Luận văn thạc sĩ hàm zêta tôpô của kì dị đường cong phẳng phức không suy biến

✣❸■ ❍➴❈ ◗❯➮❈ ●■❆ ❍⑨ ◆❐■

❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ❚Ü ◆❍■➊◆

❱ơ ❚❤à ❱➙♥

❍⑨▼ ❩❊❚❆ ❚➷P➷
❈Õ❆ ❑➐ ❉➚ ✣×❮◆● ❈❖◆● P❍➃◆● P❍Ù❈
❑❍➷◆● ❙❯❨ ❇■➌◆

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❍➔ ◆ë✐ ✲ ◆➠♠ ✷✵✷✵


✣❸■ ❍➴❈ ◗❯➮❈ ●■❆ ❍⑨ ◆❐■

❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ❚Ü ◆❍■➊◆

❱ơ ❚❤à ❱➙♥

❍⑨▼ ❩❊❚❆ ❚➷P➷
❈Õ❆ ❑➐ ❉➚ ✣×❮◆● ❈❖◆● P P

số ỵ t❤✉②➳t sè
▼➣ sè

✿ ✽✹✻✵✶✵✶✳✵✹

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

◆●×❮■ ❍×❰◆●

ị ì

ở


▲í✐ ❝↔♠ ì♥
✣➸ ❤♦➔♥ t❤➔♥❤ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❧í✐ ✤➛✉
t✐➯♥ t→❝ ❣✐↔ ①✐♥ t ỡ s s ỵ ❚❤÷í♥❣✱ ❝→♥ ❜ë
❑❤♦❛ ❚♦→♥ ✲ ❈ì ✲ ❚✐♥ ❤å❝✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ❚ü ♥❤✐➯♥ ✲ ✣↕✐ ❤å❝ ◗✉è❝
❣✐❛ ở trỹ t ữợ ❞➝♥ t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→
tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✤➸ t→❝ ❣✐↔ ❤♦➔♥ t❤✐➺♥ ❧✉➙♥ ✈➠♥ ♥➔②✳ ◆❣♦➔✐ r❛ t→❝ ❣✐↔ ①✐♥ ❝❤➙♥
t❤➔♥❤ ❝↔♠ ì♥ ❝→❝ t❤➛② ❝ỉ tr♦♥❣ ❦❤♦❛ ❚♦→♥ ✲ ❈ì ✲ ❚✐♥ ❤å❝ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ✈➔
✤â♥❣ õ ỳ ỵ qỵ t t❤➔♥❤ ❦❤â❛ ❤å❝ ✈➔ ❧✉➟♥ ✈➠♥
♥➔②✳
❈✉è✐ ❝ò♥❣ t→❝ ❣✐↔ ①✐♥ ữủ ỷ ớ ỡ t tợ ❜↕♥ ❜➧✱
♥❣÷í✐ t❤➙♥ ✤➣ ❧✉ỉ♥ ✤ë♥❣ ✈✐➯♥✱ ❝ê ✈ơ✱ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ ❝❤♦ t→❝ ❣✐↔
tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳

❍➔ ◆ë✐✱ t❤→♥❣ ✷ ♥➠♠ ✷✵✶✾
❍å❝ ✈✐➯♥ ❝❛♦ ❤å❝

❱ô ❚❤à ❱➙♥

✶


▼ư❝ ❧ư❝
▲í✐ ❝↔♠ ì♥

▲í✐ ♥â✐ ✤➛✉
✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✶
✸
✻

✶✳✶

●✐↔✐ ❦➻ ❞à ❝❤♦ ❦➻ ❞à ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✳✷

❈→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ①✉②➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✳✸

●✐↔✐ ❦➻ ❞à ❦❤æ♥❣ s✉② ❜✐➳♥ ❜➡♥❣ ❜✐➳♥ ✤ê✐ ①✉②➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵

✶✳✹

❍➔♠ ③❡t❛ tỉ♣ỉ ❝õ❛ ❦➻ ❞à ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷

✶✳✺

❱➼ ❞ö✿ ❑➻ ❞à f (x, y) = y 2 − x3 t↕✐ O ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸

✷ ❍➔♠ ③❡t❛ tỉ♣ỉ ❝õ❛ ❦➻ ❞à ✤ì♥

✻

✾

✶✼

✷✳✶

❑➻ ❞à ✤ì♥ A2n−1 ✭n ≥ 2✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼

✷✳✷

❑➻ ❞à ✤ì♥ A2n ✭n ≥ 1✮

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

✸ ❍➔♠ ③❡t❛ tæ♣æ ❝õ❛ ❦➻ ❞à ❦❤æ♥❣ s✉② ❜✐➳♥ ❝â ♣❤➛♥ ❝❤➼♥❤ tü❛
t❤✉➛♥ ♥❤➜t
✷✻
✸✳✶

❑➻ ❞à y a − xb ✈ỵ✐ (a, b) = 1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻

✸✳✷

❇✐➯♥ ◆❡✇t♦♥ ❝❤➾ ❝â ♠ët ❝↕♥❤ ❝♦♠♣➢❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵

✹ ❑➻ ❞à ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣ ♣❤ù❝ ❦❤ỉ♥❣ s✉② ❜✐➳♥

✸✺

✹✳✶

P❤➨♣ ❣✐↔✐ ①✉②➳♥ ❝❤♦ ❦➻ ❞à ❦❤æ♥❣ s✉② ❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

✹✳✷

❍➔♠ ③❡t❛ tæ♣æ ❝õ❛ ❦➻ ❞à ❦❤æ♥❣ s✉② ❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✹✷

✷


▲í✐ ♥â✐ ✤➛✉
◆➠♠ ✶✾✾✷✱ ❉❡♥❡❢ ✈➔ ▲♦❡s❡r ♣❤→t ♠✐♥❤ r❛ ởt t ợ ữủ ồ
t tổổ ✤➦❝ tr÷♥❣ ❊✉❧❡r✲P♦✐♥❝❛r➨ tỉ♣ỉ ✤÷đ❝ sû ❞ư♥❣ tr♦♥❣ ✤à♥❤
♥❣❤➽❛ ✭①❡♠ ❬✸❪✮✳ ◆â✐ ♠ët ❝→❝❤ ♥æ♠ ♥❛✱ ❤➔♠ ③❡t❛ tæ♣æ Zftop (s) ❝õ❛ ♠ët ✤❛ t❤ù❝

d ❜✐➳♥ ❤➺ sè ♣❤ù❝ f ❧➔ ♠ët ❤➔♠ ❤ú✉ t✛ ❝õ❛ s ❝❤ù❛ ♥❤ú♥❣ t❤æ♥❣ t✐♥ ✤÷đ❝ ❧➜②
r❛ tø ♠ët ♣❤➨♣ ❣✐↔✐ ❦➻ ❞à ❝õ❛ ✤❛ t↕♣ ♣❤ù❝ X0 := {x ∈ Cd | f (x) = 0}✳
❈❤ó♥❣ t❛ ♥❤➢❝ ❧↕✐ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ Zftop (s) ♥❤÷ s❛✉✳ ❈❤♦ h : Y → (X, X0 )
❧➔ ♠ët ♣❤➨♣ ❣✐↔✐ ❦➻ ❞à ❝õ❛ X0 ✳ ❑❤✐ ✤â✱ t❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ♣❤➨♣ ❣✐↔✐ ❦➻ ❞à✱

h : Y → X ❧➔ ♠ët →♥❤ ①↕ r✐➯♥❣ t❤❡♦ tæ♣æ ♣❤ù❝ ✭♥❣❤à❝❤ ↔♥❤ ❝õ❛ ♠ët t➟♣
❝♦♠♣➢❝ tr♦♥❣ X ❧➔ ♠ët t➟♣ ❝♦♠♣➢❝ tr♦♥❣ Y ✮✱ Y ❧➔ ♠ët ✤❛ t↕♣ ♣❤ù❝ trì♥✱ s❛♦
❝❤♦ →♥❤ ①↕ ❤↕♥ ❝❤➳
h : Y \ h−1 (X0 ) → X \ X0
❧➔ ♠ët ✤➥♥❣ ❝➜✉ ❣✐ú❛ ❝→❝ ✤❛ t↕♣ ✤↕✐ sè ✈➔ s❛♦ ❝❤♦ h−1 (X0 ) ❧➔ ❤ñ♣ ❝õ❛ ❝→❝

t❤➔♥❤ ♣❤➛♥ ❜➜t ❦❤↔ q✉② ♠➔ ❝❤ó♥❣ ❤♦➦❝ ❦❤ỉ♥❣ ❣✐❛♦ ♥❤❛✉ ❤♦➦❝ ❝❤➾ ❣✐❛♦ ❤♦➔♥❤
✭❣✐❛♦ ♥❤❛✉ ✈ỵ✐ ❜ë✐ ❣✐❛♦ ❜➡♥❣ 1 s❛✉ ❦❤✐ ❜ä q✉❛ sè ❜ë✐ tr➯♥ ♠é✐ t❤➔♥❤ ♣❤➛♥✮✳
❈→❝ t❤➔♥❤ ♣❤➛♥ ❜➜t ❦❤↔ q✉② ❝õ❛ h−1 (X0 ) ❝â ❤❛✐ ❧♦↕✐✿ t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t ✭♥➳✉
❝❤ó♥❣ ✤➥♥❣ ❝➜✉ ✈ỵ✐ ❦❤ỉ♥❣ ❣✐❛♥ ①↕ ↔♥❤ Pd−1
C ✮✱ t❤➔♥❤ ♣❤➛♥ t❤ü❝ sü ✭♥➳✉ ❝❤ó♥❣
d−1
✤➥♥❣ ❝➜✉ ✈ỵ✐ ❦❤æ♥❣ ❣✐❛♥ ❛❢❢✐♥❡ AC
✮✳ ●å✐ {Ei | i ∈ S} ✭✈ỵ✐ S ❧➔ ♠ët t➟♣ ❤ú✉

❤↕♥✮ ❧➔ t➟♣ t➜t ❝↔ ❝→❝ t❤➔♥❤ ♣❤➛♥ ❜➜t ❦❤↔ q✉② ❝õ❛ h−1 (X0 )✳ ●å✐ Ni ❧➔ sè ❜ë✐
❝õ❛ f ◦ h tr➯♥ Ei ✈➔ νi − 1 ❧➔ sè ❜ë✐ ❝õ❛ ❏❛❝♦❜✐❛♥ ♦❢ h tr➯♥ Ei ✳
❱ỵ✐ ♠é✐ t➟♣ ❝♦♥ I ❝õ❛ S ✱ t❛ ❦➼ ❤✐➺✉ EI ❝❤♦ t➟♣ ❣✐❛♦
❤ñ♣ EI \

Ei ✈➔ EI◦ ❝❤♦ t➟♣
j∈I Ej ✳ ❑❤✐ ✤â ❉❡♥❡❢ ✈➔ ▲♦❡s❡r ❬✸❪ ✤à♥❤ ♥❣❤➽❛ ❤➔♠ ③❡t❛ tỉ♣ỉ ❝õ❛ f

♥❤÷ s❛✉

Zftop (s) =

χ(EI◦ )
I⊆S

i∈I

i∈I

1
,

Ni s + νi

tr♦♥❣ ✤â χ ❧➔ ✤➦❝ tr÷♥❣ ❊✉❧❡r✲P♦✐♥❝❛r➨ tỉ♣ỉ✳ ❚r♦♥❣ ❬✸❪✱ ❝→❝ t→❝ ❣✐↔ ❝❤➾ r❛ r➡♥❣

Zftop (s) ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ sü ❧ü❛ ❝❤å♥ ❝õ❛ ♣❤➨♣ ❣✐↔✐ ❦➻ ❞à h ❝õ❛ X0 ✳ ❍ì♥

