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❑r✐❝❤❡✈❡r ❢♦r♠❛❧ ❣r♦✉♣s ❛♥❞ t❤❡ ❞❡❢♦r♠❡❞ ❇❛❦❡r✲❆❦❤✐❡③❡r ❢✉♥❝t✐♦♥✳ ❊❧❡♥❛ ❨✉✳ ❇✉♥❦♦✈❛ � ❜✉♥❦♦✈❛❅♠✐✳r❛s✳r✉ � ❙t❡❦❧♦✈ ▼❛t❤❡♠❛t✐❝❛❧ ■♥st✐t✉t❡✱ ❘✉ss✐❛♥ ❆❝❛❞❡♠② ♦❢ s❝✐❡♥❝❡s✱ ▲❛❜♦r❛t♦r② ♦❢ ❣❡♦♠❡tr✐❝ ♠❡t❤♦❞s ✐♥ ♠❛t❤❡♠❛t✐❝❛❧ ♣❤②s✐❝s✱ ▼❙❯ ▼♦s❝♦✇ ✐♥ ❝♦❧❧❛❜♦r❛t✐♦♥ ✇✐t❤ ❱✳▼✳❇✉❝❤st❛❜❡r ■♥t❡r♥❛t✐♦♥❛❧ ❝♦♥❢❡r❡♥❝❡ ” ●❡♦♠❡tr②✱ ■♥t❡❣r❛❜✐❧✐t② ❛♥❞ ◗✉❛♥t✐③❛t✐♦♥ “ ✱ ✽ ❥✉♥❡ ✷✵✶✶

◆❡✇ r❡s✉❧ts ❛r❡ ♣✉❜❧✐s❤❡❞ ✐♥ ❬✶❪ ❱✳ ▼✳ ❇✉❝❤st❛❜❡r ❛♥❞ ❊✳ ❨✉✳ ❇✉♥❦♦✈❛✱ ❑r✐❝❤❡✈❡r ❢♦r♠❛❧ ❣r♦✉♣ ❧❛✇ ✱ ❋✉♥❝t✐♦♥❛❧✳ ❆♥❛❧✳ ❆♣♣❧✳✱ ✹✺✿✷ ✭✷✵✶✶✮✳ ❬✷❪ ❊✳ ❨✉✳ ❇✉♥❦♦✈❛✱ ❚❤❡ ❛❞❞✐t✐♦♥ t❤❡♦r❡♠ ❢♦r t❤❡ ❞❡❢♦r♠❡❞ ❇❛❦❡r✲❆❦❤✐❡③❡r ❢✉♥❝t✐♦♥ ✱ ❘✉ss✐❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙✉r✈❡②s✱ ✷✵✶✵✱ ✻✺✿✻✳ ❬✸❪ ❱✳ ▼✳ ❇✉❝❤st❛❜❡r✱ ❚❤❡ ❣❡♥❡r❛❧ ❑r✐❝❤❡✈❡r ❣❡♥✉s ✱ ❘✉ss✐❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙✉r✈❡②s✱ ✷✵✶✵✱ ✻✺✿✺✱ ✾✼✾✕✾✽✶✳ ❬✹❪ ❱✐❝t♦r ▼✳ ❇✉❝❤st❛❜❡r✱ ❊❧❡♥❛ ❨✉✳ ❇✉♥❦♦✈❛✱ ” ❊❧❧✐♣t✐❝ ❢♦r♠❛❧ ❣r♦✉♣ ❧❛✇s✱ ✐♥t❡❣r❛❧ ❍✐r③❡❜r✉❝❤ ❣❡♥❡r❛ ❛♥❞ ❑r✐❝❤❡✈❡r ❣❡♥❡r❛ “ ✱ ❛r❳✐✈✿✶✵✶✵✳✵✾✹✹ ❬✺❪ ❱✳ ▼✳ ❇✉❝❤st❛❜❡r ❛♥❞ ❊✳ ❨✉✳ ❇✉♥❦♦✈❛✱ ” ❆❞❞✐t✐♦♥ ❚❤❡♦r❡♠s✱ ❋♦r♠❛❧ ●r♦✉♣ ▲❛✇s ❛♥❞ ■♥t❡❣r❛❜❧❡ ❙②st❡♠s “ ✱ ❆■P ❈♦♥❢❡r❡♥❝❡ Pr♦❝❡❡❞✐♥❣s ❱♦❧✉♠❡ ✶✸✵✼✱ ❳❳■❳ ❲♦r❦s❤♦♣ ♦♥ ●❡♦♠❡tr✐❝ ▼❡t❤♦❞s ✐♥ P❤②s✐❝s✳ ✶

