Classifying algebraic varieties

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Classifying algebraic varieties

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■♥tr♦❞✉❝t✐♦♥ ✷ ❡q✉❛t✐♦♥s ✐♥ ✷ ✈❛r✐❛❜❧❡s ✶ ❡q✉❛t✐♦♥ ✐♥ ✷ ✈❛r✐❛❜❧❡s ■rr❡❞✉❝✐❜❧❡ s✉❜s❡ts ❈✉r✈❡s ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ Pr❡✈✐❡✇✳ ❈❧❛ss✐❢②✐♥❣ ❆❧❣❡❜r❛✐❝ ❱❛r✐❡t✐❡s ■ ❈❤r✐st♦♣❤❡r ❍❛❝♦♥ ❯♥✐✈❡rs✐t② ♦❢ ❯t❛❤ ▼❛r❝❤✱ ✷✵✵✽ ❈❤r✐st♦♣❤❡r ❍❛❝♦♥ ❈❧❛ss✐❢②✐♥❣ ❆❧❣❡❜r❛✐❝ ❱❛r✐❡t✐❡s ■ ■♥tr♦❞✉❝t✐♦♥ ✷ ❡q✉❛t✐♦♥s ✐♥ ✷ ✈❛r✐❛❜❧❡s ✶ ❡q✉❛t✐♦♥ ✐♥ ✷ ✈❛r✐❛❜❧❡s ■rr❡❞✉❝✐❜❧❡ s✉❜s❡ts ❈✉r✈❡s ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ Pr❡✈✐❡✇✳ ❖✉t❧✐♥❡ ♦❢ t❤❡ t❛❧❦ ✶ ■♥tr♦❞✉❝t✐♦♥ ❈❤r✐st♦♣❤❡r ❍❛❝♦♥ ❈❧❛ss✐❢②✐♥❣ ❆❧❣❡❜r❛✐❝ ❱❛r✐❡t✐❡s ■ ■♥tr♦❞✉❝t✐♦♥ ✷ ❡q✉❛t✐♦♥s ✐♥ ✷ ✈❛r✐❛❜❧❡s ✶ ❡q✉❛t✐♦♥ ✐♥ ✷ ✈❛r✐❛❜❧❡s ■rr❡❞✉❝✐❜❧❡ s✉❜s❡ts ❈✉r✈❡s ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ Pr❡✈✐❡✇✳ ❖✉t❧✐♥❡ ♦❢ t❤❡ t❛❧❦ ✶ ■♥tr♦❞✉❝t✐♦♥ ✷ ✷ ❡q✉❛t✐♦♥s ✐♥ ✷ ✈❛r✐❛❜❧❡s ❈❤r✐st♦♣❤❡r ❍❛❝♦♥ ❈❧❛ss✐❢②✐♥❣ ❆❧❣❡❜r❛✐❝ ❱❛r✐❡t✐❡s ■ ■♥tr♦❞✉❝t✐♦♥ ✷ ❡q✉❛t✐♦♥s ✐♥ ✷ ✈❛r✐❛❜❧❡s ✶ ❡q✉❛t✐♦♥ ✐♥ ✷ ✈❛r✐❛❜❧❡s ■rr❡❞✉❝✐❜❧❡ s✉❜s❡ts ❈✉r✈❡s ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ Pr❡✈✐❡✇✳ ❖✉t❧✐♥❡ ♦❢ t❤❡ t❛❧❦ ✶ ■♥tr♦❞✉❝t✐♦♥ ✷ ✷ ❡q✉❛t✐♦♥s ✐♥ ✷ ✈❛r✐❛❜❧❡s ✸ ✶ ❡q✉❛t✐♦♥ ✐♥ ✷ ✈❛r✐❛❜❧❡s ❈❤r✐st♦♣❤❡r ❍❛❝♦♥ ❈❧❛ss✐❢②✐♥❣ ❆❧❣❡❜r❛✐❝ ❱❛r✐❡t✐❡s ■ ■♥tr♦❞✉❝t✐♦♥ ✷ ❡q✉❛t✐♦♥s ✐♥ ✷ ✈❛r✐❛❜❧❡s ✶ ❡q✉❛t✐♦♥ ✐♥ ✷ ✈❛r✐❛❜❧❡s ■rr❡❞✉❝✐❜❧❡ s✉❜s❡ts ❈✉r✈❡s ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ Pr❡✈✐❡✇✳ ❖✉t❧✐♥❡ ♦❢ t❤❡ t❛❧❦ ✶ ■♥tr♦❞✉❝t✐♦♥ ✷ ✷ ❡q✉❛t✐♦♥s ✐♥ ✷ ✈❛r✐❛❜❧❡s ✸ ✶ ❡q✉❛t✐♦♥ ✐♥ ✷ ✈❛r✐❛❜❧❡s ✹ ■rr❡❞✉❝✐❜❧❡ s✉❜s❡ts ❈❤r✐st♦♣❤❡r ❍❛❝♦♥ ❈❧❛ss✐❢②✐♥❣ ❆❧❣❡❜r❛✐❝ ❱❛r✐❡t✐❡s ■ ■♥tr♦❞✉❝t✐♦♥ ✷ ❡q✉❛t✐♦♥s ✐♥ ✷ ✈❛r✐❛❜❧❡s ✶ ❡q✉❛t✐♦♥ ✐♥ ✷ ✈❛r✐❛❜❧❡s ■rr❡❞✉❝✐❜❧❡ s✉❜s❡ts ❈✉r✈❡s ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ Pr❡✈✐❡✇✳ ❖✉t❧✐♥❡ ♦❢ t❤❡ t❛❧❦ ✶ ■♥tr♦❞✉❝t✐♦♥ ✷ ✷ ❡q✉❛t✐♦♥s ✐♥ ✷ ✈❛r✐❛❜❧❡s ✸ ✶ ❡q✉❛t✐♦♥ ✐♥ ✷ ✈❛r✐❛❜❧❡s ✹ ■rr❡❞✉❝✐❜❧❡ s✉❜s❡ts ✺ ❈✉r✈❡s ❈❤r✐st♦♣❤❡r ❍❛❝♦♥ ❈❧❛ss✐❢②✐♥❣ ❆❧❣❡❜r❛✐❝ ❱❛r✐❡t✐❡s ■ ■♥tr♦❞✉❝t✐♦♥ ✷ ❡q✉❛t✐♦♥s ✐♥ ✷ ✈❛r✐❛❜❧❡s ✶ ❡q✉❛t✐♦♥ ✐♥ ✷ ✈❛r✐❛❜❧❡s ■rr❡❞✉❝✐❜❧❡ s✉❜s❡ts ❈✉r✈❡s ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ Pr❡✈✐❡✇✳ ❖✉t❧✐♥❡ ♦❢ t❤❡ t❛❧❦ ✶ ■♥tr♦❞✉❝t✐♦♥ ✷ ✷ ❡q✉❛t✐♦♥s ✐♥ ✷ ✈❛r✐❛❜❧❡s ✸ ✶ ❡q✉❛t✐♦♥ ✐♥ ✷ ✈❛r✐❛❜❧❡s ✹ ■rr❡❞✉❝✐❜❧❡ s✉❜s❡ts ✺ ❈✉r✈❡s ✻ ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ Pr❡✈✐❡✇✳ ❈❤r✐st♦♣❤❡r ❍❛❝♦♥ ❈❧❛ss✐❢②✐♥❣ ❆❧❣❡❜r❛✐❝ ❱❛r✐❡t✐❡s ■ ■♥tr♦❞✉❝t✐♦♥ ✷ ❡q✉❛t✐♦♥s ✐♥ ✷ ✈❛r✐❛❜❧❡s ✶ ❡q✉❛t✐♦♥ ✐♥ ✷ ✈❛r✐❛❜❧❡s ■rr❡❞✉❝✐❜❧❡ s✉❜s❡ts ❈✉r✈❡s ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ Pr❡✈✐❡✇✳ ❖✉t❧✐♥❡ ♦❢ t❤❡ t❛❧❦ ✶ ■♥tr♦❞✉❝t✐♦♥ ✷ ✷ ❡q✉❛t✐♦♥s ✐♥ ✷ ✈❛r✐❛❜❧❡s ✸ ✶ ❡q✉❛t✐♦♥ ✐♥ ✷ ✈❛r✐❛❜❧❡s ✹ ■rr❡❞✉❝✐❜❧❡ s✉❜s❡ts ✺ ❈✉r✈❡s ✻ ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ Pr❡✈✐❡✇✳ ❈❤r✐st♦♣❤❡r ❍❛❝♦♥ ❈❧❛ss✐❢②✐♥❣ ❆❧❣❡❜r❛✐❝ ❱❛r✐❡t✐❡s ■ ■♥tr♦❞✉❝t✐♦♥ ✷ ❡q✉❛t✐♦♥s ✐♥ ✷ ✈❛r✐❛❜❧❡s ✶ ❡q✉❛t✐♦♥ ✐♥ ✷ ✈❛r✐❛❜❧❡s ■rr❡❞✉❝✐❜❧❡ s✉❜s❡ts ❈✉r✈❡s ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ Pr❡✈✐❡✇✳ P♦❧②♥♦♠✐❛❧ ❡q✉❛t✐♦♥s ❆❧❣❡❜r❛✐❝ ●❡♦♠❡tr② ✐s ❝♦♥❝❡r♥❡❞ ✇✐t❤ t❤❡ st✉❞② ♦❢ s♦❧✉t✐♦♥s ♦❢ ♣♦❧②♥♦♠✐❛❧ ❡q✉❛t✐♦♥s s❛② ♥✲❡q✉❛t✐♦♥s ✐♥ ♠✲✈❛r✐❛❜❧❡s ✽ P ✭① ❀ ✿ ✿ ✿ ❀ ① ✮ ❂ ✵ ❃❁ ♠ ✿✿✿ ❃✿ ✿ ✿ ✿ P♥ ✭① ❀ ✿ ✿ ✿ ❀ ①♠ ✮ ❂ ✵✿ ✶ ✶ ✶ ❈❤r✐st♦♣❤❡r ❍❛❝♦♥ ❈❧❛ss✐❢②✐♥❣ ❆❧❣❡❜r❛✐❝ ❱❛r✐❡t✐❡s ■ ■♥tr♦❞✉❝t✐♦♥ ✷ ❡q✉❛t✐♦♥s ✐♥ ✷ ✈❛r✐❛❜❧❡s ✶ ❡q✉❛t✐♦♥ ✐♥ ✷ ✈❛r✐❛❜❧❡s ■rr❡❞✉❝✐❜❧❡ s✉❜s❡ts ❈✉r✈❡s ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ Pr❡✈✐❡✇✳ ✶ ❡q✉❛t✐♦♥ ✐♥ ✶ ✈❛r✐❛❜❧❡ ❋♦r ❡①❛♠♣❧❡✱ ✶ ❡q✉❛t✐♦♥ ✐♥ ✶ ✈❛r✐❛❜❧❡ P ✭① ✮ ❂ ❛❞ ① ❞ ✰ ❛❞  ✶ ① ❞  ✶ ✰ ✿ ✿ ✿ ✰ ❛✶ ① ✰ ❛✵ ✭❛ ♣♦❧②♥♦♠✐❛❧ ♦❢ ❞❡❣r❡❡ ❞ ✐♥ ① ✮✳ ❈❤r✐st♦♣❤❡r ❍❛❝♦♥ ❈❧❛ss✐❢②✐♥❣ ❆❧❣❡❜r❛✐❝ ❱❛r✐❡t✐❡s ■ ❛❞ ❂ ✻ ✵ ■♥tr♦❞✉❝t✐♦♥ ✷ ❡q✉❛t✐♦♥s ✐♥ ✷ ✈❛r✐❛❜❧❡s ✶ ❡q✉❛t✐♦♥ ✐♥ ✷ ✈❛r✐❛❜❧❡s ■rr❡❞✉❝✐❜❧❡ s✉❜s❡ts ❈✉r✈❡s ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ Pr❡✈✐❡✇✳ ❍♦❧♦♠♦r♣❤✐❝ ✶✲❢♦r♠s ■■ ❚❤❡r❡ ✐s ❛ ♠♦r❡ ❛❜str❛❝t✴♥❛t✉r❛❧ ♣♦✐♥t ♦❢ ✈✐❡✇✿ ❉❡☞♥❡ t❤❡ ❝❛♥♦♥✐❝❛❧ r✐♥❣ ▼ ✵ ❘ ✭❳ ✮ ❂ ❍ ✭❳ ❀ ✭✦❳ ✮✡❦ ✮✿ ❦ ✕✵ ❈❤r✐st♦♣❤❡r ❍❛❝♦♥ ❈❧❛ss✐❢②✐♥❣ ❆❧❣❡❜r❛✐❝ ❱❛r✐❡t✐❡s ■ ■♥tr♦❞✉❝t✐♦♥ ✷ ❡q✉❛t✐♦♥s ✐♥ ✷ ✈❛r✐❛❜❧❡s ✶ ❡q✉❛t✐♦♥ ✐♥ ✷ ✈❛r✐❛❜❧❡s ■rr❡❞✉❝✐❜❧❡ s✉❜s❡ts ❈✉r✈❡s ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ Pr❡✈✐❡✇✳ ❍♦❧♦♠♦r♣❤✐❝ ✶✲❢♦r♠s ■■ ❚❤❡r❡ ✐s ❛ ♠♦r❡ ❛❜str❛❝t✴♥❛t✉r❛❧ ♣♦✐♥t ♦❢ ✈✐❡✇✿ ❉❡☞♥❡ t❤❡ ❝❛♥♦♥✐❝❛❧ r✐♥❣ ▼ ✵ ❘ ✭❳ ✮ ❂ ❍ ✭❳ ❀ ✭✦❳ ✮✡❦ ✮✿ ❦ ✕✵ ❚❤❡♥ ❘ ✭❳ ✮ ✐s ☞♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❈✲❛❧❣❡❜r❛ ❛♥❞ ❳ ❂ Pr♦❥❘ ✭❳ ✮✳ ❈❤r✐st♦♣❤❡r ❍❛❝♦♥ ❈❧❛ss✐❢②✐♥❣ ❆❧❣❡❜r❛✐❝ ❱❛r✐❡t✐❡s ■ ■♥tr♦❞✉❝t✐♦♥ ✷ ❡q✉❛t✐♦♥s ✐♥ ✷ ✈❛r✐❛❜❧❡s ✶ ❡q✉❛t✐♦♥ ✐♥ ✷ ✈❛r✐❛❜❧❡s ■rr❡❞✉❝✐❜❧❡ s✉❜s❡ts ❈✉r✈❡s ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ Pr❡✈✐❡✇✳ ❍♦❧♦♠♦r♣❤✐❝ ✶✲❢♦r♠s ■■ ❚❤❡r❡ ✐s ❛ ♠♦r❡ ❛❜str❛❝t✴♥❛t✉r❛❧ ♣♦✐♥t ♦❢ ✈✐❡✇✿ ❉❡☞♥❡ t❤❡ ❝❛♥♦♥✐❝❛❧ r✐♥❣ ▼ ✵ ❘ ✭❳ ✮ ❂ ❍ ✭❳ ❀ ✭✦❳ ✮✡❦ ✮✿ ❦ ✕✵ ❚❤❡♥ ❘ ✭❳ ✮ ✐s ☞♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❈✲❛❧❣❡❜r❛ ❛♥❞ ❳ ❂ Pr♦❥❘ ✭❳ ✮✳ ■♥ ❝♦♦r❞✐♥❛t❡s t❤✐s ♠❛② ❜❡ ❞❡s❝r✐❜❡❞ ❛s ❢♦❧❧♦✇s✳ ■❢ r✵ ❀ ✁ ✁ ✁ ❀ r♠ ❛r❡ ❣❡♥❡r❛t♦rs ♦❢ ❘ ✭❳ ✮ ❛♥❞ ❑ ❂ ❦❡r ✭❈❬①✵ ❀ ✿ ✿ ✿ ❀ ①♠ ❪ ✦ ❘ ✭❳ ✮✮ ❀ t❤❡♥ ❳ ✚ P♠ C ✐s ❞❡☞♥❡❞ ❜② t❤❡ ❡q✉❛t✐♦♥s ✐♥ ❑ ✳ ❈❤r✐st♦♣❤❡r ❍❛❝♦♥ ❈❧❛ss✐❢②✐♥❣ ❆❧❣❡❜r❛✐❝ ❱❛r✐❡t✐❡s ■ ■♥tr♦❞✉❝t✐♦♥ ✷ ❡q✉❛t✐♦♥s ✐♥ ✷ ✈❛r✐❛❜❧❡s ✶ ❡q✉❛t✐♦♥ ✐♥ ✷ ✈❛r✐❛❜❧❡s ■rr❡❞✉❝✐❜❧❡ s✉❜s❡ts ❈✉r✈❡s ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ Pr❡✈✐❡✇✳ ❍♦❧♦♠♦r♣❤✐❝ ✶✲❢♦r♠s ■■ ❚❤❡r❡ ✐s ❛ ♠♦r❡ ❛❜str❛❝t✴♥❛t✉r❛❧ ♣♦✐♥t ♦❢ ✈✐❡✇✿ ❉❡☞♥❡ t❤❡ ❝❛♥♦♥✐❝❛❧ r✐♥❣ ▼ ✵ ❘ ✭❳ ✮ ❂ ❍ ✭❳ ❀ ✭✦❳ ✮✡❦ ✮✿ ❦ ✕✵ ❚❤❡♥ ❘ ✭❳ ✮ ✐s ☞♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❈✲❛❧❣❡❜r❛ ❛♥❞ ❳ ❂ Pr♦❥❘ ✭❳ ✮✳ ■♥ ❝♦♦r❞✐♥❛t❡s t❤✐s ♠❛② ❜❡ ❞❡s❝r✐❜❡❞ ❛s ❢♦❧❧♦✇s✳ ■❢ r✵ ❀ ✁ ✁ ✁ ❀ r♠ ❛r❡ ❣❡♥❡r❛t♦rs ♦❢ ❘ ✭❳ ✮ ❛♥❞ ❑ ❂ ❦❡r ✭❈❬①✵ ❀ ✿ ✿ ✿ ❀ ①♠ ❪ ✦ ❘ ✭❳ ✮✮ ❀ t❤❡♥ ❳ ✚ P♠ C ✐s ❞❡☞♥❡❞ ❜② t❤❡ ❡q✉❛t✐♦♥s ✐♥ ❑ ✳ ❚❤❡r❡ ❛r❡ ♠❛♥② ✐♥t❡r❡st✐♥❣ ♦♣❡♥ ♣r♦❜❧❡♠s ♦♥ t❤❡ str✉❝t✉r❡ ♦❢ ❘ ✭❳ ✮ ✭❡✈❡♥ ✐❢ ❞✐♠ ❳ ❂ ✶✮✳ ❈❤r✐st♦♣❤❡r ❍❛❝♦♥ ❈❧❛ss✐❢②✐♥❣ ❆❧❣❡❜r❛✐❝ ❱❛r✐❡t✐❡s ■ ■♥tr♦❞✉❝t✐♦♥ ✷ ❡q✉❛t✐♦♥s ✐♥ ✷ ✈❛r✐❛❜❧❡s ✶ ❡q✉❛t✐♦♥ ✐♥ ✷ ✈❛r✐❛❜❧❡s ■rr❡❞✉❝✐❜❧❡ s✉❜s❡ts ❈✉r✈❡s ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ Pr❡✈✐❡✇✳ ❖✉t❧✐♥❡ ♦❢ t❤❡ t❛❧❦ ✶ ■♥tr♦❞✉❝t✐♦♥ ✷ ✷ ❡q✉❛t✐♦♥s ✐♥ ✷ ✈❛r✐❛❜❧❡s ✸ ✶ ❡q✉❛t✐♦♥ ✐♥ ✷ ✈❛r✐❛❜❧❡s ✹ ■rr❡❞✉❝✐❜❧❡ s✉❜s❡ts ✺ ❈✉r✈❡s ✻ ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ Pr❡✈✐❡✇✳ ❈❤r✐st♦♣❤❡r ❍❛❝♦♥ ❈❧❛ss✐❢②✐♥❣ ❆❧❣❡❜r❛✐❝ ❱❛r✐❡t✐❡s ■ ■♥tr♦❞✉❝t✐♦♥ ✷ ❡q✉❛t✐♦♥s ✐♥ ✷ ✈❛r✐❛❜❧❡s ✶ ❡q✉❛t✐♦♥ ✐♥ ✷ ✈❛r✐❛❜❧❡s ■rr❡❞✉❝✐❜❧❡ s✉❜s❡ts ❈✉r✈❡s ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ Pr❡✈✐❡✇✳ ❚❤❡ ❈❛♥♦♥✐❝❛❧ ❘✐♥❣ ■♥ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❧❡❝t✉r❡s ■ ✇✐❧❧ ❡①♣❧❛✐♥ ❤♦✇ ♦♥❡ ♠❛② ✉♥❞❡rst❛♥❞ s♦❧✉t✐♦♥ s❡ts ♦❢ ❞✐♠❡♥s✐♦♥ ✕ ✷✳ ❈❤r✐st♦♣❤❡r ❍❛❝♦♥ ❈❧❛ss✐❢②✐♥❣ ❆❧❣❡❜r❛✐❝ ❱❛r✐❡t✐❡s ■ ■♥tr♦❞✉❝t✐♦♥ ✷ ❡q✉❛t✐♦♥s ✐♥ ✷ ✈❛r✐❛❜❧❡s ✶ ❡q✉❛t✐♦♥ ✐♥ ✷ ✈❛r✐❛❜❧❡s ■rr❡❞✉❝✐❜❧❡ s✉❜s❡ts ❈✉r✈❡s ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ Pr❡✈✐❡✇✳ ❚❤❡ ❈❛♥♦♥✐❝❛❧ ❘✐♥❣ ■♥ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❧❡❝t✉r❡s ■ ✇✐❧❧ ❡①♣❧❛✐♥ ❤♦✇ ♦♥❡ ♠❛② ✉♥❞❡rst❛♥❞ s♦❧✉t✐♦♥ s❡ts ♦❢ ❞✐♠❡♥s✐♦♥ ✕ ✷✳ ▲❡t ❳ ❜❡ ❛♥ ✐rr❡❞✉❝✐❜❧❡ ❩❛r✐s❦✐ ❝❧♦s❡❞ s❡t ♦❢ ❞✐♠❡♥s✐♦♥ ❞ ✳ ❚❤❡ s❡❝t✐♦♥s s ✷ ❍ ✵ ✭❳ ❀ ✦❳✡❦ ✮ ♠❛② ❜❡ ❧♦❝❛❧❧② ✇r✐tt❡♥ ❛s ❢ ✭①✶ ❀ ✿ ✿ ✿ ❀ ①❞ ✮✭❞①✶ ❫ ✿ ✿ ✿ ❫ ❞①❞ ✮✡❦ ✳ ❈❤r✐st♦♣❤❡r ❍❛❝♦♥ ❈❧❛ss✐❢②✐♥❣ ❆❧❣❡❜r❛✐❝ ❱❛r✐❡t✐❡s ■ ■♥tr♦❞✉❝t✐♦♥ ✷ ❡q✉❛t✐♦♥s ✐♥ ✷ ✈❛r✐❛❜❧❡s ✶ ❡q✉❛t✐♦♥ ✐♥ ✷ ✈❛r✐❛❜❧❡s ■rr❡❞✉❝✐❜❧❡ s✉❜s❡ts ❈✉r✈❡s ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ Pr❡✈✐❡✇✳ ❚❤❡ ❈❛♥♦♥✐❝❛❧ ❘✐♥❣ ■♥ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❧❡❝t✉r❡s ■ ✇✐❧❧ ❡①♣❧❛✐♥ ❤♦✇ ♦♥❡ ♠❛② ✉♥❞❡rst❛♥❞ s♦❧✉t✐♦♥ s❡ts ♦❢ ❞✐♠❡♥s✐♦♥ ✕ ✷✳ ▲❡t ❳ ❜❡ ❛♥ ✐rr❡❞✉❝✐❜❧❡ ❩❛r✐s❦✐ ❝❧♦s❡❞ s❡t ♦❢ ❞✐♠❡♥s✐♦♥ ❞ ✳ ❚❤❡ s❡❝t✐♦♥s s ✷ ❍ ✵ ✭❳ ❀ ✦❳✡❦ ✮ ♠❛② ❜❡ ❧♦❝❛❧❧② ✇r✐tt❡♥ ❛s ❢ ✭①✶ ❀ ✿ ✿ ✿ ❀ ①❞ ✮✭❞①✶ ❫ ✿ ✿ ✿ ❫ ❞①❞ ✮✡❦ ✳ ❚❤❡ ❈❛♥♦♥✐❝❛❧ ❘✐♥❣ ♦❢ ❳ ❣✐✈❡♥ ❜② ❘ ✭❳ ✮ ❂ ❈❤r✐st♦♣❤❡r ❍❛❝♦♥ ▼ ❦ ✕✵ ❍ ✵ ✭❳ ❀ ✦❳✡❦ ✮✿ ❈❧❛ss✐❢②✐♥❣ ❆❧❣❡❜r❛✐❝ ❱❛r✐❡t✐❡s ■ ■♥tr♦❞✉❝t✐♦♥ ✷ ❡q✉❛t✐♦♥s ✐♥ ✷ ✈❛r✐❛❜❧❡s ✶ ❡q✉❛t✐♦♥ ✐♥ ✷ ✈❛r✐❛❜❧❡s ■rr❡❞✉❝✐❜❧❡ s✉❜s❡ts ❈✉r✈❡s ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ Pr❡✈✐❡✇✳ ❙✉r❢❛❝❡s ❛♥❞ ✸✲❢♦❧❞s ❚❤❡ ❣❡♦♠❡tr② ♦❢ ❝♦♠♣❧❡① s✉r❢❛❝❡s ✭❞ ❂ ✷✮ ✇❛s ✇❡❧❧ ✉♥❞❡rst♦♦❞ ✐♥ t❡r♠s ♦❢ ❘ ✭❳ ✮ ❜② t❤❡ ■t❛❧✐❛♥ ❙❝❤♦♦❧ ♦❢ ❆❧❣❡❜r❛✐❝ ●❡♦♠❡tr② ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ✷✵✲t❤ ❝❡♥t✉r②✳ ❈❤r✐st♦♣❤❡r ❍❛❝♦♥ ❈❧❛ss✐❢②✐♥❣ ❆❧❣❡❜r❛✐❝ ❱❛r✐❡t✐❡s ■ ■♥tr♦❞✉❝t✐♦♥ ✷ ❡q✉❛t✐♦♥s ✐♥ ✷ ✈❛r✐❛❜❧❡s ✶ ❡q✉❛t✐♦♥ ✐♥ ✷ ✈❛r✐❛❜❧❡s ■rr❡❞✉❝✐❜❧❡ s✉❜s❡ts ❈✉r✈❡s ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ Pr❡✈✐❡✇✳ ❙✉r❢❛❝❡s ❛♥❞ ✸✲❢♦❧❞s ❚❤❡ ❣❡♦♠❡tr② ♦❢ ❝♦♠♣❧❡① s✉r❢❛❝❡s ✭❞ ❂ ✷✮ ✇❛s ✇❡❧❧ ✉♥❞❡rst♦♦❞ ✐♥ t❡r♠s ♦❢ ❘ ✭❳ ✮ ❜② t❤❡ ■t❛❧✐❛♥ ❙❝❤♦♦❧ ♦❢ ❆❧❣❡❜r❛✐❝ ●❡♦♠❡tr② ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ✷✵✲t❤ ❝❡♥t✉r②✳ ❚❤❡ ❣❡♦♠❡tr② ♦❢ ❝♦♠♣❧❡① ✸✲❢♦❧❞s ✭❞ ❂ ✸✮ ✇❛s ✉♥❞❡rst♦♦❞ ✐♥ t❤❡ ✶✾✽✵✬s ❜② ✇♦r❦ ♦❢ ❙✳ ▼♦r✐✱ ❨✳ ❑❛✇❛♠❛t❛✱ ❏✳ ❑♦❧❧✓❛r✱ ❱✳ ❙❤♦❦✉r♦✈ ❛♥❞ ♦t❤❡rs✳ ❈❤r✐st♦♣❤❡r ❍❛❝♦♥ ❈❧❛ss✐❢②✐♥❣ ❆❧❣❡❜r❛✐❝ ❱❛r✐❡t✐❡s ■ ■♥tr♦❞✉❝t✐♦♥ ✷ ❡q✉❛t✐♦♥s ✐♥ ✷ ✈❛r✐❛❜❧❡s ✶ ❡q✉❛t✐♦♥ ✐♥ ✷ ✈❛r✐❛❜❧❡s ■rr❡❞✉❝✐❜❧❡ s✉❜s❡ts ❈✉r✈❡s ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ Pr❡✈✐❡✇✳ ❙✉r❢❛❝❡s ❛♥❞ ✸✲❢♦❧❞s ❚❤❡ ❣❡♦♠❡tr② ♦❢ ❝♦♠♣❧❡① s✉r❢❛❝❡s ✭❞ ❂ ✷✮ ✇❛s ✇❡❧❧ ✉♥❞❡rst♦♦❞ ✐♥ t❡r♠s ♦❢ ❘ ✭❳ ✮ ❜② t❤❡ ■t❛❧✐❛♥ ❙❝❤♦♦❧ ♦❢ ❆❧❣❡❜r❛✐❝ ●❡♦♠❡tr② ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ✷✵✲t❤ ❝❡♥t✉r②✳ ❚❤❡ ❣❡♦♠❡tr② ♦❢ ❝♦♠♣❧❡① ✸✲❢♦❧❞s ✭❞ ❂ ✸✮ ✇❛s ✉♥❞❡rst♦♦❞ ✐♥ t❤❡ ✶✾✽✵✬s ❜② ✇♦r❦ ♦❢ ❙✳ ▼♦r✐✱ ❨✳ ❑❛✇❛♠❛t❛✱ ❏✳ ❑♦❧❧✓❛r✱ ❱✳ ❙❤♦❦✉r♦✈ ❛♥❞ ♦t❤❡rs✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✐❢ ❞✐♠ ❳ ❂ ✸✱ t❤❡♥ ❘ ✭❳ ✮ ✐s ☞♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❈❤r✐st♦♣❤❡r ❍❛❝♦♥ ❈❧❛ss✐❢②✐♥❣ ❆❧❣❡❜r❛✐❝ ❱❛r✐❡t✐❡s ■ ■♥tr♦❞✉❝t✐♦♥ ✷ ❡q✉❛t✐♦♥s ✐♥ ✷ ✈❛r✐❛❜❧❡s ✶ ❡q✉❛t✐♦♥ ✐♥ ✷ ✈❛r✐❛❜❧❡s ■rr❡❞✉❝✐❜❧❡ s✉❜s❡ts ❈✉r✈❡s ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ Pr❡✈✐❡✇✳ ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s ❘❡❝❡♥t❧② t❤❡r❡ ❤❛✈❡ ❜❡❡♥ s♦♠❡ ✈❡r② ❡①❝✐t✐♥❣ ❞❡✈❡❧♦♣♠❡♥ts ✐♥ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s ✭❞ ✕ ✹✮✳ ■♥ ♣❛rt✐❝✉❧❛r ✇❡ ❤❛✈❡✿ ❈❤r✐st♦♣❤❡r ❍❛❝♦♥ ❈❧❛ss✐❢②✐♥❣ ❆❧❣❡❜r❛✐❝ ❱❛r✐❡t✐❡s ■ ■♥tr♦❞✉❝t✐♦♥ ✷ ❡q✉❛t✐♦♥s ✐♥ ✷ ✈❛r✐❛❜❧❡s ✶ ❡q✉❛t✐♦♥ ✐♥ ✷ ✈❛r✐❛❜❧❡s ■rr❡❞✉❝✐❜❧❡ s✉❜s❡ts ❈✉r✈❡s ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ Pr❡✈✐❡✇✳ ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s ❘❡❝❡♥t❧② t❤❡r❡ ❤❛✈❡ ❜❡❡♥ s♦♠❡ ✈❡r② ❡①❝✐t✐♥❣ ❞❡✈❡❧♦♣♠❡♥ts ✐♥ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s ✭❞ ✕ ✹✮✳ ■♥ ♣❛rt✐❝✉❧❛r ✇❡ ❤❛✈❡✿ ❚❤❡♦r❡♠ ✭❇✐r❦❛r✲❈❛s❝✐♥✐✲❍❛❝♦♥✲▼❝ ❑❡r♥❛♥✱ ❙✐✉✮ ❚❤❡ ❝❛♥♦♥✐❝❛❧ r✐♥❣ ❘ ✭❳ ✮ ❂ ▼ ❦ ✕✵ ❍ ✵ ✭❳ ❀ ✦❳✡❦ ✮ ✐s ☞♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❈❤r✐st♦♣❤❡r ❍❛❝♦♥ ❈❧❛ss✐❢②✐♥❣ ❆❧❣❡❜r❛✐❝ ❱❛r✐❡t✐❡s ■ ■♥tr♦❞✉❝t✐♦♥ ✷ ❡q✉❛t✐♦♥s ✐♥ ✷ ✈❛r✐❛❜❧❡s ✶ ❡q✉❛t✐♦♥ ✐♥ ✷ ✈❛r✐❛❜❧❡s ■rr❡❞✉❝✐❜❧❡ s✉❜s❡ts ❈✉r✈❡s ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ Pr❡✈✐❡✇✳ ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s ❘❡❝❡♥t❧② t❤❡r❡ ❤❛✈❡ ❜❡❡♥ s♦♠❡ ✈❡r② ❡①❝✐t✐♥❣ ❞❡✈❡❧♦♣♠❡♥ts ✐♥ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s ✭❞ ✕ ✹✮✳ ■♥ ♣❛rt✐❝✉❧❛r ✇❡ ❤❛✈❡✿ ❚❤❡♦r❡♠ ✭❇✐r❦❛r✲❈❛s❝✐♥✐✲❍❛❝♦♥✲▼❝ ❑❡r♥❛♥✱ ❙✐✉✮ ❚❤❡ ❝❛♥♦♥✐❝❛❧ r✐♥❣ ❘ ✭❳ ✮ ❂ ▼ ❦ ✕✵ ❍ ✵ ✭❳ ❀ ✦❳✡❦ ✮ ✐s ☞♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ■t ❢♦❧❧♦✇s t❤❛t t❤❡r❡ ✐s ❛ ♥❛t✉r❛❧ ♠❛♣ ✣ ✿ ❳ ✾✾❑ Pr♦❥❘ ✭❳ ✮ ✘ ❂ P♠ C✿ ❈❤r✐st♦♣❤❡r ❍❛❝♦♥ ❈❧❛ss✐❢②✐♥❣ ❆❧❣❡❜r❛✐❝ ❱❛r✐❡t✐❡s ■ ■♥tr♦❞✉❝t✐♦♥ ✷ ❡q✉❛t✐♦♥s ✐♥ ✷ ✈❛r✐❛❜❧❡s ✶ ❡q✉❛t✐♦♥ ✐♥ ✷ ✈❛r✐❛❜❧❡s ■rr❡❞✉❝✐❜❧❡ s✉❜s❡ts ❈✉r✈❡s ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳ Pr❡✈✐❡✇✳ ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥s ❘❡❝❡♥t❧② t❤❡r❡ ❤❛✈❡ ❜❡❡♥ s♦♠❡ ✈❡r② ❡①❝✐t✐♥❣ ❞❡✈❡❧♦♣♠❡♥ts ✐♥ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s ✭❞ ✕ ✹✮✳ ■♥ ♣❛rt✐❝✉❧❛r ✇❡ ❤❛✈❡✿ ❚❤❡♦r❡♠ ✭❇✐r❦❛r✲❈❛s❝✐♥✐✲❍❛❝♦♥✲▼❝ ❑❡r♥❛♥✱ ❙✐✉✮ ❚❤❡ ❝❛♥♦♥✐❝❛❧ r✐♥❣ ❘ ✭❳ ✮ ❂ ▼ ❦ ✕✵ ❍ ✵ ✭❳ ❀ ✦❳✡❦ ✮ ✐s ☞♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ■t ❢♦❧❧♦✇s t❤❛t t❤❡r❡ ✐s ❛ ♥❛t✉r❛❧ ♠❛♣ ✣ ✿ ❳ ✾✾❑ Pr♦❥❘ ✭❳ ✮ ✘ ❂ P♠ C✿ ■♥ ♠♦st ❝❛s❡s✱ ✣ ❝❛rr✐❡s ❛ ❧♦t ♦❢ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ❳ ❛♥❞ ✐t ❝❛♥ ❜❡ ✉s❡❞ t♦ st✉❞② t❤❡ ❣❡♦♠❡tr② ♦❢ ❳ ✐♥ ❛ ❝♦♦r❞✐♥❛t❡ ❢r❡❡ ♠❛♥♥❡r✳ ❈❤r✐st♦♣❤❡r ❍❛❝♦♥ ❈❧❛ss✐❢②✐♥❣ ❆❧❣❡❜r❛✐❝ ❱❛r✐❡t✐❡s ■

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Mục lục

  • Introduction

  • 2 equations in 2 variables

  • 1 equation in 2 variables

  • Irreducible subsets

  • Curves

  • Higher dimensions. Preview.

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