✸


✹

♥ú❛✱ ❤➔♠ ③❡t❛ tæ♣æ ♥➔② ❝á♥ ❧➔ ♠ët ❜➜t ❜✐➳♥ r➜t t❤ó ✈à✱ ❧✐➯♥ q✉❛♥ ✤➳♥ ●✐↔
t❤✉②➳t ✤ì♥ ✤↕♦ ✭♠ët tt q trồ tr ỵ tt ❍➻♥❤ ❤å❝
✤↕✐ sè✱ ✤÷đ❝ ♣❤→t ❜✐➸✉ ❜ð✐ ♥❤➔ t♦→♥ ❤å❝ ◆❤➟t ❇↔♥ ❏✉♥✲■❝❤✐ ■❣✉s❛ ♥❤ú♥❣ ♥➠♠
✶✾✽✵✮✳
◆➳✉ x ❧➔ ♠ët ✤✐➸♠ ❝õ❛ X0 ✱ t❛ ✤à♥❤ ♥❣❤➽❛ Sx := {i ∈ S | h(Ei ) = x}✳ ❑❤✐
✤â ❤➔♠ ③❡t❛ tæ♣æ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ f t↕✐ ✤✐➸♠ x ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉
top
Zf,x
(s) =

χ(EI◦ )
I⊆S,I∩Sx =∅

i∈I

1
.
Ni s + νi

●✐↔ t❤✉②➳t ✤ì♥ ✤↕♦ ✭♣❤✐➯♥ ❜↔♥ tæ♣æ✮ ❦➳t ♥è✐ ❝→❝ ❝ü❝ ❝õ❛ ❤➔♠ t tổổ ợ
t q trồ ừ ỵ tt ❦➻ ❞à✳ ❚ø ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ❝→❝ ❤➔♠ ③❡t❛
top
tæ♣æ✱ ♠å✐ ❝ü❝ ❝õ❛ Zftop (s) ✈➔ Zf,x
(s) ✤➲✉ ❝â ❞↕♥❣ − Nνii ✈ỵ✐ i ∈ S ✳ ❚✉② ♥❤✐➯♥✱

❜➡♥❣ r➜t ♥❤✐➲✉ ✈➼ ❞ư✱ ♥❣÷í✐ t❛ ♥❤➟♥ t❤➜② r➡♥❣ ❝â r➜t ♥❤✐➲✉ sè ❤ú✉ t✛ − Nνii ✭✈ỵ✐

top
(s)✳ ❉♦ ✤â✱ ✤➸ ❝❤ù♥❣ ♠✐♥❤ ●✐↔
i ∈ S ✮ ❦❤æ♥❣ ♣❤↔✐ ❧➔ ❝ü❝ ❝õ❛ Zftop (s) ✈➔ Zf,x
top
top
(s) ❧➔ ♠ët ❜➔✐ t♦→♥ r➜t
t❤✉②➳t ✤ì♥ ✤↕♦✱ ❜➔✐ t♦→♥ t➻♠ ❝ü❝ ❝õ❛ Zf (s) ✈➔ Zf,x
q✉❛♥ trå♥❣✳ ◆â✐ ❝❤✉♥❣✱ ❝❤♦ ✤➳♥ ♥❛②✱ ❜➔✐ t♦→♥ ♥➔② ❝❤÷❛ ✤÷đ❝ ❣✐↔✐ tr♦♥❣ tr÷í♥❣
❤đ♣ tê♥❣ q✉→t✳

❈â r➜t ♥❤✐➲✉ ❜➔✐ ❜→♦ ✈✐➳t ✈➲ ❤➔♠ ③❡t❛ tỉ♣ỉ ✈➔ ●✐↔ t❤✉②➳t ✤ì♥ ✤↕♦ ❝❤♦ ❦➻
❞à ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣ ✭tù❝ ❧➔ ❝❤♦ tr÷í♥❣ ❤đ♣ d = 2✮✱ ❝❤➥♥❣ ❤↕♥ ❬✼❪✱ ❬✶✶❪✱ ❬✶✵❪✱
❬✹❪✱ ❬✻❪✳ ❚r÷í♥❣ ❤đ♣ d = 2✱ ●✐↔ t❤✉②➳t ✤ì♥ ✤↕♦ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ tr♦♥❣ ❬✼❪✱
❬✶✵❪✱ ❬✻❪✳ ◆â✐ r✐➯♥❣✱ ♣❤÷ì♥❣ ♣❤→♣ tr♦♥❣ ❬✻❪ ❞ü❛ tr➯♥ ♥❤ú♥❣ ❤✐➸✉ ❜✐➳t q✉❛♥ trå♥❣
✈➲ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ①✉②➳♥✱ ♣❤➨♣ ❣✐↔✐ ①✉②➳♥✱ ❤➻♥❤ ❤å❝ ①✉②➳♥ ✈➔ t❤→♣
srs ữớ ữủ ợ t trữợ õ tr

r ú tỉ✐ t➻♠ ❤✐➸✉ ♣❤÷ì♥❣ ♣❤→♣ t➼♥❤ ❤➔♠ ③❡t❛ tỉ♣ỉ ✤à❛
♣❤÷ì♥❣ tr♦♥❣ ❜➔✐ ❜→♦ ❝õ❛ ❚❤÷í♥❣ ✈➔ ❍÷♥❣ ❬✻❪✳ ▼➦❝ ❞ị ❜➔✐ ❜→♦ ❬✻❪ ✤➲ ❝➟♣ ✤➳♥
❤➔♠ ③❡t❛ tæ♣æ ❝❤♦ ❦➻ ❞à ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣ tê♥❣ q✉→t✱ tr♦♥❣ ❧✉➟♥ ✈➠♥ ❝❤ó♥❣
tỉ✐ ❝❤➾ ✤å❝ ❤✐➸✉ ✈➔ tr➻♥❤ ❜➔② ❧↕✐ ♣❤÷ì♥❣ ♣❤→♣ ♥➔② ❝❤♦ ❝→❝ ❦➻ ❞à ✤÷í♥❣ ❝♦♥❣
♣❤➥♥❣ tø ❝→❝ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ♥❤÷ ❦➻ ❞à ✤ì♥ ✤➳♥ ❦➻ ❞à ❦❤ỉ♥❣ s✉② ❜✐➳♥✳

▲✉➟♥ ✈➠♥ ❝❤✐❛ t❤➔♥❤ ❜è♥ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶ tr➻♥❤ ❜➔② ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✈➲
♣❤➨♣ ❜✐➳♥ ✤ê✐ ①✉②➳♥ ✈➔ ❤➔♠ ③❡t❛ tỉ♣ỉ ❝õ❛ ❦➻ ❞à ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣✳ ❈❤÷ì♥❣ ✷
❝❤♦ ❝→❝ t➼♥❤ t♦→♥ ❝ö t❤➸ ✈➲ ❣✐↔✐ ❦➻ ❞à ✈➔ ❤➔♠ ③❡t❛ tỉ♣ỉ ❝õ❛ ❦➻ ❞à ✤ì♥✳ ❈❤÷ì♥❣
✸ tr➻♥❤ ❜➔② ✈➲ ❤➔♠ ③❡t❛ tæ♣æ ❝õ❛ ❦➻ ❞à ❦❤æ♥❣ s✉② ❜✐➳♥ ❝â ♣❤➛♥ ❝❤➼♥❤ ❧➔ ♠ët
✤❛ t❤ù❝ tü❛ t❤✉➛♥ ♥❤➜t✳ ❈✉è✐ ❝ị♥❣✱ tr♦♥❣ ❈❤÷ì♥❣ ✹✱ ❝❤ó♥❣ tỉ✐ ❦❤↔♦ s→t ❦➻ ❞à


✺

✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣ ❦❤ỉ♥❣ s✉② ❜✐➳♥ ❜➡♥❣ ❝→❝❤ sû ❞ư♥❣ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ①✉②➳♥✱
tø ✤â ♠æ t↔ ❤➔♠ ③❡t❛ tæ♣æ ❝õ❛ ❦➻ ❞à ♥➔② t❤æ♥❣ q✉❛ ✤❛ ❞✐➺♥ ◆❡✇t♦♥ ❝õ❛ ♥â✳


❈❤÷ì♥❣ ✶

❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
✶✳✶ ●✐↔✐ ❦➻ ❞à ❝❤♦ ❦➻ ❞à ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣
●✐↔✐ ❦➻ ❞à ❧➔ ♠ët ❝ỉ♥❣ ❝ư q trồ ỵ tt õ r ❤å❝ ✤↕✐
sè ♥â✐ ❝❤✉♥❣✳ ◆â ❝❤♦ ♣❤➨♣ ❝❤✉②➸♥ t❤æ♥❣ t✐♥ ❤➻♥❤ ❤å❝ ❝õ❛ ✤✐➸♠ ❦➻ ❞à t❤➔♥❤
❝→❝ t❤æ♥❣ t✐♥ tê ❤đ♣ ♥❤÷ ♠ët tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❝õ❛ s➢♣ ①➳♣ s✐➯✉ ♣❤➥♥❣✱
❝❤♦ ♣❤➨♣ ♥❣❤✐➯♥ ❝ù✉ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ✤❛ t❤ù❝ tổ q ự
ữỡ tr ỡ tự ỹ tỗ t↕✐ ❝õ❛ ❣✐↔✐ ❦➻ ❞à tr➯♥ tr÷í♥❣ ✤➦❝ sè ✵ ✤÷đ❝
❝❤ù♥❣ ♠✐♥❤ ❜ð✐ ❍✐r♦♥❛❦❛✳ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ t❛ ❝❤➾ ✤➲ ❝➟♣ ✤➳♥ ❣✐↔✐ ❦➻ ❞à ❝õ❛
✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣ ♣❤ù❝✳
●å✐ O ❧➔ ❣è❝ tå❛ ✤ë ❝õ❛ C2 ✳ ❚❛ s➩ ❦➼ ❤✐➺✉ ❜ð✐ C{x, y} ✈➔♥❤ ❝→❝ ❝❤✉é✐ ❧✉ÿ
t❤ø❛ ❤❛✐ ❜✐➳♥ ❤➺ sè ♣❤ù❝ ❤ë✐ tö tr♦♥❣ ♠ët ❧➙♥ ❝➟♥ ❝õ❛ O tr♦♥❣ C2 ✳ ❳➨t ♠ët
♣❤➛♥ tû f (x, y) ❝õ❛ C{x, y}✳ ✣➦t C = {(x, y) ∈ C2 | f (x, y) = 0}✳ ✣✐➸♠ O
✤÷đ❝ ❣å✐ ❧➔

✤✐➸♠ ❦➻ ❞à ❝õ❛ C ✭❤♦➦❝ ❝õ❛ f ✮ ♥➳✉

f (O) =

∂f
∂f
(O) =
(O) = 0.
∂x
∂y

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ❈❤♦ f (x, y) ❧➔ ♠ët ❝❤✉é✐ ❧ơ② t❤ø❛ ❤ë✐ t❤ư tr➯♥ ♠ët ❧➙♥ ❝➟♥