❚❤❡ ❢♦r♠❛❧ ❣r♦✉♣✳ ❆ ❝♦♠♠✉t❛t✐✈❡ ♦♥❡✲❞✐♠ ❢♦r♠❛❧ ❣r♦✉♣ ❧❛✇ ✭♦r s❤♦rt❧② ❢♦r♠❛❧ ❣r♦✉♣✮ ♦✈❡r ❛ ❝♦♠♠✉t❛t✐✈❡ ❛ss♦❝✐❛t✐✈❡ r✐♥❣ A ✐s t❤❡ ❢♦r♠❛❧ s❡r✐❡s � a i,j u i v j , F ( u, v ) = u + v + a i,j ∈ A, i > 0 , j > 0 , s❛t✐s❢②✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥s � � � � F ( u, v ) = F ( v, u ) , F u, F ( v, w ) = F F ( u, v ) , w . ▲❡t F a ( u, v ) = u + v ✳ ❋♦r ❛♥② ❢♦r♠❛❧ ❣r♦✉♣ F ( u, v ) ∈ A [[ u, v ]] t❤❡r❡ ❡①✐sts ❛♥ ✐s♦♠♦r♣❤✐s♠ f : F a → F ♦✈❡r A ⊗ Q ✳ ❚❤❡ s❡r✐❡s f ( t ) ∈ A ⊗ Q [[ t ]] ✉♥✐q✉❡❧② ❞❡✜♥❡❞ ❜② t❤❡ ❝♦♥❞✐t✐♦♥s � � f ′ (0) = 1 f ( t 1 + t 2 ) = F f ( t 1 ) , f ( t 2 ) f (0) = 0 , , ✐s t❤❡ ❡①♣♦♥❡♥t✐❛❧ ♦❢ t❤❡ ❢♦r♠❛❧ ❣r♦✉♣ F ( u, v ) ✳ ✷

❊①❛♠♣❧❡s ♦❢ ❢♦r♠❛❧ ❣r♦✉♣s ❛♥❞ t❤❡✐r ❡①♣♦♥❡♥t✐❛❧s✳ � � f ( t ) = 1 F ( u, v ) = u + v − µ 1 uv, 1 − exp( − µ 1 t ) . µ 1 � √ µ 2 t � u + v 1 F ( u, v ) = 1+ µ 2 uv , f ( t ) = √ µ 2 th ✳ √ √ 1 − 2 δv 2 + εv 4 + v 1 − 2 δu 2 + εu 4 F ( u, v ) = u , f ( t ) = sn ( t ) ✱ 1 − εu 2 v 2 ✇❤❡r❡ sn ( t ) ✐s t❤❡ ❏❛❝♦❜✐ ❡❧❧✐♣t✐❝ s✐♥❡✿ ( f ′ ) 2 = 1 − 2 δf 2 + εf 4 . ✸

