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▼❛♣s t❤❛t t❛❦❡ ❧✐♥❡s t♦ ♣❧❛♥❡ ❝✉r✈❡s ❱s❡✈♦❧♦❞ P❡tr✉s❝❤❡♥❦♦✱ ❱❧❛❞❧❡♥ ❚✐♠♦r✐♥ ∗ ∗ ❋❛❝✉❧t② ♦❢ ▼❛t❤❡♠❛t✐❝s ◆❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❯♥✐✈❡rs✐t② ❍✐❣❤❡r ❙❝❤♦♦❧ ♦❢ ❊❝♦♥♦♠✐❝s ❋✐❡❧❞s ■♥st✐t✉t❡✱ ❚♦r♦♥t♦✱ ◆♦✈❡♠❜❡r ✷✽✱ ✷✵✶✹

P❧❛♥❛r✐③❛t✐♦♥s ❉❡✜♥✐t✐♦♥ ❆ ♣❧❛♥❛r✐③❛t✐♦♥ ✐s ❛ s✉✣❝✐❡♥t❧② s♠♦♦t❤ ♠❛♣♣✐♥❣ f : U ⊂ R P 2 → R P 3 s✉❝❤ t❤❛t✱ ❢♦r ❡✈❡r② ❧✐♥❡ L ⊂ R P 2 ✱ t❤❡ s❡t f ( U ∩ L ) ✐s ♣❧❛♥❛r✳ ❉❡✜♥✐t✐♦♥ ❚✇♦ ♣❧❛♥❛r✐③❛t✐♦♥s f : U → R P 3 ❛♥❞ g : V → R P 3 ❛r❡ ❡q✉✐✈❛❧❡♥t ✐❢ t❤❡r❡ ✐s ❛ ♥♦♥❡♠♣t② ♦♣❡♥ s✉❜s❡t W ⊂ U ∩ V s✉❝❤ t❤❛t f = g ♦♥ W ✱ ✉♣ t♦ ♣r♦❥❡❝t✐✈❡ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ s♦✉r❝❡ ❛♥❞ t❛r❣❡t s♣❛❝❡s✳ Pr♦❜❧❡♠ ❈❧❛ss✐❢② ♣❧❛♥❛r✐③❛t✐♦♥s ❛❝❝♦r❞✐♥❣ t♦ t❤✐s ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥✳

P❧❛♥❛r✐③❛t✐♦♥s ❉❡✜♥✐t✐♦♥ ❆ ♣❧❛♥❛r✐③❛t✐♦♥ ✐s ❛ s✉✣❝✐❡♥t❧② s♠♦♦t❤ ♠❛♣♣✐♥❣ f : U ⊂ R P 2 → R P 3 s✉❝❤ t❤❛t✱ ❢♦r ❡✈❡r② ❧✐♥❡ L ⊂ R P 2 ✱ t❤❡ s❡t f ( U ∩ L ) ✐s ♣❧❛♥❛r✳ ❉❡✜♥✐t✐♦♥ ❚✇♦ ♣❧❛♥❛r✐③❛t✐♦♥s f : U → R P 3 ❛♥❞ g : V → R P 3 ❛r❡ ❡q✉✐✈❛❧❡♥t ✐❢ t❤❡r❡ ✐s ❛ ♥♦♥❡♠♣t② ♦♣❡♥ s✉❜s❡t W ⊂ U ∩ V s✉❝❤ t❤❛t f = g ♦♥ W ✱ ✉♣ t♦ ♣r♦❥❡❝t✐✈❡ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ s♦✉r❝❡ ❛♥❞ t❛r❣❡t s♣❛❝❡s✳ Pr♦❜❧❡♠ ❈❧❛ss✐❢② ♣❧❛♥❛r✐③❛t✐♦♥s ❛❝❝♦r❞✐♥❣ t♦ t❤✐s ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥✳

P❧❛♥❛r✐③❛t✐♦♥s ❉❡✜♥✐t✐♦♥ ❆ ♣❧❛♥❛r✐③❛t✐♦♥ ✐s ❛ s✉✣❝✐❡♥t❧② s♠♦♦t❤ ♠❛♣♣✐♥❣ f : U ⊂ R P 2 → R P 3 s✉❝❤ t❤❛t✱ ❢♦r ❡✈❡r② ❧✐♥❡ L ⊂ R P 2 ✱ t❤❡ s❡t f ( U ∩ L ) ✐s ♣❧❛♥❛r✳ ❉❡✜♥✐t✐♦♥ ❚✇♦ ♣❧❛♥❛r✐③❛t✐♦♥s f : U → R P 3 ❛♥❞ g : V → R P 3 ❛r❡ ❡q✉✐✈❛❧❡♥t ✐❢ t❤❡r❡ ✐s ❛ ♥♦♥❡♠♣t② ♦♣❡♥ s✉❜s❡t W ⊂ U ∩ V s✉❝❤ t❤❛t f = g ♦♥ W ✱ ✉♣ t♦ ♣r♦❥❡❝t✐✈❡ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ s♦✉r❝❡ ❛♥❞ t❛r❣❡t s♣❛❝❡s✳ Pr♦❜❧❡♠ ❈❧❛ss✐❢② ♣❧❛♥❛r✐③❛t✐♦♥s ❛❝❝♦r❞✐♥❣ t♦ t❤✐s ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥✳

❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ Pr♦❥❡❝t✐✈❡ ●❡♦♠❡tr② ❚❤❡♦r❡♠ ✭▼☎ ♦❜✐✉s✱ ✶✽✷✼✮ ❙✉♣♣♦s❡ t❤❛t f : R P n → R P n ✐s ❛ ❝♦♥t✐♥✉♦✉s ♦♥❡✲t♦✲♦♥❡ ♠❛♣ t❛❦✐♥❣ ❛❧❧ str❛✐❣❤t ❧✐♥❡s t♦ str❛✐❣❤t ❧✐♥❡s✳ ❚❤❡♥ f ✐s ❛ ♣r♦❥❡❝t✐✈❡ tr❛♥s❢♦r♠❛t✐♦♥✱ ✐✳❡✳✱ ❛ ♣r♦❥❡❝t✐✈✐③❛t✐♦♥ ♦❢ ❛ ❧✐♥❡❛r ✐s♦♠♦r♣❤✐s♠ R n +1 → R n +1 ✳ ❚❤❡♦r❡♠ ✭✈♦♥ ❙t❛✉❞t✮ ❚❤❡ ❝♦♥t✐♥✉✐t② ❛ss✉♠♣t✐♦♥ ✐s s✉♣❡r✢✉♦✉s✳ ❘❡♠❛r❦ ❚❤✐s t❤❡♦r❡♠ ❤❛s ❧♦❝❛❧ ✈❡rs✐♦♥s✳

❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ Pr♦❥❡❝t✐✈❡ ●❡♦♠❡tr② ❚❤❡♦r❡♠ ✭▼☎ ♦❜✐✉s✱ ✶✽✷✼✮ ❙✉♣♣♦s❡ t❤❛t f : R P n → R P n ✐s ❛ ❝♦♥t✐♥✉♦✉s ♦♥❡✲t♦✲♦♥❡ ♠❛♣ t❛❦✐♥❣ ❛❧❧ str❛✐❣❤t ❧✐♥❡s t♦ str❛✐❣❤t ❧✐♥❡s✳ ❚❤❡♥ f ✐s ❛ ♣r♦❥❡❝t✐✈❡ tr❛♥s❢♦r♠❛t✐♦♥✱ ✐✳❡✳✱ ❛ ♣r♦❥❡❝t✐✈✐③❛t✐♦♥ ♦❢ ❛ ❧✐♥❡❛r ✐s♦♠♦r♣❤✐s♠ R n +1 → R n +1 ✳ ❚❤❡♦r❡♠ ✭✈♦♥ ❙t❛✉❞t✮ ❚❤❡ ❝♦♥t✐♥✉✐t② ❛ss✉♠♣t✐♦♥ ✐s s✉♣❡r✢✉♦✉s✳ ❘❡♠❛r❦ ❚❤✐s t❤❡♦r❡♠ ❤❛s ❧♦❝❛❧ ✈❡rs✐♦♥s✳

❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ Pr♦❥❡❝t✐✈❡ ●❡♦♠❡tr② ❚❤❡♦r❡♠ ✭▼☎ ♦❜✐✉s✱ ✶✽✷✼✮ ❙✉♣♣♦s❡ t❤❛t f : R P n → R P n ✐s ❛ ❝♦♥t✐♥✉♦✉s ♦♥❡✲t♦✲♦♥❡ ♠❛♣ t❛❦✐♥❣ ❛❧❧ str❛✐❣❤t ❧✐♥❡s t♦ str❛✐❣❤t ❧✐♥❡s✳ ❚❤❡♥ f ✐s ❛ ♣r♦❥❡❝t✐✈❡ tr❛♥s❢♦r♠❛t✐♦♥✱ ✐✳❡✳✱ ❛ ♣r♦❥❡❝t✐✈✐③❛t✐♦♥ ♦❢ ❛ ❧✐♥❡❛r ✐s♦♠♦r♣❤✐s♠ R n +1 → R n +1 ✳ ❚❤❡♦r❡♠ ✭✈♦♥ ❙t❛✉❞t✮ ❚❤❡ ❝♦♥t✐♥✉✐t② ❛ss✉♠♣t✐♦♥ ✐s s✉♣❡r✢✉♦✉s✳ ❘❡♠❛r❦ ❚❤✐s t❤❡♦r❡♠ ❤❛s ❧♦❝❛❧ ✈❡rs✐♦♥s✳

❈❧❛ss✐❝❛❧ ❣❡♦♠❡t❡rs ❆✉❣✉st ▼☎ ♦❜✐✉s ❑❛r❧ ●❡♦r❣ ❈❤r✐st✐❛♥ ✈♦♥ ❙t❛✉❞t ✶✼✾✵✕✶✽✻✽ ✶✼✾✽✕✶✽✻✼