W ❝õ❛ O tr♦♥❣ C2 s❛♦ ❝❤♦ O ❧➔ ♠ët ✤✐➸♠ ❦➻ ❞à ❝õ❛ f ✳ ▼ët →♥❤ ①↕ π : Y → W
✤÷đ❝ ❣å✐ ❧➔ ♠ët ♣❤➨♣ ❣✐↔✐ tèt ❝õ❛ C t↕✐ O ✭❤❛② ❝õ❛ (C, O)✱ ❝õ❛ (f, O) ỡ
ỡ ừ f tỗ t↕✐ ♠ët ❧➙♥ ❝➟♥ V ❝õ❛ O s❛♦ ❝❤♦ V ⊆ W ✈➔ ❝→❝ ✤✐➲✉
s❛✉ ✤➙② ✤÷đ❝ t❤ä❛ ♠➣♥✳
✭❛✮ Y ❧➔ ♠ët ✤❛ t↕♣ trì♥✱ tù❝ ❧➔ Y ✤÷đ❝ ♣❤õ ỗ ữỡ ộ
ỗ ổ ợ (C2 ; x, y) ỗ ữủ ✈ỵ✐ ♥❤❛✉ ❜➡♥❣ ❝→❝ →♥❤
①↕ trì♥✳

✻


❈❤÷ì♥❣ ✶✳

❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✼

✭❜✮ π ❧➔ ♠ët →♥❤ ①↕ ①→❝ ✤à♥❤ ❜ð✐ ❝→❝ ❝❤✉é✐ ❧ơ② t❤ø❛ ❤ë✐ tư tr➯♥ V ✭tù❝ ❧➔ ♠ët
→♥❤ ①↕ ❣✐↔✐ t➼❝❤✮✱ π ❧➔ r✐➯♥❣ ✭♥❣❤à❝❤ ↔♥❤ ❝õ❛ ♠ët t➟♣ ❝♦♠♣➢❝ ❧➔ ♠ët t➟♣

❝♦♠♣➢❝✮✱ t♦➔♥ →♥❤ ✈➔ π|π−1 (V )\π−1 (O) : π −1 (V ) \ π −1 (O) → V \ {O} ❧➔ ởt
t
ìợ div( f ) := π −1 (C ∩ V ) ❝❤➾ ❝â ❦➻ ❞à ❧➔ ❝→❝ ✤✐➸♠ ❣✐❛♦ ❤♦➔♥❤ ❝õ❛ ❝→❝
t❤➔♥❤ ♣❤➛♥ ❜➜t ❦❤↔ q✉② ❝õ❛ ♥â✱ ❝→❝ t❤➔♥❤ ♣❤➛♥ ❜➜t ❦❤↔ q✉② ♥➔② ❧➔ trì♥
tr♦♥❣ π −1 (V )✳
❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❝❤ó♥❣ t❛ s➩ ❣å✐ ♠ët ♣❤➨♣ ❣✐↔✐ tèt ❝õ❛ f ❧➔ ♠ët ♣❤➨♣ ❣✐↔✐

❦➻ ❞à ❝õ❛ f ✳ ▼é✐ t❤➔♥❤ ♣❤➛♥ ❜➜t ❦❤↔ q✉② ❝õ❛ π−1(O) ✤÷đ❝ ❣å✐ ❧➔ ♠ët t❤➔♥❤
♣❤➛♥ ❝→ ❜✐➺t ❝õ❛ ♣❤➨♣ ❣✐↔✐ ❦➻ ❞à π✳ ▼é✐ t❤➔♥❤ ♣❤➛♥ ❜➜t ❦❤↔ q✉② ❝õ❛ ❜❛♦ ✤â♥❣
π −1 (C \ {O}) tr♦♥❣ Y ❝õ❛ π −1 (C \ {O}) ✤÷đ❝ ❣å✐ ❧➔ ♠ët t❤➔♥❤ ♣❤➛♥ t❤ü❝ sü
❝õ❛ ♣❤➨♣ ❣✐↔✐ ❦➻ ❞à π ✳
❚ø ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ♣❤➨♣ ❣✐↔✐ ❦➻ ❞à t❛ t❤➜② ♠é✐ t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t ❝õ❛ π
✤➥♥❣ ❝➜✉ ✈ỵ✐ ❦❤ỉ♥❣ ❣✐❛♥ ①↕ ↔♥❤ ♣❤ù❝ ♠ët ❝❤✐➲✉ P1 ✱ ộ t tỹ sỹ
ợ ữớ t ự C✳ ●✐↔ sû (U ; u, v) ❧➔ ♠ët ❜↔♥ ỗ ữỡ
tr Y s tr õ f ❝â ❞↕♥❣

π ∗ f (u, v) = λ(u, v)um v n ,
tr♦♥❣ ✤â λ(u, v) ❦❤→❝ 0 ✈ỵ✐ ♠å✐ (u, v) ∈ U ✳ ❑❤✐ ✤â u = 0 ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ①→❝
✤à♥❤ ♠ët t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t E õ ừ tr ỗ (U ; u, v) ỗ t
t t ởt ỗ ự ởt ừ E E tr ỗ ✤â
❜➡♥❣ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ✤à❛ ♣❤÷ì♥❣ t❤❡♦ ❝→❝❤ t÷ì♥❣ tü✳ ố m ởt t

tr ồ ỗ ợ E õ ữủ ồ số

ở ừ f tr➯♥ E ✱ t❛ s➩ ❦➼ ❤✐➺✉

sè ♥➔② ❜ð✐ N (E)✳ ◆➳✉ Cj ❧➔ ♠ët t❤➔♥❤ ♣❤➛♥ t❤ü❝ sü ✈➔ ♥➳✉ f ❧➔ rót ❣å♥ ✭❝→❝
t❤➔♥❤ ♣❤➛♥ ❜➜t ❦❤↔ q✉② ❝õ❛ f ✤➲✉ ❝â ❧ô② t❤ø❛ ❜➡♥❣ 1✮ t❤➻ sè ❜ë✐ tr➯♥ t❤➔♥❤
♣❤➛♥ ❝→ ❜✐➺t ❧✉æ♥ ❧➔ N (Cj ) = 1✳ ◆➳✉ ❦➼ ❤✐➺✉ Ei ✱ ✈ỵ✐ i t❤✉ë❝ ♠ët t➟♣ ❤ú✉ ❤↕♥

S ✱ ❧➔ t➜t ❝↔ ❝→❝ t❤➔♥❤ ♣❤➛♥ ❜➜t ❦❤↔ q✉② ❝õ❛ π −1 (C)✱ ✈➔ ❦➼ ❤✐➺✉ Ni t❤❛② ❝❤♦
N (Ei )✱ t❛ ❝â
div(π ∗ f ) = 1 (C) =
N i Ei .
iS

ú ỵ r➡♥❣✱ Ei ❝â t❤➸ ❧➔ ♠ët t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t✱ ❝ô♥❣ ❝â t❤➸ ❧➔ ♠ët t❤➔♥❤
♣❤➛♥ t❤ü❝ sü✳
❈ô♥❣ tr♦♥❣ ♠ët ỗ (U ; u, v) ữ det Jacπ ❝â ❞↕♥❣

det Jacπ (u, v) = δ(u, v)up v q ,


❈❤÷ì♥❣ ✶✳

❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✽

tr♦♥❣ ✤â δ(u, v) ❦❤→❝ 0 ✈ỵ✐ ♠å✐ (u, v) ∈ U ✳ ◆➳✉ E ❧➔ ởt t t
tr ỗ ữủ ①→❝ ✤à♥❤ ❜ð✐ ♣❤÷ì♥❣ tr➻♥❤ u = 0 t❤➻✱ t÷ì♥❣ tü ♥❤÷
tr➯♥✱ p ❧➔ ♠ët ❜➜t ❜✐➳♥ ❝❤➾ ♣❤ư t❤✉ë❝ E ổ ử tở ỗ
❦➼ ❤✐➺✉ ν(E) := p + 1✳ ❚÷ì♥❣ tü ♥❤÷ tr➯♥ t❛ ❝â

(νi − 1)Ei ,

Kπ := KY /W := div(det Jacπ ) =
i∈S

tr♦♥❣ ✤â νi := ν(Ei ) ✈ỵ✐ ♠å✐ i ∈ S ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ✣➦t
X = (x1 , x2 ; y1 : y2 ) ∈ C2 × P1 | x1 y2 = x2 y1 .
❑❤✐ ✤â

♣❤➨♣ ♥ê t➙♠ O ❝õ❛ C2 ❧➔ →♥❤ ①↕ ρ : X → C2 ①→❝ ✤à♥❤ ❜ð✐
ρ(x1 , x2 ; y1 : y2 ) = (x1 , x2 ).

✣❛ t↕♣

ρ−1 (O) = (x1 , x2 ; y1 : y2 ) ∈ C2 × P1 | (x1 , x2 ) = O, x1 y2 = x2 y1 ∼
= P1
✤÷đ❝ ❣å✐ ❧➔

t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t ❝õ❛ ♣❤➨♣ ♥ê ρ✳

❚r♦♥❣ ✤à♥❤ ♥❣❤➽❛ ✈➲ ♣❤➨♣ ♥ê ρ ð tr➯♥✱ ✤❛ t↕♣ X ✤÷đ❝ ❞→♥ tø ❤❛✐ ỗ
ữỡ

U1 = {(x2 y1 , x2 ; y1 : 1) | x2 , y1 ∈ C}
✈➔

U2 = {(x1 , x1 y2 ; 1 : y2 ) | x1 , y2 ∈ C}
t❤❡♦ ♣❤÷ì♥❣ tr➻♥❤ ❞→♥ y1 y2 = 1 ỗ t U1 ợ {(x2 , y1 ) | x2 , y1 C} ỗ
t U2 ợ {(x1 , y2 ) | x1 , y2 ∈ C}✳ ❑❤✐ ✤â ❜✐➸✉ t❤ù❝ t÷í♥❣ ♠✐♥❤ ❝õ❛ ρ tr➯♥ U1
❧➔

ρ(x2 , y1 ) = (x2 y1 , x2 ),
❜✐➸✉ t❤ù❝ t÷í♥❣ ♠✐♥❤ ❝õ❛ ρ tr➯♥ U2 ❧➔

ρ(x1 , y2 ) = (x1 , x1 y2 ).
ỵ ừ ❍✐r♦♥❛❦❛ →♣ ❞ư♥❣ ✈➔♦ ❦➻ ❞à ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣ ♣❤ù❝
❝â t❤➸ ♣❤→t ❜✐➸✉ ♥❤÷ s❛✉✿ ▼é✐ ♣❤➨♣ ❣✐↔✐ ❦➻ ❞à ❝õ❛ C t↕✐ O ❧➔ ♠ët ♣❤➨♣ ❤ñ♣
t❤➔♥❤ ❝õ❛ ♠ët sè ❤ú✉ ❤↕♥ ❝→❝ ♣❤➨♣ ♥ê ❝â ❞↕♥❣ ρ|ρ−1 (V ) : ρ−1 (V ) → V ✱ ✈ỵ✐ V
❧➔ ♠ët t➟♣ ❝♦♥ ♠ð ❝❤ù❛ O ❝õ❛ C2 ✈➔ ρ ①→❝ ✤à♥❤ ♥❤÷ tr➯♥✳


❈❤÷ì♥❣ ✶✳

❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✾

✶✳✷ ❈→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ①✉②➳♥
❳➨t ❝→❝ ❞➔♥ s❛✉ ✤➙② N = {(a, b)t | a, b ∈ Z}✱ N + = {(a, b)t | a, b ∈ Z≥0 } ✈➔