❚❤❡ ❣❡♥❡r❛❧ ❲❡✐❡rstr❛ss ♠♦❞❡❧ ♦❢ t❤❡ ❡❧❧✐♣t✐❝ ❝✉r✈❡ Y 2 Z + µ 1 XY Z + µ 3 Y Z 2 = X 3 + µ 2 X 2 Z + µ 4 XZ 2 + µ 6 Z 3 ❞❡♣❡♥❞s ♦♥ 5 ♣❛r❛♠❡t❡rs µ = ( µ 1 , µ 2 , µ 3 , µ 4 , µ 6 ) ✳ ❚❤❡ ❣❡♦♠❡tr✐❝ ❣r♦✉♣ str✉❝t✉r❡ ” ✰ “ ♦♥ t❤❡ ❡❧❧✐♣t✐❝ ❝✉r✈❡✿ ❚❤❡ ♣♦✐♥ts P ✱ Q ✱ R ♦❢ t❤❡ ❝✉r✈❡ ❛r❡ ♦♥ ❛ str❛✐❣❤t ❧✐♥❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ P + Q + R = 0 ✳ ▲❡t O = (0 : 1 : 0) ❜❡ t❤❡ ③❡r♦ ♦❢ t❤❡ ❣❡♦♠❡tr✐❝ ❣r♦✉♣ str✉❝t✉r❡✳ ❋♦r P + Q + R = 0 ❛♥❞ R + R + O = 0 ✇❡ ❣❡t P + Q = R ✳ ✹

■♥ t❤❡ ❝❤❛rt Z � = 0 ✐♥ ❲❡✐❡rstr❛ss ❝♦♦r❞✐♥❛t❡s x = X/Z ❛♥❞ y = Y/Z t❤❡ ❝✉r✈❡ t❛❦❡s t❤❡ ❢♦r♠ y 2 + µ 1 xy + µ 3 y = x 3 + µ 2 x 2 + µ 4 x + µ 6 . ■♥ t❤❡ ❝❤❛rt Y � = 0 ✐♥ ❚❛t❡ ❝♦♦r❞✐♥❛t❡s u = − X/Y ❛♥❞ s = − Z/Y t❤❡ ❝✉r✈❡ t❛❦❡s t❤❡ ❢♦r♠ s = u 3 + µ 1 us + µ 2 u 2 s + µ 3 s 2 + µ 4 us 2 + µ 6 s 3 . ■♥ t❤❡ ❝❤❛rt X � = 0 ✇✐t❤ ❝♦♦r❞✐♥❛t❡s v = Y/X ❛♥❞ w = Z/X t❤❡ ❝✉r✈❡ t❛❦❡s t❤❡ ❢♦r♠ vw ( v + µ 1 + µ 3 w ) = 1 + µ 2 w + µ 4 w 2 + µ 6 w 3 . ❚❤❡ ❣r❛❞✐♥❣s ❛r❡ deg X = − 4 ✱ deg Y = − 6 ✱ deg Z = 0 ✱ deg µ i = − 2 i ✳ ✺

❚❤❡ ❚❛t❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ❡❧❧✐♣t✐❝ ❝✉r✈❡✳ ■♥ t❤❡ ❝❤❛rt { Y � = 0 } ✇✐t❤ u = − X/Y ❛♥❞ s = − Z/Y t❤❡ ❝✉r✈❡ t❛❦❡s t❤❡ ❢♦r♠ s = u 3 + µ 1 us + µ 2 u 2 s + µ 3 s 2 + µ 4 us 2 + µ 6 s 3 . ❲❡ ❣❡t t❤❡ ❢♦r♠❛❧ s❡r✐❡s s ( u ) ∈ Z [ µ ][[ u ]] ✿ s = u 3 + µ 1 u 4 + ( µ 2 1 + µ 2 ) u 5 + ( µ 3 1 + 2 µ 1 µ 2 + µ 3 ) u 6 + 2 + 3 µ 1 µ 3 + µ 4 ) u 7 + ... + ( µ 4 1 + 3 µ 2 1 µ 2 + µ 2 ❚❤❡ ❝♦♦r❞✐♥❛t❡s ( u, s ( u )) ❛r❡ t❤❡ ❛r✐t❤♠❡t✐❝ ❚❛t❡ ❝♦♦r❞✐♥❛t❡s✳ ❚❤❡② ❞❡✜♥❡ t❤❡ ❚❛t❡ ✉♥✐❢♦♠✐③❛t✐♦♥ ♦❢ t❤❡ ❡❧❧✐♣t✐❝ ❝✉r✈❡✳ ✻

