✰✶ ✓ ✁ that ✚ ✰✟ ,✚ ✰ ✝ ✝ ✥ . This implies ❃ ❁ ❃ ✍ ❁ ❞ r ✍ ✭ ★❢✆✔✭❀✿ ✔ ❞❞ ✆✆✩✹✹★✪✒❄ ✹ ✿ ✔ ❞❞ ✆✩✆ ✹✹★☞✪ ❄ ✹ ★❢✆♠✰ ❁ ✹✺❂ ❁ ✹✩❅ ❞ ✚ ✘ ✚ ✘ ✘ ❞ Show that multiplying this by ✤ ✁ ✦ ✰✬ ❆ gives an element of ✏✽✷ for ❇✜❈ ✠ . (By being where ✰ is homogeneous of degree ✟ .
13]. 2. ✭✁ ✍ ✖✄✂ for ✞ ✝☎✂✝✆ . 2. Here are some properties of relative interiors. Let ✂ ✄✟✞✡✠ be a cone. ✂ ✄ (a) Show that if ✣☛✝✄✂ ✆ , then ✣☛✝☎☛⑩❄ ☛⑥④✌☞✎✍✙✤✏✂✺✬ if and only if ✞✛✷ ✣ ✖ ✠ for all ✞ ✝☎✂ ✆ ✖✑✂✓✒ . (b) Let ✙ be a face of ✂ and fix ✞ ☎ ✝ ✂ ✆ . 1. 7. 3. 12. 4. Let ✂ ✄✗✞ ✠ be a polyhedral cone. 10 to prove that has a unique minimal face with respect to inclusion. Let ✤ denote this minimal face. ✭ ✤✏✂ ✆ ✬✘✒ . ✂ ★ ✂✺✬ . (d) Prove that ✤ ✭✙✂ ✖ ✤♣✚ (e) Fix ✞ ✝☎✂ ✆ . Prove that ✞ ✝☎☛⑩❄✳☛ ④✖☞✎✍✫✤✛✂ ✆ ✬ if and only if ✤ ✭✙ ✍ ✖✄✂ .
Toric varieties by Cox D., Little J., Schenck H.
by Anthony
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