NR = N ⊗Z R = {(a, b)t | a, b ∈ R}✳ ❚➟♣ ❝♦♥ NR+ = {(a, b)t | a, b ∈ R≥0 } ❝õ❛
NR ✤÷đ❝ ❣å✐ ❧➔ ♥â♥ ❞÷ì♥❣ tr♦♥❣ NR ✳ ▼é✐ ♣❤➛♥ tû ❝õ❛ N ✤÷đ❝ ❣å✐ ❧➔ ❝→❝ ✈➨❝tì
trå♥❣ ♥❣✉②➯♥✳

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳ ▼ët ♣❤➨♣ ♣❤➙♥ ❝❤✐❛ ♥â♥ ✤ì♥ ❤➻♥❤ Σ

∗

❝õ❛ NR+ ❧➔ ♠ët ❞➣②

(T1 , ..., Tm ) ❝→❝ ✈➨❝tì trå♥❣ ♥❣✉②➯♥ sì ✭❤❛✐ t❤➔♥❤ ♣❤➛♥ ❝õ❛ ✈➨❝tì ♥❣✉②➯♥ tè
❝ị♥❣ ♥❤❛✉✮ tr♦♥❣ N + s❛♦ ❝❤♦ det(Ti , Ti+1 ) ≥ 1 ợ ồ 0 i m ợ q
ữợ T0 = (1, 0)t ✱ Tm+1 = (0, 1)t ✳ P❤➨♣ ♣❤➙♥ ❝❤✐❛ Σ∗ ✤÷đ❝ ❣å✐ ❧➔ ❝❤➼♥❤ q✉② ♥➳✉
det(Ti , Ti+1 ) = 1 ✈ỵ✐ ♠å✐ 0 ≤ i ≤ m✳

❚❤❡♦ ✤à♥❤ ♥❣❤➽❛✱ NR+ ✤÷đ❝ ♣❤õ ❜ð✐ m + 1 ♥â♥ C(Ti , Ti+1 ) = {xTi + yTi+1 |

x, y ≥ 0} ❝õ❛ Σ∗ ✳ ✣➸ ✤ì♥ ❣✐↔♥✱ tr t s ỗ t C(Ti , Ti+1 )
a b
✈ỵ✐ ♠❛ tr➟♥ ✈✉ỉ♥❣ ❝➜♣ ❤❛✐ σi = (Ti , Ti+1 )✳ ▼é✐ ♠❛ tr➟♥ σ =
❝â ✤à♥❤
c d
t❤ù❝ ❜➡♥❣ 1 ❤♦➦❝ −1 ①→❝ ✤à♥❤ ♠ët →♥❤ ①↕ Φσ : C2 → C2 ❝❤♦ ❜ð✐
Φσ (x, y) = (xa y b , xc y d ).
❱ỵ✐ ♠ët ♣❤➨♣ ♣❤➙♥ ❝❤✐❛ ♥❤â♠ ✤ì♥ ❤➻♥❤ ❝❤➼♥❤ q✉② Σ∗ ❝â ❝→❝ ✤➾♥❤ T1 , . . . , Tm ✱
t❛ ①➨t ❝→❝ ♥â♥ σi = (Ti , Ti+1 )✱ 0 ≤ i m ỗ tữỡ ự

{(C2i ; xi , yi )}✱ ✈ỵ✐ 0 ≤ i ≤ m✳ ❳➨t ❝→❝ →♥❤ ①↕ πσi : C2σi → C2 ❝❤♦ ❜ð✐ πσi = Φσi ✳
❳➨t ❤đ♣ rí✐
m

C2σi ; xi , yi ,
i=0

tr➯♥ ✤â ①➙② ❞ü♥❣ q✉❛♥ ❤➺ ∼ ♥❤÷ s❛✉✿ (xi , yi ) ∼ (xj , yj ) ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ Φσ−1 σi
j

①→❝ ✤à♥❤ t↕✐ (xi , yi ) ✈➔

Φσj−1 σi (xi , yi ) = (xj , yj ).
❚❤❡♦ ❬✾❪✱ q✉❛♥ ❤➺ ∼ ❧➔ ♠ët q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣✳ ✣➦t
m

C2σi ; xi , yi

X=

/∼

i=0

✈➔ tr❛♥❣ ❜à tỉ♣ỉ t❤÷ì♥❣ ❝❤♦ X ✳ ❑❤✐ ✤â X ❧➔ ♠ët ổ tổổ ỡ ỳ
ợ ồ ỗ tr➯♥ t❛ ❝â t❤➸ ①➙② ❞ü♥❣ ♠ët ❝➜✉ tró❝ ✈✐ ♣❤➙♥ tr➯♥ X


❈❤÷ì♥❣ ✶✳

❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✶✵

s❛♦ ❝❤♦ X ❧➔ ♠ët ✤❛ t↕♣ trì♥✳ ❈ư t❤➸ ❤ì♥✱ t❛ ❝â t❤➸ ✈✐➳t
m

C2σi ; xi , yi ,

X=
i=0

✈ỵ✐ C2σi ❧➔ ❝→❝ t➟♣ ♠ð ❝õ❛ X tr õ ỗ (C2i ; xi , yi ) ✈➔ (C2σj ; xj , yj )
♥➳✉ ❝❤ó♥❣ ❣✐❛♦ ♥❤❛✉ ❦❤→❝ ré♥❣ t❤➻ ❝❤ó♥❣ ✤÷đ❝ ❞→♥ t❤❡♦ ❝→❝❤ s❛✉ ✤➙②✿

(xi , yj ) ≡ (xj , yj ) ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ (xi , yi ) ∼ (xj , yj ).

✭✶✳✶✮

❘ã r➔♥❣ ♣❤➨♣ ❞→♥ ❧➔ trì♥ ✈➔ i tữỡ t ợ ❑❤✐
✤â t❛ ✤➦t

π(xi , yi ) = πσi (xi , yi ),

✣à♥❤ ♥❣❤➽❛ ✶✳✹✳ ⑩♥❤ ①↕ π : X → C

2

0 ≤ i ≤ m.

①→❝ ✤à♥❤ ♥❤÷ tr➯♥ ✤÷đ❝ ❣å✐ ❧➔ ♣❤➨♣

✤ê✐ ①✉②➳♥ ❧✐➯♥ ❦➳t ✈ỵ✐ ♣❤➨♣ ♣❤➙♥ ❝❤✐❛ ♥â♥ ✤ì♥ ❤➻♥❤ ❝❤➼♥❤ q✉② Σ ✳

❜✐➳♥

∗

◆❤➟♥ ①➨t r➡♥❣✱ t❤❡♦ ❬✺❪✱ ♠ët ♣❤➨♣ ❜✐➳♥ ✤ê✐ ①✉②➳♥ ❜➡♥❣ ❤ñ♣ t❤➔♥❤ ❝õ❛ ♠ët
sè ỳ ờ tr ìợ π −1 (0) ❧➔ ❤ñ♣ ❝õ❛ m t❤➔♥❤
♣❤➛♥ ❝→ ❜✐➺t E(Ti )✱ ✈ỵ✐ 1 ≤ i ≤ m ✭t❤➔♥❤ ♣❤➛♥ t E(Ti ) tữỡ ự ợ
t Ti ❝õ❛ ♣❤➨♣ ♣❤➙♥ ❝❤✐❛ ♥â♥ ✤ì♥ ❤➻♥❤ ❝❤➼♥❤ q✉② Σ∗ ✮✳ ▼é✐ t❤➔♥❤
♣❤➛♥ ❝→ ❜✐➺t E(Ti ) ✤÷đ❝ ♣❤õ ❜ð✐ ỗ C2i1 C2i ữỡ tr ừ
õ tr ỗ õ ữủt yi1 = 0 ✈➔ xi = 0✳ ❉♦ ✤â✱ ❝❤➾ ❝→❝ ✤÷í♥❣

E(Ti ) ✈➔ E(Ti+1 ) ❣✐❛♦ ♥❤❛✉ ✈➔ ✤â ❧➔ ❣✐❛♦ t ố ừ ỗ C2i
t ❦❤æ♥❣ ❝♦♠♣➢❝ E(T0 ) = {x0 = 0} ✈➔ E(Tm+1 ) = {ym = 0} ✤➥♥❣
❝➜✉ ✈ỵ✐ ❝→❝ trư❝ t♦↕ ✤ë x = 0 ✈➔ y = 0 t÷ì♥❣ ù♥❣✳

✶✳✸ ●✐↔✐ ❦➻ ❞à ❦❤æ♥❣ s✉② ❜✐➳♥ ❜➡♥❣ ❜✐➳♥ ✤ê✐ ①✉②➳♥
◆❤➢❝ ❧↕✐ r➡♥❣ C{x, y} ❧➔ ✈➔♥❤ ❝→❝ ❝❤✉é✐ ❧✉ÿ t❤ø❛ ❤❛✐ ❜✐➳♥ ❤➺ sè tr➯♥ C ❤ë✐
tö tr♦♥❣ ♠ët ❧➙♥ ❝➟♥ ❝õ❛ O tr♦♥❣ C2 ✳ ❚❤❡♦ ❬✷❪✱ ✈➔♥❤ ♥➔② ✤➥♥❣ ❝➜✉ ✈ỵ✐ ✈➔♥❤
❝→❝ ❤➔♠ ❣✐↔✐ t➼❝❤ ❤❛✐ ❜✐➳♥ tr➯♥ C2 ✱ ❞♦ ✤â t❛ ❝â t❤➸ ①❡♠ ❤❛✐ ✈➔♥❤ ❧➔ ♠ët✳
❚r♦♥❣ ♠ö❝ ♥➔② t❛ ①➨t ♠ët ❤➔♠ f (x, y) =
s❛♦ ❝❤♦ f (O) = 0✳

y tr♦♥❣ C{x, y}
f (x, y) ỗ ừ

(a,b)N2 c x

t Γ = Γ(f ; x, y) ❝õ❛

α β

t➟♣ ❤ñ♣

(α, β) + R2≥0
cαβ =0

tr♦♥❣ R2≥0 ✳ ❉➵ t❤➜② ❜✐➯♥ ❝õ❛ Γ ❝❤ù❛ ♠ët sè ❤ú✉ ❤↕♥ ❝→❝ ❝↕♥❤ ✭♠ët ❝❤✐➲✉✮✱
♠é✐ ❝↕♥❤ ♥➔② ❤♦➔♥ t♦➔♥ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ♠ët ✈➨❝tì trå♥❣ ♥❣✉②➯♥ sì P =


❈❤÷ì♥❣ ✶✳

❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✶✶

(a, b)t ∈ N + ✱ ợ (a, b) ởt tỡ t ừ ữớ t❤➥♥❣ ❝❤ù❛ ❝↕♥❤
♥➔②✳ ❑➼ ❤✐➺✉ ∂Γ ❧➔ ❜✐➯♥ ❝õ❛ Γ✳ ợ ộ P = (a, b)t ữ tr t t


d(P, f ) = min{aα + bβ | (α, β) ∈ Γ},
∆(P, f ) = {(α, β) ∈ ∂Γ | aα + bβ = d(P, f )}.