✼

❚❤❡ ❡❧❧✐♣t✐❝ ❢♦r♠❛❧ ❣r♦✉♣✳ � � � � � � � � ▲❡t P = u, s ( u ) ✱ Q = v, s ( v ) ✱ R = w, s ( w ) ❛♥❞ R = w, s ( ¯ ¯ w ) ✳ ❚❤❡ ❣❡♦♠❡tr✐❝ ❣r♦✉♣ str✉❝t✉r❡ ♦♥ t❤❡ ❡❧❧✐♣t✐❝ ❝✉r✈❡ ❞❡✜♥❡s t❤❡ s❡r✐❡s F El ( u, v ) ♦✈❡r Z [ µ ] ✿ F El ( u, v ) = ¯ w. ❚❤❡♦r❡♠✳ ✭●❡♥❡r❛❧ ❡❧❧✐♣t✐❝ ❢♦r♠❛❧ ❣r♦✉♣✮ � � u + v − uv ( µ 1 + µ 3 m ) + ( µ 4 + 2 µ 6 m ) k F El ( u, v ) = × (1 − µ 3 k − µ 6 k 2 ) (1 + µ 2 m + µ 4 m 2 + µ 6 m 3 ) × (1 + µ 2 n + µ 4 n 2 + µ 6 n 3 )(1 − µ 3 k − µ 6 k 2 ) , � � m = s ( u ) − s ( v ) ✇❤❡r❡ u, s ( u ) ∈ V µ ❛♥❞ , u − v n = m + uv (1 + µ 2 m + µ 4 m 2 + µ 6 m 3 ) k = us ( v ) − vs ( u ) , . (1 − µ 3 k − µ 6 k 2 ) u − v ✽

❋♦r♠❛❧ ❣r♦✉♣s ❢♦r ❝✉r✈❡s ✇✐t❤ ❛✉t♦♠♦r♣❤✐s♠s✳ ❚❤❡ ♠❛♣ u → αu ❞❡✜♥❡s ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ❛♥ ❡❧❧✐♣t✐❝ ❢♦r♠❛❧ ❣r♦✉♣✳ α = 2 ✱ µ = (0 , 0 , µ 2 , µ 4 , µ 6 ) ✳ ❚❤❡♥ ( µ 2 + 2 µ 4 m + 3 µ 6 m 2 ) F ( u, v ) = u + v + b (1 + µ 2 m + µ 4 m 2 + µ 6 m 3 ) . α = 3 ✱ µ = (0 , 0 , µ 3 , 0 , µ 6 ) ✳ ❚❤❡♥ ( u + v )(1 + µ 6 m 3 ) + µ 3 m 2 + 3 µ 6 m 2 b F ( u, v ) = . (1 + µ 6 m 3 )(1 − µ 3 b ) − bm uv µ 3 (1 − µ 3 b − µ 6 b 2 ) α = 4 ✱ µ = (0 , 0 , 0 , µ 4 , 0) � = 0 ✳ α = 6 ✱ µ = (0 , 0 , 0 , 0 , µ 6 ) � = 0 f 2 ( f ′ 2 + 4 µ 4 f 4 − 1) = 0 , f ′ 3 + 3 f ′ 2 + 27 µ 6 f 6 − 4 = 0 , F ( u, v ) = u + v + 3 µ 6 m 2 b 2 µ 4 mb F ( u, v ) = u + v + 1 + µ 4 m 2 . 1 + µ 6 m 3 . ✾