❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛
Pr♦❣r❛♠❛ ❞❡ Pós✕●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛
▼❡str❛❞♦ ❡♠ ▼❛t❡♠át✐❝❛
❙♦❜r❡ ♦ ♥ú♠❡r♦ ♠á①✐♠♦ ❞❡ r❡t❛s ❡♠
s✉♣❡r❢í❝✐❡s ♥ã♦ s✐♥❣✉❧❛r ❞❡ ❣r❛✉
4
❡♠
P
3
.
❚❤✐❛❣♦ ▲✉✐③ ❞❡ ❖❧✐✈❡✐r❛ ❞♦ ❘ê❣♦
Pr♦❣r❛♠❛ ❞❡ Pós✕●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛
▼❡str❛❞♦ ❡♠ ▼❛t❡♠át✐❝❛
❙♦❜r❡ ♦ ♥ú♠❡r♦ ♠á①✐♠♦ ❞❡ r❡t❛s
♥✉♠❛ s✉♣❡r❢í❝✐❡ ♥ã♦ s✐♥❣✉❧❛r ❞❡ ❣r❛✉
4
❡♠
P
3
.
♣♦r
❚❤✐❛❣♦ ▲✉✐③ ❞❡ ❖❧✐✈❡✐r❛ ❞♦ ❘ê❣♦
s♦❜ ❛ ♦r✐❡♥t❛çã♦ ❞❛
Pr♦❢✳ ❉r
❛✳ ❏❛❝q✉❡❧✐♥❡ ❋❛❜✐♦❧❛ ❘♦❥❛s ❆r❛♥❝✐❜✐❛
R343s Rêgo, Thiago Luiz de Oliveira do.
Sobre o número máximo de retas numa superfície não singular de grau 4 em ℙ³/ Thiago Luiz de Oliveira do Rêgo.- João Pessoa, 2016.
87f. : il.
Orientadora: Jacqueline Fabiola Rojas Arancibia Dissertação (Mestrado) - UFPB/CCEN
1. Matemática. 2. Número máximo de retas. 3. Superfície quártica não singular. 4. Curva residual. 5. Característica de Euler.
❙✳ ❞❡ ❖❧✐✈❡✐r❛ ❞♦ ❘❡❣♦✱ q✉❡ ♠❡ ❞❡✉ s❡✉ ❛♠♦r ❡ ❝♦♥✜♦✉ q✉❡ ❡✉ ♣♦❞❡r✐❛
✐r ❛❧é♠✳
✧P♦rq✉❡ ◆❡❧❡ ✈✐✈❡♠♦s✱ ❡ ♥♦s ♠♦✈❡♠♦s✱ ❡ ❡①✐st✐♠♦s❀ ❝♦♠♦ t❛♠❜é♠ ❛❧❣✉♥s ❞♦s ✈♦ss♦s ♣♦❡t❛s ❞✐ss❡r❛♠✿ P♦✐s s♦♠♦s t❛♠❜é♠ s✉❛ ❣❡r❛çã♦✳✧
❆❣r❛❞❡ç♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s ♣♦r s✉❛ ✐♥✜♥✐t❛ ❣r❛ç❛ ❡♠ ♠✐♥❤❛ ✈✐❞❛✳ ▼❡❞✐t❛♥❞♦ ♥♦ ❛♠♦r ❉❡❧❡ ♣♦r ♠✐♠ ❡ ♥❛ s✉❛ ❣r❛♥❞✐♦s❛ ♣r♦♠❡ss❛ ♠❡ t♦r♥♦ ♠❡❧❤♦r✳