●✐↔ sû Γ ❝â m ợ trồ tữỡ ự ữủ số t tự tỹ

Pi ự trữợ Pi+1 tự det(Pi , Pi+1 ) ≥ 1✳ ❈→❝ ❤➔♠ fPi (x, y) t÷ì♥❣ ự ợ Pi
ữủ ữ s fPi (x, y) = (α,β)∈∆(Pi ,f ) cαβ xα y β ✳ ❈❤ó♥❣ ❧➔ ❝→❝ ✤❛
t❤ù❝ ❤➺ sè ♣❤ù❝ ❤❛✐ ❜✐➳♥ tü❛ t❤✉➛♥ t ổ ữủ t ữợ
ki

(y ai + i,j xbi )Ai,j ,

✭✶✳✸✮

tr♦♥❣ ✤â c˜i , ξi,j ∈ C∗ ✈ỵ✐ ♠å✐ i, j ✱ ✈➔ ξi,j = ξi,j ♥➳✉ j = j ✳

P❤➛♥ ❝❤➼♥❤ ◆❡✇t♦♥

ri si

fPi (x, y) = c˜i x y

j=1

✭❤♦➦❝ ❣å✐ ✤ì♥ ❣✐↔♥

♣❤➛♥ ❝❤➼♥❤✮ ❝õ❛ f (x, y) ❧➔ ❤➔♠ ①→❝ ✤à♥❤ ❜ð✐
cαβ xα y β .

N (f )(x, y) :=
(α,β)∈∂Γ

✣à♥❤ ♥❣❤➽❛ ✶✳✺✳ ❍➔♠ f (x, y) ✤÷đ❝ ❣å✐ ❧➔ ❦❤æ♥❣ s✉② ❜✐➳♥ ♥➳✉ A
i, j ✳ ❍➔♠ f (x, y) ✤÷đ❝ ❣å✐

✣à♥❤ ♥❣❤➽❛ ✶✳✻✳ ❈❤♦ Σ

∗

s✉② ❜✐➳♥ tỗ t i, j s Ai,j

i,j

= 1 ợ ♠å✐
2✳

❧➔ ♠ët ♣❤➨♣ ♣❤➙♥ ❝❤✐❛ ♥â♥ ✤ì♥ ❤➻♥❤ ❝❤➼♥❤ q✉② ✈ỵ✐

❝→❝ ✤➾♥❤ T1 , . . . , Tm ✳ ❇ê s✉♥❣ T0 = (1, 0)t ✈➔ Tm+1 = (0, 1)t ✳ ●✐↔ sû P1 , . . . , Pk
❧➔ t➜t ❝↔ ❝→❝ ✈➨❝tì trå♥❣ ♥❣✉②➯♥ sì ❞÷ì♥❣ t÷ì♥❣ ù♥❣ ✈ỵ✐ ❝→❝ ❝↕♥❤ ❝õ❛ ✤❛
❞✐➺♥ ◆❡✇t♦♥ Γ ❝õ❛ f ✳ ❑❤✐ ✤â Σ∗ ✤÷đ❝ ❣å✐ ❧➔

❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ✤è✐ ✈ỵ✐ f

♥➳✉

{P1 , . . . , Pk } ⊆ {T0 , T1 , . . . , Tm+1 }✳
❈❤♦ π : X → C2 ❧➔ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ①✉②➳♥ ❧✐➯♥ ❦➳t ✈ỵ✐ ♣❤➨♣ ♣❤➙♥ ❝❤✐❛ ♥â♥
✤ì♥ ❤➻♥❤ ❝❤➼♥❤ q✉② Σ∗ ✳ ❑❤✐ ✤â π ✤÷đ❝ ❣å✐ ❧➔

❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ố ợ f

ữủ ố ợ f ✳
❚❛ s➩ ❧✐➺t ❦➯ ❝→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ờ
ữủ ố ợ f ✳ ❚❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t E(Tj ) ❣✐❛♦ ❦❤→❝ ré♥❣ ✈ỵ✐ ❝→❝ t❤➔♥❤ ♣❤➛♥
t❤ü❝ sü C ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ tỗ t 1 i k s Tj = Pi ✳ ❙û ❞ư♥❣ ❦➼ ❤✐➺✉
♥❤÷ tr♦♥❣ ✭✶✳✸✮✱ t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t E(Pi ) ✭Pi = Tj ✮ ❣✐❛♦ ✈ỵ✐ ❝→❝ t❤➔♥❤ ♣❤➛♥
t❤ü❝ sü C t↕✐ ki ✤✐➸♠ (0, −ξi,s ) ∈ C2σj ✱ ✈ỵ✐ 1 ≤ s ≤ ki ✳ ❍ì♥ ♥ú❛✱ sè ❜ë✐ ❝õ❛ π ∗ f
tr➯♥ E(Tj ) ✤÷đ❝ t➼♥❤ ♥❤÷ s❛✉

N (Tj ) := N (E(Tj )) = d(Tj , f ),


ữỡ

tự



ợ d(Tj , f ) ữủ ✤à♥❤ ♥❣❤➽❛ tr♦♥❣ ✭✶✳✷✮✳ ❉♦ ✤â t❛ ❝â
m
∗

div(π f ) =

k

ki

Ci,s ,

d(Tj , f )E(Tj ) +
i=1 s=1

j=1

tr♦♥❣ ✤â Ci,s ❧➔ ❝→❝ t❤➔♥❤ ♣❤➛♥ t❤ü❝ sü ✤✐ q✉❛ ❝→❝ ✤✐➸♠ (0, −ξi,s ) ∈ C2σi ♥â✐
tr➯♥✳ ◆➳✉ f (x, y) ❜➜t ❦❤↔ q✉②✱ t❤➻ k = 1✱ k1 = 1✱ f (x, y) õ t t ữủ
ữợ

f (x, y) = (y a1 + ξ1 xb1 )A2 + ✭❝→❝ sè ❤↕♥❣ ❝❛♦ ❤ì♥✮.
◆➳✉ f ❦❤ỉ♥❣ s✉② ❜✐➳♥✱ t❤➻ Ci,j trỡ Ci,j ợ E(Pi ) r
trữớ ❤đ♣ ♥➔②✱ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ①✉②➳♥ π ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ✤è✐ ✈ỵ✐ f ❧➔ ♠ët ♣❤➨♣
❣✐↔✐ ❦➻ ❞à ❝õ❛ f ✳

✶✳✹ ❍➔♠ ③❡t❛ tỉ♣ỉ ❝õ❛ ❦➻ ❞à ✤÷í♥❣ ❝♦♥❣ ♣❤➥♥❣
❈❤♦ f (x, y) ❧➔ ♠ët ♣❤➛♥ tû ❝õ❛ C{x, y} s❛♦ ❝❤♦ O ❧➔ ♠ët ✤✐➸♠ ❦➻ ❞à ❝õ❛
♥â✳ ❑➼ ❤✐➺✉ C = {(x, y) ∈ C2 | f (x, y) = 0}✳ ❳➨t ♠ët ♣❤➨♣ ❣✐↔✐ ❦➻ ❞à tò② þ

π : Y → C2 ❝õ❛ C t↕✐ O ✈ỵ✐ ❝→❝ ❞ú ❧✐➺✉ tê ❤đ♣ ♥❤÷ s❛✉
div(π ∗ f ) := π −1 (C) =

Ni Ei
i∈S

✈➔

div(π ∗ dx ∧ dy) := div(det Jacπ ) =

(νi − 1)Ei ,
i∈S

✈ỵ✐ Ei ✱ i ∈ S ✱ ❧➔ ❝→❝ t❤➔♥❤ ♣❤➛♥ ❜➜t ❦❤↔ q✉② ❝õ❛ π −1 (C)❀ tr♦♥❣ sè ♥➔②✱ ♥➳✉

Ei ❧➔ ♠ët t❤➔♥❤ ♣❤➛♥ ❜➜t ❦❤↔ q✉② ❝õ❛ π −1 (O) t❤➻ ♥â ✤÷đ❝ ❣å✐ ❧➔ ♠ët t❤➔♥❤
♣❤➛♥ ❝→ ❜✐➺t ❝õ❛ π✱ ♥➳✉ Ei ❧➔ ♠ët t❤➔♥❤ ♣❤➛♥ ❜➜t ❦❤↔ q✉② ❝õ❛ π−1(C \ {O})
t❤➻ ✤÷đ❝ ♥â ✤÷đ❝ ❣å✐ ❧➔ ♠ët t❤➔♥❤ ♣❤➛♥ t❤ü❝ sü ❝õ❛ π ✳ ❈→❝ sè Ni i
số ữỡ ợ ồ i S ✱ ✤➦t
Ei◦ = Ei \

Ej .
j∈S,j=i

✣à♥❤ ♥❣❤➽❛ ✶✳✼✳ ❍➔♠ ③❡t❛ tæ♣æ ❝õ❛ ❦➻ ❞à f ✭❤♦➦❝ C ✮ t↕✐ O ❧➔ ❤➔♠ ❤ú✉ t✛

s❛✉ ✤➙②

top
Zf,O
(s) =
i∈S

χ(Ei◦ )
χ(Ei ∩ Ej )

+
.
Ni s + νi i,j∈S,i=j (Ni s + νi )(Nj s + νj )


❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

❈❤÷ì♥❣ ✶✳

✶✸

top
◆❤➢❝ ❧↕✐ r➡♥❣✱ ❉❡♥❡❢✲▲♦❡s❡r ✤➣ ❝❤ù♥❣ ♠✐♥❤ ❤➔♠ ③❡t❛ tỉ♣ỉ Zf,O
(s) ❦❤ỉ♥❣

♣❤ư t❤✉ë❝ ✈➔♦ ♣❤➨♣ ❣✐↔✐ ❦➻ ❞à ❝õ❛ f t↕✐ O✳

✶✳✺ ❱➼ ❞ö✿ ❑➻ ❞à f (x, y) = y

2

− x3

t↕✐ O

❉➵ t❤➜② f (x, y) ❦❤ỉ♥❣ s✉② ❜✐➳♥ ✤è✐ ✈ỵ✐ ✤❛ ❞✐➺♥ ◆❡✇t♦♥ Γ ❝õ❛ ♥â✳ ✣❛ ❞✐➺♥

Γ ❝â ♠ët ❝↕♥❤ ❝♦♠♣➢❝ ❞✉② ♥❤➜t ❧➔ ✤♦↕♥ t❤➥♥❣ ♥è✐ ❤❛✐ ✤✐➸♠ (0, 2) ✈➔ (3, 0)✱
✈➨❝tì ❞÷ì♥❣ ♥❣✉②➯♥ sì ❧➔ ♣❤→♣ t✉②➳♥ ❝õ❛ ✤÷í♥❣ t❤➥♥❣ ♥è✐ ❤❛✐ ✤✐➸♠ ♥➔② ❧➔
(2, 3)✳ ✣➦t P1 = (2, 3)t ✳ ❉♦ ✤â✱ ♠ët ♣❤➨♣ ♣❤➙♥ ❝❤✐❛ ♥â♥ ✤ì♥ q

ữủ ố ợ f ✈ỵ✐ ❝→❝ ✤➾♥❤ ✭t➼♥❤ ❝↔ ❝→❝ ✤➾♥❤ ❜ê s✉♥❣ T0
Tm+1 t q ữợ
T0 =

1
, T1 =
0

1
, T2 = P1 =
1

2
, T3 =
3

1
, T4 =
2

0
.
1

✭❱➟② tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② m = 3 ✈➔ k = 1✳✮ P❤➨♣ ♣❤➙♥ ❝❤✐❛ ♥â♥ ✤ì♥ ❤➻♥❤ Σ∗
❝❤♦ 4 ♠❛ tr➟♥ s❛✉ ✤➙② ❧➟♣ tø ❝→❝ ✤➾♥❤ ❦➳ t✐➳♣ ♥❤❛✉✿

σ0 =

1 1

, σ1 =
0 1

1 2
, σ2 =
1 3

2 1
, σ3 =
3 2

1 0
.
2 1

❑❤æ♥❣ ỗ ừ ờ ữủ ố ợ f
t trỡ

X = C20 ; x0 , y0 ∪ C2σ1 ; x1 , y1 ∪ C2σ2 ; x2 , y2 ∪ C2σ3 ; x3 , y3 ,
tr õ ỗ C2i ; xi , yi ữủ trỡ ợ t t
r ỗ C20 ; x0 , y0 tự t÷í♥❣ ♠✐♥❤ ❝õ❛ π ❧➔

π(x0 , y0 ) = Φσ0 (x0 , y0 ) = (x0 y0 , y0 ).
❑➼ ❤✐➺✉ dx ∧ dy ❧➔ ❞↕♥❣ ✈✐ ♣❤➙♥ ❝❤➼♥❤ t➢❝ tr➯♥ C2 ✳ ❑❤✐ ✤â

π ∗ f (x0 , y0 ) = y02 − (x0 y0 )3 = y02 (1 − x30 y0 ),
π ∗ (dx ∧ dy)(x0 , y0 ) = d(x0 y0 ) ∧ dy0 = (y0 dx0 + x0 dy0 ) ∧ dy0
= y0 dx0 ∧ dy0 .
❇✐➸✉ t❤ù❝ ❝õ❛ π tr➯♥ C2σ1 ; x1 , y1 ❧➔

π(x1 , y1 ) = Φσ1 (x1 , y1 ) = (x1 y12 , x1 y13 ).