▼♦t✐✈❛t✐♦♥ • ❆♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ Pr♦❥❡❝t✐✈❡ ●❡♦♠❡tr② • ▲❡t L ❜❡ ❛ ❧✐♥❡❛r s②st❡♠ ♦❢ ❝✉r✈❡s ✭❡✳❣✳✱ t❤❡ ❢❛♠✐❧② ♦❢ ❛❧❧ ❧✐♥❡s✱ ❝✐r❝❧❡s✱ ❝♦♥✐❝s✱ ❡t❝✳✮✳ ❙t✉❞②✐♥❣ ♠❛♣♣✐♥❣s f : U ⊂ R P 2 → R P 2 t❛❦✐♥❣ ❧✐♥❡ s❡❣♠❡♥ts t♦ ❝✉r✈❡s ❢r♦♠ L ✐s r❡❧❛t❡❞ ✇✐t❤ st✉❞②✐♥❣ ♣❧❛♥❛r✐③❛t✐♦♥s✳

▼♦t✐✈❛t✐♦♥ • ❆♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ Pr♦❥❡❝t✐✈❡ ●❡♦♠❡tr② • ▲❡t L ❜❡ ❛ ❧✐♥❡❛r s②st❡♠ ♦❢ ❝✉r✈❡s ✭❡✳❣✳✱ t❤❡ ❢❛♠✐❧② ♦❢ ❛❧❧ ❧✐♥❡s✱ ❝✐r❝❧❡s✱ ❝♦♥✐❝s✱ ❡t❝✳✮✳ ❙t✉❞②✐♥❣ ♠❛♣♣✐♥❣s f : U ⊂ R P 2 → R P 2 t❛❦✐♥❣ ❧✐♥❡ s❡❣♠❡♥ts t♦ ❝✉r✈❡s ❢r♦♠ L ✐s r❡❧❛t❡❞ ✇✐t❤ st✉❞②✐♥❣ ♣❧❛♥❛r✐③❛t✐♦♥s✳

❚r✐✈✐❛❧ ❝❛s❡s ❉❡✜♥✐t✐♦♥ ❆ ♣❧❛♥❛r✐③❛t✐♦♥ f : U → R P 3 ✐s tr✐✈✐❛❧ ✐❢ f ( U ) ❧✐❡s ✐♥ ❛ ♣❧❛♥❡✳ ❉❡✜♥✐t✐♦♥ ❆ ♣❧❛♥❛r✐③❛t✐♦♥ f : U → R P 3 ✐s ❝♦✲tr✐✈✐❛❧ ✐❢ t❤❡r❡ ❡①✐sts ❛ ♣♦✐♥t a ∈ R P 3 s✉❝❤ t❤❛t f ( U ∩ L ) ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ♣❧❛♥❡ t❤r♦✉❣❤ a ✱ ❢♦r ❡✈❡r② ❧✐♥❡ L ⊂ R P 2 ✳

❚r✐✈✐❛❧ ❝❛s❡s ❉❡✜♥✐t✐♦♥ ❆ ♣❧❛♥❛r✐③❛t✐♦♥ f : U → R P 3 ✐s tr✐✈✐❛❧ ✐❢ f ( U ) ❧✐❡s ✐♥ ❛ ♣❧❛♥❡✳ ❉❡✜♥✐t✐♦♥ ❆ ♣❧❛♥❛r✐③❛t✐♦♥ f : U → R P 3 ✐s ❝♦✲tr✐✈✐❛❧ ✐❢ t❤❡r❡ ❡①✐sts ❛ ♣♦✐♥t a ∈ R P 3 s✉❝❤ t❤❛t f ( U ∩ L ) ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ♣❧❛♥❡ t❤r♦✉❣❤ a ✱ ❢♦r ❡✈❡r② ❧✐♥❡ L ⊂ R P 2 ✳

❈♦✲tr✐✈✐❛❧ ♣❧❛♥❛r✐③❛t✐♦♥s

◆♦♥✲tr✐✈✐❛❧ ❡①❛♠♣❧❡s ❉❡✜♥✐t✐♦♥ ❆ q✉❛❞r❛t✐❝ r❛t✐♦♥❛❧ ♠❛♣♣✐♥❣ ✐s ❛ r❛t✐♦♥❛❧ ♠❛♣♣✐♥❣ f : R P 2 ��� R P 2 ❣✐✈❡♥ ✐♥ ❤♦♠♦❣❡♥❡♦✉s ❝♦♦r❞✐♥❛t❡s ❜② ❤♦♠♦❣❡♥❡♦✉s ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ✷✿ 2 � a i , j f [ x 0 : x 1 : x 2 ] = [ y 0 : y 1 : y 2 : y 3 ] , y α = α x i x j . i , j =0 ❊①❛♠♣❧❡ ❆♥② q✉❛❞r❛t✐❝ r❛t✐♦♥❛❧ ♠❛♣♣✐♥❣ ✐s ❛ ♣❧❛♥❛r✐③❛t✐♦♥❀ ✐t t❛❦❡s ❧✐♥❡s t♦ ❝♦♥✐❝s✳