✭✶✳✹✮


❈❤÷ì♥❣ ✶✳

❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✶✹

❑❤✐ ✤â

π ∗ f (x1 , y1 ) = (x1 y13 )2 − (x1 y12 )3 = x21 y16 (1 − x1 ),
π ∗ (dx ∧ dy)(x1 , y1 ) = d(x1 y12 ) ∧ d(x1 y13 ) = x1 y14 dx1 ∧ dy1 .

✭✶✳✺✮

❍❛✐ ❜↔♥ ỗ C20 ; x0 , y0 C21 ; x1 , y1 ♣❤õ t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t E(T1 )❀ tr♦♥❣
❜↔♥ ỗ tự t E(T1 ) ữỡ tr y0 = 0 tr ỗ
tự E(T1 ) ①→❝ ✤à♥❤ ❜ð✐ ♣❤÷ì♥❣ tr➻♥❤ x1 = 0✳ ❚❤❡♦ ❝→❝ t➼♥❤ t♦→♥ ✭✶✳✹✮ ✈➔
✭✶✳✺✮✱ ❝→❝ ❞ú ❧✐➺✉ tê ❤ñ♣ N (T1 ) ✈➔ ν(T1 ) ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿

N (T1 ) = 2,

ν(T1 ) = 1 + 1 = 2.

❇✐➸✉ t❤ù❝ ❝õ❛ π tr➯♥ C2σ2 ; x2 , y2 ❧➔

π(x2 , y2 ) = Φσ2 (x2 , y2 ) = (x22 y2 , x32 y22 ).

❑❤✐ ✤â

π ∗ f (x2 , y2 ) = (x32 y22 )2 − (x22 y2 )3 = x62 y23 (y2 − 1),
π ∗ (dx ∧ dy)(x2 , y2 ) = d(x22 y2 ) ∧ d(x32 y22 ) = x42 y22 dx2 ∧ dy2 .

✭✶✳✻✮

❍❛✐ ỗ C21 ; x1 , y1 C22 ; x2 , y2 ♣❤õ t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t E(T2 ) = E(P1 )
tr ỗ tự t E(T2 ) ữỡ tr y1 = 0 tr
ỗ t❤ù ❤❛✐✱ E(T2 ) ①→❝ ✤à♥❤ ❜ð✐ ♣❤÷ì♥❣ tr➻♥❤ x2 = 0✳ ❚❤❡♦ ❝→❝ t➼♥❤ t♦→♥ ✭✶✳✺✮
✈➔ ✭✶✳✻✮✱ ❝→❝ ❞ú ❧✐➺✉ tê ❤ñ♣ N (T2 ) = N (P1 ) ✈➔ ν(T2 ) = ν(P1 ) ✤÷đ❝ ①→❝ ✤à♥❤
♥❤÷ s❛✉✿

N (T2 ) = N (P1 ) = 6,

ν(T2 ) = ν(P1 ) = 4 + 1 = 5.

❱➻ P1 t÷ì♥❣ ù♥❣ ✈ỵ✐ ❝↕♥❤ ❝♦♠♣➢❝ ❞✉② ♥❤➜t ❝õ❛ Γ ♥➯♥ E(T2 ) = E(P1 ) ❧➔
t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t ❞✉② ♥❤➜t ❣✐❛♦ ✈ỵ✐ t❤➔♥❤ ♣❤➛♥ t❤ü❝ sü ✭❦➼ ❤✐➺✉ E0 ✮ ❝õ❛ ♣❤➨♣
❣✐↔✐ ❦➻ ❞à π ❀ ✤✐➸♠ ❣✐❛♦ ❞✉② ♥❤➜t õ tồ ở (1, 0) tr ỗ C21 ; x1 , y1 ✱ ✈➔ ❝â
tå❛ ✤ë (0, 1) tr♦♥❣ ỗ C22 ; x2 , y2 ró r r (x1 = 1, y1 = 0) ≡ (x2 =

0, y2 = 1) t❤❡♦ ❧✉➟t ❞→♥ ✭✶✳✶✮✮✳ ❚ø ❝→❝ ❜✐➸✉ t❤ù❝ ♥➔② t❛ ❝ô♥❣ ❝â N (E0 ) = 1 ✈➔
ν(E0 ) = 0 + 1 = 1✳
❇✐➸✉ t❤ù❝ ❝õ❛ π tr➯♥ C2σ3 ; x3 , y3 ❧➔

π(x3 , y3 ) = Φσ3 (x3 , y3 ) = (x3 , x23 y3 ).
❑❤✐ ✤â

π ∗ f (x3 , y3 ) = (x23 y3 )2 − x33 = x33 (x3 y32 − 1),

π ∗ (dx ∧ dy)(x3 , y3 ) = dx3 ∧ d(x23 y3 ) = x23 dx3 ∧ dy3 .

✭✶✳✼✮


ữỡ

tự



ỗ C22 ; x2 , y2 ✈➔ C2σ3 ; x3 , y3 ♣❤õ t❤➔♥❤ t E(T3 ) tr
ỗ tự t E(T3 ) ①→❝ ✤à♥❤ ❜ð✐ ♣❤÷ì♥❣ tr➻♥❤ y2 = 0✱ ✈➔ tr ỗ
tự E(T3 ) ữỡ tr➻♥❤ x3 = 0✳ ❚❤❡♦ ❝→❝ t➼♥❤ t♦→♥ ✭✶✳✻✮ ✈➔
✭✶✳✼✮✱ ❝→❝ ❞ú ❧✐➺✉ tê ❤ñ♣ N (T3 ) ✈➔ ν(T3 ) ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿

N (T3 ) = 3,

ν(T3 ) = 2 + 1 = 3.

❙❛✉ ❦❤✐ ♣❤➙♥ t➼❝❤ ✈➲ ❤➻♥❤ ❤å❝ ✈➔ tê ❤ñ♣ ❝õ❛ ♣❤➨♣ ❣✐↔✐ ❦➻ ❞à π ❝õ❛ ❦➻ ❞à ❝õ❛

C := {(x, y) ∈ C2 | y 2 − x3 = 0 t↕✐ O ♥❤÷ tr➯♥✱ t❛ ❝â t❤➸ ♠✐♥❤ ❤å❛ ✈✐➺❝ s➢♣
①➳♣ ❝→❝ t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t E(T1 )✱ E(T2 ) = E(P1 )✱ E(T3 ) ✈➔ t❤➔♥❤ ♣❤➛♥ t❤ü❝
sü ✭❦➼ ❤✐➺✉ E0 tr ữủ ỗ s r ộ ừ ữủ ỗ tổ
số E (N, )


E(P1 ) (6, 5)

E(T1 ) (2, 2)

E0 (1, 1)

E(T3 ) (3, 3)

top
(s) ❝õ❛ f (x, y) = y 2 − x3 t↕✐ O✳ ❚❛
❙❛✉ ✤➙② t❛ s➩ t➼♥❤ ❤➔♠ ③❡t❛ tæ♣æ Zf,O

❝â ❝→❝ ỗ ổ s ừ ổ tổổ

E(T1 ) ∼
= E(T3 )◦ ∼
= R2 , E(P1 )◦ ∼
= R2 \ (✷ ✤✐➸♠), E0◦ ∼
= R2 \ (✶ ✤✐➸♠).
P❤➨♣ t❛♠ ❣✐→❝ ♣❤➙♥ ✤ì♥ ❣✐↔♥ ♥❤➜t ❝õ❛ R2 ❝❤♦ 1 ♠➦t ✭❝❤➼♥❤ ❧➔ R2 ✮✱ 0 ❝↕♥❤✱

0 ✤➾♥❤❀ ❝❤♦ ♥➯♥ ✤➦❝ tr÷♥❣ ❊✉❧❡r ❝õ❛ ♥â ❜➡♥❣ 1✳ ▼ët ♣❤➨♣ t❛♠
❝õ❛ R2 \ (✷ ✤✐➸♠) ❝❤♦ 2 ♠➦t✱ 3 ❝↕♥❤✱ 0 ✤➾♥❤❀ ❝❤♦ ♥➯♥ ✤➦❝ tr÷♥❣
R2 \ (✷ ✤✐➸♠) ❜➡♥❣ 2 − 3 + 0 = −1✳ ❈✉è✐ ❝ò♥❣✱ ♠ët ♣❤➨♣ t❛♠
❝õ❛ R2 \ (✶ ✤✐➸♠) ❝❤♦ 2 ♠➦t✱ 2 ❝↕♥❤✱ 0 ✤➾♥❤❀ ❝❤♦ ♥➯♥ ✤➦❝ tr÷♥❣
R2 \ (✶ ✤✐➸♠) ❜➡♥❣ 2 2 + 0 = 0 st ữủ ỗ tr➯♥ t❛ ❝â
top
Zf,O
(s)

❣✐→❝ ♣❤➙♥
❊✉❧❡r ❝õ❛

❣✐→❝ ♣❤➙♥
❊✉❧❡r ❝õ❛

χ(E(T1 )◦ ) χ(E(P1 )◦ ) χ(E(T3 )◦ ) χ(E0◦ )
=
+
+
+
+
2s + 2
6s + 5
3s + 3
s+1
χ(E(T1 ) ∩ E(P1 )) χ(E(T3 ) ∩ E(P1 )) χ(E0 ∩ E(P1 ))
+
+
+
(2s + 2)(6s + 5)
(3s + 3)(6s + 5)
(s + 1)(6s + 5)
1
1
1
=
−
+
+
2s + 2 6s + 5 3s + 3
1
1

1
+
+
+
(2s + 2)(6s + 5) (3s + 3)(6s + 5) (s + 1)(6s + 5)


❈❤÷ì♥❣ ✶✳

❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

5(6s + 5) + 11 − 6(s + 1)
6(s + 1)(6s + 5)
24s + 30
=
6(s + 1)(6s + 5)
4s + 5
=
.
(s + 1)(6s + 5)
=

✶✻


❈❤÷ì♥❣ ✷

❍➔♠ ③❡t❛ tỉ♣ỉ ❝õ❛ ❦➻ ❞à ✤ì♥
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ t❛ t➼♥❤ ❤➔♠ ③❡t❛ tỉ♣ỉ ❝❤♦ ❦➻ ❞à ✤ì♥ Ak ✭♣❤÷ì♥❣ tr➻♥❤
✤à❛ ♣❤÷ì♥❣ y 2 − xk+1 = 0✮✱ tø ✤â t❛ ❝â t❤➸ ①→❝ ✤à♥❤ ✤÷đ❝ ❝ü❝ ❝õ❛ ♥â✳

✷✳✶ ❑➻ ❞à ✤ì♥ A

2n−1

✭n ≥ 2✮

❑➳t q✉↔ ❝❤➼♥❤ ❝õ❛ ♠ư❝ ỵ s

ỵ t tæ♣æ ❝õ❛ ❦➻ ❞à f (x, y) = y
top
(s) =
Zf,O

2

− x2n

t↕✐ O ∈ C2 ❜➡♥❣

(n − 1)s + n + 1
.
(s + 1)(2ns + n + 1)

❈❤ù♥❣ ♠✐♥❤✳ ❑à ❞à ✤ì♥ f (x, y) = y2 − x2n t↕✐ O ❧➔ ♠ët ❦➻ ❞à ❦❤ỉ♥❣ s✉② ❜✐➳♥
✤è✐ ✈ỵ✐ ✤❛ ❞✐➺♥ ◆❡✇t♦♥ Γ ❝õ❛ ♥â✳ ❚❤ü❝ r❛✱ Γ ❝❤➾ ❝â ♠ët ❝↕♥❤ ❝♦♠♣➢❝ ❞✉②
♥❤➜t ✈ỵ✐ ✈➨❝tì ♣❤→♣ t✉②➳♥ (1, n) ✭t❛ ✤➦t P1 = (1, n)t ✮✳ ❚❛ ❝â t❤➸ ❣✐↔✐ ❦➻ ❞à ♥➔②
❜➡♥❣ ♠ët ♣❤➨♣ ❜✐➳♥ ✤ê✐ ①✉②➳♥ ❝❤➜♣ ♥❤➟♥ ữủ ố ợ f t
õ ỡ q ợ ữ s

T0 =

1
, T1 =
0

1
, T2 =
1

1
, . . . , Tn =
2

1
, Tn+1 =
n

0
.
1

P❤➨♣ ♣❤➙♥ ❝❤✐❛ ♥â♥ ✤ì♥ ❤➻♥❤ Σ∗ ❝❤♦ n + 1 ♠❛ tr➟♥ s❛✉ ✤➙② ❧➟♣ tø ❝→❝ ✤➾♥❤
❦➳ t✐➳♣ ♥❤❛✉✿

σ0 =

1 1
, σ1 =
0 1

1 1
, . . . , σn−1 =
1 2

1
1
, σn =
n−1 n

❳➨t ♣❤➨♣ ❜✐➳♥ ờ t ợ ữ s
n

C2i ; xi , yi → C2 ; x, y ,

π : X :=
i=0

✶✼

1 0
.
n 1


ữỡ

t tổổ ừ ỡ

ợ X ởt t ự trỡ tứ ỗ ✤à❛ ♣❤÷ì♥❣ C2σi ; xi , yi ✱

0 ≤ i ≤ n✱ t❤❡♦ ❧✉➟t ❞→♥ ✭✶✳✶✮✳ ❚❛ t➼♥❤ div(π ∗ f ) ✈➔ div(π ∗ (dx ∧ dy)) tr➯♥ ♠é✐
❜↔♥ ỗ ữỡ ợ 0 i n 1 tự tồ ở ừ tr ỗ
C2i ; xi , yi ❧➔
π(xi , yi ) = Φσi (xi , yi ) = (xi yi , xii yii+1 ).
õ tr ỗ

f (xi , yi ) = (xii yii+1 )2 − (xi yi )2n
2i+2
n−i−1
n−i−1
= x2i
(1 + xn−i
)(1 − xn−i
),
i yi
i yi
i yi

π ∗ (dx ∧ dy)(xi , yi ) = d(xi yi ) ∧ d(xii yii+1 )
= xii yii+1 dxi ∧ dyi .
❇✐➸✉ t❤ù❝ tồ ở ừ tr ỗ C2n ; xn , yn ❧➔

π(xn , yn ) = Φσn (xn , yn ) = (xn , xnn yn ).
❉♦ ✤â✱ tr➯♥ C2σn ; xn , yn ✱ t❛ ❝â

π ∗ f (xn , yn ) = (xnn yn )2 − x2n
n

= x2n
n (yn + 1)(yn − 1),
π ∗ (dx ∧ dy)(xn , yn ) = dxn ∧ d(xnn yn )
= xn dxn ∧ dyn .
❱ỵ✐ ♠é✐ 1 ≤ i ≤ n✱ ỗ (C2i1 ; xi1 , yi1 ) (C2σi ; xi , yi ) ♣❤õ t❤➔♥❤
♣❤➛♥ ❝→ ❜✐➺t E(Ti ) tr ỗ tự t E(Ti ) ữỡ tr

yi1 = 0 tr ỗ t❤ù ❤❛✐✱ E(Ti ) ①→❝ ✤à♥❤ ❜ð✐ ♣❤÷ì♥❣ tr➻♥❤ xi = 0✳
❚❤❡♦ ❝→❝ t➼♥❤ t♦→♥ tr➯♥✱ ✈ỵ✐ ♠å✐ 1 ≤ i ≤ n✱ ❝→❝ ❞ú ❧✐➺✉ tê ❤ñ♣ N (Ti ) ✈➔ ν(Ti )
✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿
N (Ti ) = 2i,

ν(Ti ) = i + 1.

❈→❝ t❤➔♥❤ ♣❤➛♥ t❤ü❝ sü ❝õ❛ π ❧➔ ❝→❝ t❤➔♥❤ ♣❤➛♥ ❜➜t ❦❤↔ q✉② ừ ữớ
tr X tr ỗ C2i ; xi , yi ✱ ✈ỵ✐ 0 ≤ i ≤ n − 1✱ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐
n−i−1
n−i−1
(1 + xn−i
)(1 − xni
) = 0,
i yi
i yi

tr ỗ C2n ; xn , yn ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐

(yn + 1)(yn − 1) = 0.
❉♦ ✤â ❝❤➾ ❝â ❤❛✐ t❤➔♥❤ ♣❤➛♥ t❤ü❝ sü ❝õ❛ π ✱ t❛ s➩ ❦➼ ❤✐➺✉ ❧➛♥ ❧÷đt ❧➔ E01 ✈➔

E02 ✳ P❤÷ì♥❣ tr➻♥❤ ❝õ❛ E01 ✈➔ E02 ✤÷đ❝ ❝❤♦ tr♦♥❣ ❜↔♥❣ s❛✉ ✤➙②✿

ữỡ

t tổổ ừ ỡ



ỗ
0in1

C2i ; xi , yi

i=n

C2σn ; xn , yn

Pt✳ ❝õ❛ E01

Pt✳ ❝õ❛ E02

n−i−1
n−i−1
1 + xn−i
= 0 1 − xn−i
=0
i yi
i yi

1 + yn = 0

1 − yn = 0

❘ã r➔♥❣ N (E01 ) = 1✱ N (E02 ) = 1✱ ν(E01 ) = 1 ✈➔ ν(E02 ) = 1✳ ❍ì♥ ♥ú❛✱ E01
✈➔ E02 ❝❤➾ ❣✐❛♦ ✈ỵ✐ t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t E(Tn ) = E(P1 ) t↕✐ ❤❛✐ ✤✐➸♠ ❣✐❛♦ ❦❤→❝
♥❤❛✉✱ ❝❤➥♥❣ ❤↕♥ tr ỗ C2n ; xn , yn ❣✐❛♦ ✤â ❝â tå❛ ✤ë ❧➛♥ ❧÷đt
❧➔ (0, −1) ✈➔ (0, 1)✳
❚❛ ❝â t❤➸ ♠✐♥❤ ❤å❛ ✈✐➺❝ s➢♣ ①➳♣ ❝→❝ t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t E(Ti )✱ 1 ≤ i ≤ n✱
✈➔ ❤❛✐ t❤➔♥❤ ♣❤➛♥ t❤ü❝ sü E01 ✈➔ E02 tr♦♥❣ ữủ ỗ s r ộ
ừ ữủ ỗ ❝→❝ t❤æ♥❣ sè E (N, ν)✳

E01 (1, 1)
✲

E(Ti ) (2i, i + 1)

E(Tn−1 )

✳✳✳

✲

E02 (1, 1)

E(T1 ) (2, 2)

E(P1 ) (2n, n + 1)
◆❤÷ tr♦♥❣ ▼ư❝ ✶✳✺✱ t❛ ❞➵ ❞➔♥❣ t➼♥❤ ✤÷đ❝ ✤➦❝ tr÷♥❣ ❊✉❧❡r ❝õ❛ ❣✐❛♦ ❝õ❛ ❝→❝
t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t ✈➔ t❤➔♥❤ ♣❤➛♥ t❤ü❝ sü✳ ❙❛✉ ✤➙② ❧➔ ❜↔♥❣ t➼♥❤ t♦→♥✿

●✐❛♦

✣➦❝ tr÷♥❣ ❊✉❧❡r

✣✐➲✉ ❦✐➺♥

E(T1 )◦

1

E(Tn )◦ = E(P1 )◦

−1

E(Ti )◦

0

◦
◦
E01
✱ E02
✱ E01 ∩ E02 = ∅

0

E(Ti ) ∩ E(Ti+1 ) = (1 ✤✐➸♠)

1

1≤i≤n−1

E(Ti ) ∩ E(Tj ) = ∅

0

|i − j| ≥ 2

E01 ∩ E(Ti ) = E02 ∩ E(Ti ) = ∅

0

1≤i≤n−1

E01 ∩ E(Tn )✱ E02 ∩ E(Tn )

1

2≤i≤n−1

❉♦ ✤â ❤➔♠ ③❡t❛ tæ♣æ ❝õ❛ f (x, y) = y 2 − x2n t↕✐ O ❧➔
top
Zf,O
(s)

χ(E(T1 )◦ )
χ(E(Tn )◦ )
=
+
+

2s + 2
2ns + n + 1

n−1

i=1

χ(E(Ti ) ∩ E(Ti+1 ))
(2is + i + 1)((2i + 2)s + i + 2)


❈❤÷ì♥❣ ✷✳

❍➔♠ ③❡t❛ tỉ♣ỉ ❝õ❛ ❦➻ ❞à ✤ì♥

✷✵

χ(E01 ∩ E(Tn ))
χ(E02 ∩ E(Tn ))
+
(s + 1)(2ns + n + 1) (s + 1)(2ns + n + 1)
1
1
n
1
=
−
+
−
2s + 2 2ns + n + 1

2ns + n + 1 2s + 2
2
+
(s + 1)(2ns + n + 1)
(n − 1)s + n + 1
=
.
(s + 1)(2ns + n + 1)
+

ỵ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳

✷✳✷ ❑➻ ❞à ✤ì♥ A ✭n ≥ 1✮
2n

❑➳t q ừ ử ỵ s

ỵ t tổổ ừ f (x, y) = y

2

− x2n+1

t↕✐ O ∈ C2 ❜➡♥❣

2(n + 1)s + 2n + 3
.
(s + 1)((4n + 2)s + 2n + 3)

top

Zf,O
(s) =

❈❤ù♥❣ ♠✐♥❤✳ ❑à ❞à ✤ì♥ f (x, y) = y2 − x2n+1 t↕✐ O ❦❤æ♥❣ s✉② ❜✐➳♥ ✈➻ ✤❛ ❞✐➺♥
◆❡✇t♦♥ Γ ❝õ❛ ♥â ❝❤➾ ❝â ♠ët ❝↕♥❤ ❝♦♠♣➢❝✳ ❱➨❝tì ♣❤→♣ t✉②➳♥ ❝õ❛ ❝↕♥❤ ♥➔② ❧➔

(2, 2n + 1)✱ t❛ ✤➦t P1 = (2, 2n + 1)t ✳ ❳➨t ♣❤➨♣ ♣❤➙♥ ❝❤✐❛ ♥â♥ ✤ì♥ ❤➻♥❤ ❝❤➼♥❤
q✉② Σ∗ ❝❤➜♣ ♥❤➟♥ ữủ ố ợ f õ ữ s
Ti =

1
i

(0 ≤ i ≤ n), Tn+1 = P1 =
Tn+2 =

1
, Tn+3 =
n+1

2
,
2n + 1

0
.
1

❚❛ ✤➦t

σi =

1
1
i i+1

✈ỵ✐ 0 ≤ i ≤ n − 1,

✈➔

σn =

1
2
, σn+1 =
n 2n + 1

2
1
, σn+2 =
2n + 1 n + 1

❳➨t ♣❤➨♣ ❣✐↔✐ ❦➻ ❞à ①✉②➳♥ ❧✐➯♥ ❦➳t ✈ỵ✐ Σ∗ s❛✉ ✤➙②

π : X → C2 ; x, y ,

1
0
.
n+1 1

ữỡ

t tổổ ừ ỡ



ợ X ởt t ự trỡ tứ ỗ C2σi ; xi , yi ✱ 0 ≤ i ≤ n + 2✱
t❤❡♦ ❧✉➟t ❞→♥ ✭✶✳✶✮✳ ❇✐➸✉ t❤ù❝ ❝õ❛ π tr➯♥ C2σi ; xi , yi ✱ 0 ≤ i ≤ n − 1✱ ❧➔

π(xi , yi ) = Φσi (xi , yi ) = (xi yi , xii yii+1 ).
õ tr ỗ

f (xi , yi ) = (xii yii+1 )2 − (xi yi )2n+1
2i+2
= x2i
(1 − x2n−2i+1
yi2n−2i−1 ),
i yi
i

π ∗ (dx ∧ dy)(xi , yi ) = d(xi yi ) ∧ d(xii yii+1 )
= xii yii+1 dxi ∧ dyi .
❇✐➸✉ t❤ù❝ ❝õ❛ π tr➯♥ ỗ C2n ; xn , yn

(xn , yn ) = (xn yn2 , xnn yn2n+1 ).
❉♦ ✤â✱ tr➯♥ ỗ C2n ; xn , yn t õ

∗ f (xn , yn ) = (xnn yn2n+1 )2 − (xn yn2 )2n+1

4n+2
= x2n
(1 − xn ),
n yn

π ∗ (dx ∧ dy)(xn , yn ) = d(xn yn2 ) ∧ d(xnn yn2n+1 )
= xnn yn2n+2 dxn ∧ dyn .
❱ỵ✐ ộ 1 i n ỗ (C2i1 ; xi−1 , yi−1 ) ✈➔ (C2σi ; xi , yi ) ừ E(Ti )
tr ỗ tự t E(Ti ) ①→❝ ✤à♥❤ ❜ð✐ ♣❤÷ì♥❣ tr➻♥❤ yi−1 = 0✱ ✈➔ tr
ỗ tự E(Ti ) xi = 0✳ ❚❤❡♦ ❝→❝ t➼♥❤ t♦→♥ tr➯♥✱ ✈ỵ✐ ♠å✐

1 ≤ i ≤ n✱ ❝→❝ ❞ú ❧✐➺✉ tê ❤ñ♣ N (Ti ) ✈➔ ν(Ti ) ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿
N (Ti ) = 2i,

ν(Ti ) = i + 1.

❇✐➸✉ t❤ù❝ ❝õ❛ tr ỗ (C2n+1 ; xn+1 , yn+1 ) ❧➔
n+1
π(xn+1 , yn+1 ) = (x2n+1 yn+1 , x2n+1
n+1 yn+1 ).

õ tr ỗ (C2n+1 ; xn+1 , yn+1 )✱ t❛ ❝â
n+1 2
2
2n+1
π ∗ f (xn+1 , yn+1 ) = (x2n+1
n+1 yn+1 ) − (xn+1 yn+1 )
2n+1
= x4n+2
n+1 yn+1 (yn+1 − 1),

n+1
π ∗ (dx ∧ dy)(xn+1 , yn+1 ) = d(x2n+1 yn+1 ) ∧ d(x2n+1
n+1 yn+1 )
n+1
= x2n+2
n+1 yn+1 dxn+1 ∧ dyn+1 .


❈❤÷ì♥❣ ✷✳

❍➔♠ ③❡t❛ tỉ♣ỉ ❝õ❛ ❦➻ ❞à ✤ì♥

✷✷

❚❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t E(Tn+1 ) = E(P1 ) ✤÷đ❝ ♣❤õ ❜ð✐ ❤❛✐ ỗ C2n ; xn , yn

C2n+1 ; xn+1 , yn+1 tr ỗ tự t ữỡ tr ①→❝ ✤à♥❤ E(Tn+1 ) ❧➔
yn = 0✱ ✈➔ tr♦♥❣ ❜↔♥ ỗ tự ữỡ tr E(Tn+1 ) xn+1 = 0✳
❈→❝ ❞ú ❧✐➺✉ tê ❤ñ♣ N (Tn+1 ) ✈➔ ν(Tn+1 ) ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿
N (Tn+1 ) = N (P1 ) = 4n + 2,
ν(Tn+1 ) = ν(P1 ) = (2n + 2) + 1 = 2n + 3.
❱➻ P1 ❧➔ ù♥❣ ✈ỵ✐ ❝↕♥❤ ❝♦♠♣➢❝ ❞✉② ♥❤➜t ❝õ❛ Γ ♥➯♥ E(Tn+1 ) = E(P1 ) ❧➔ t❤➔♥❤
♣❤➛♥ ❝→ ❜✐➺t ❞✉② ♥❤➜t ❣✐❛♦ ✈ỵ✐ t❤➔♥❤ ♣❤➛♥ t❤ü❝ sü✳ ❉➵ t❤➜② r➡♥❣ ❝❤➾ ❝â ❞✉②
♥❤➜t ♠ët t❤➔♥❤ ♣❤➛♥ t❤ü❝ ✭❦➼ ❤✐➺✉ E0 ✮ ❝õ❛ π ✈➔

N (E0 ) = 1,

ν(E0 ) = 1.

❈✉è✐ ❝ò♥❣✱ ❜✐➸✉ t❤ù❝ ❝õ❛ π tr➯♥ ỗ C2n+2 ; xn+2 , yn+2

(xn+2 , yn+2 ) = (xn+2 , xn+1
n+2 yn+2 ).
❉♦ ✤â✱ tr➯♥ ❜↔♥ ỗ C2n+2 ; xn+2 , yn+2 t õ
2
2n+1
f (xn+2 , yn+2 ) = (xn+1
n+2 yn+2 ) − xn+2
2
= x2n+1
n+2 (xn+2 yn+2 − 1),

π ∗ (dx ∧ dy)(xn+2 , yn+2 ) = dxn+2 ∧ d(xn+1
n+2 yn+2 )
= xn+1
n+2 dxn+2 ∧ dyn+2 .
❚❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t E(Tn+2 ) ✤÷đ❝ ừ ỗ ữỡ

(C2n+1 ; xn+1 , yn+1 ) ✈➔ (C2σn+2 ; xn+2 , yn+2 )❀ tr♦♥❣ ỗ tự t ữỡ tr
E(Tn+2 ) yn+1 = 0 tr ỗ tự ữỡ tr➻♥❤ ①→❝ ✤à♥❤
E(Tn+2 ) ❧➔ xn+2 = 0✳ ❈→❝ ❞ú ❧✐➺✉ tê ❤đ♣ N (Tn+2 ) ✈➔ ν(Tn+2 ) ✤÷đ❝ ①→❝ ✤à♥❤
♥❤÷ s❛✉✿
N (Tn+2 ) = 2n + 1,
ν(Tn+1 ) = (n + 1) + 1 = n + 2.
❚❛ ♠✐♥❤ ❤å❛ ✈✐➺❝ s➢♣ ①➳♣ ❝→❝ t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t E(Ti )✱ 1 ≤ i ≤ n + 2✱ ✈➔
t❤➔♥❤ tỹ sỹ E0 tr ữủ ỗ s r ộ ừ ữủ ỗ
tổ số E (N, ν)✳


❈❤÷ì♥❣ ✷✳

❍➔♠ ③❡t❛ tỉ♣ỉ ❝õ❛ ❦➻ ❞à ✤ì♥

✷✸

E(Tn ) (2n, n + 1)

✳✳✳

E(Ti ) (2i, i + 1)

✲E0

(1, 1)

E(Tn+2 ) (2n + 1, n + 2)

E(T1 ) (2, 2)

E(P1 ) (4n + 2, 2n + 3)
❙❛✉ ✤➙② ❧➔ ❜↔♥❣ t➼♥❤ ✤➦❝ tr÷♥❣ ❊✉❧❡r ❝õ❛ ❣✐❛♦ ❝õ❛ ❝→❝ t❤➔♥❤ ♣❤➛♥ ❝→ ❜✐➺t
✈➔ t❤➔♥❤ ♣❤➛♥ t❤ü❝ sü✱ t÷ì♥❣ tü ♥❤÷ tr♦♥❣ ▼ư❝ ✶✳✺✳

●✐❛♦

✣➦❝ tr÷♥❣ ❊✉❧❡r

✣✐➲✉ ❦✐➺♥

E(T1 )◦ ✱ E(Tn+2 )◦

1

E(Ti )◦

0

E(Tn+1 )◦ = E(P1 )◦

−1

E0◦

0

E(Ti ) ∩ E(Ti+1 ) = (1 ✤✐➸♠)

1

1≤i≤n+1

E(Ti ) ∩ E(Tj ) = ∅

0

|i − j| ≥ 2

E(Ti ) ∩ E0 = ∅

0

i=n+1

E(Tn+1 ) ∩ E0 = (1 ✤✐➸♠)

1

2≤i≤n

top
(s) ❝õ❛ f (x, y) = y 2 − x2n+1 t↕✐ O ❧➔ tê♥❣ ❝õ❛ ❝→❝
❉♦ ✤â ❤➔♠ ③❡t❛ tæ♣æ Zf,O

♣❤➙♥ t❤ù❝

1
1
−1
,
,
,
2s + 2 (2n + 1)s + n + 2 (4n + 2)s + 2n + 3
n−1

i=1

1
,
(2is + i + 1)(2(i + 1)s + i + 2)

1
1
,
,
(2ns + n + 1)((4n+2)s + 2n + 3) ((4n+2)s + 2n + 3)((2n+1)s + n + 2)
1
.
(s + 1)((4n + 2)s + 2n + 3)
❇ð✐ ✈➻

1
i+1
i
=
−
(2is + i + 1)(2(i + 1)s + i + 2) 2(i + 1)s + i + 2 2is + i + 1
t❛ s✉② r❛
n−1

i=1

1
n
1
=
−
.
(2is + i + 1)(2(i + 1)s + i + 2) 2ns + n + 1 2s + 2