❆ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ❡♠ ❡s♣❡❝✐❛❧ ❛ ♠✐♥❤❛ s♦❣r❛ ❆✉r✐♥❡t❡ ❉✐❛s ❡ ❛ ♠✐♥❤❛ ♠ã❡ ❆♥❛ ❈❧❛✉❞✐❛✳
❆ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛ Pr♦❢✳ ❉r❛✳ ❏❛❝q✉❡❧✐♥❡ ❋❛❜✐♦❧❛ ❘♦❥❛s ❆r❛♥❝✐❜✐❛✳ ❙✉❛ ✐♠❡♥s❛
♣❛❝✐ê♥❝✐❛ ❡ ❛❥✉❞❛ t♦r♥♦✉ ❡ss❡ tr❛❜❛❧❤♦ ♣♦ssí✈❡❧✳
❆♦s ♣r♦❢❡ss♦r❡s ❞❛ ❜❛♥❝❛ ❆♥❞r❡ ▼❡✐r❡❧❡s✱ ▼✐r✐❛♠ ❞❛ ❙✐❧✈❛ ❡ ❋❡r♥❛♥❞♦ ❙♦✉③❛✳
❆♦s ♠❡✉s ❝♦❧❡❣❛s ❞♦ ♠❡str❛❞♦ ❡ ❞♦✉t♦r❛❞♦✳ ❊♠ ❡s♣❡❝✐❛❧ à ❙❛❧❧② ❆♥❞r✐❛ ❡ à ❉❛②❛♥❡ ❙❛♥t♦s✱ ♠✐♥❤❛s ✐r♠ãs ❛❝❛❞ê♠✐❝❛s✳
❆♦s ♠❡✉s ❛♠✐❣♦s ❞♦ ❧❛❜♦r❛tór✐♦ ▼✐❧ê♥✐♦ q✉❡ ❡st✐✈❡r❛♠ ❛♦ ♠❡✉ ❧❛❞♦✱ ♠✉✐t♦s ❞❡s❞❡ ❛ ❣r❛❞✉❛çã♦✱ ❝♦♠♣❛rt✐❧❤❛♥❞♦ ❝♦♥❤❡❝✐♠❡♥t♦s✱ ❞✐✜❝✉❧❞❛❞❡s ❡ ❛❧❡❣r✐❛s✳
❘❡s✉♠♦
❊♠ ✶✾✹✸✱ ❇❡♥✐❛♠✐♥♦ ❙❡❣r❡ ❛❝r❡❞✐t♦✉ t❡r ❞❡♠♦♥str❛❞♦ q✉❡ ♦ ♥ú♠❡r♦ ♠á①✐♠♦ ❞❡ r❡t❛s ❝♦♥t✐❞❛s ♥✉♠❛ s✉♣❡r❢í❝✐❡ q✉árt✐❝❛ ♥ã♦ s✐♥❣✉❧❛r ❡♠ P3 é 64, ✭❬✶✻❪✮✳ ▼❛s
r❡❝❡♥t❡♠❡♥t❡✱ ❤♦✉✈❡ ✉♠❛ r❡✈✐r❛✈♦❧t❛ ♥❡ss❡ t❡♠❛✱ q✉❛♥❞♦ ♦s ♠❛t❡♠át✐❝♦s ❙➟❛✇♦♠✐r ❘❛♠s ❡ ▼❛tt❤✐❛s ❙❝❤ütt ❝♦♥st❛t❛r❛♠ q✉❡ ❙❡❣r❡ t✐♥❤❛ ❝♦♠❡t✐❞♦ ✉♠ ❡rr♦ ❡♠ s❡✉ tr❛❜❛❧❤♦ ❛♦ ❡sq✉❡❝❡r ❛s q✉árt✐❝❛s ❞❛ ❢❛♠í❧✐❛ Z, ✭❬✶✹❪✮✱ q✉❡ ❝♦rr❡s♣♦♥❞❡♠ ❡ss❡♥❝✐❛❧♠❡♥t❡ ❛s
q✉árt✐❝❛s q✉❡ ♣♦ss✉❡♠ r❡t❛s q✉❡ ♣♦❞❡♠ s❡r ✐♥❝✐❞❡♥t❡s ❛ ♠❛✐s ❞❡ 18 r❡t❛s ❝♦♥t✐❞❛s
♥❛ s✉♣❡r❢í❝✐❡✳ ◆❡st❡ tr❛❜❛❧❤♦✱ t❡♥❞♦ ❝♦♠♦ ❜❛s❡ ❬✶✹❪✱ ♠♦str❛♠♦s q✉❡ t♦❞❛ q✉árt✐❝❛ ♥ã♦ s✐♥❣✉❧❛r✱ q✉❡ ♥ã♦ ♣❡rt❡♥❝❡ ❛ ❢❛♠í❧✐❛ Z, ❝♦♥té♠ ♥♦ ♠á①✐♠♦ 64 r❡t❛s✳ ❯♠❛ ❞❛s
❢❡rr❛♠❡♥t❛s ♠❛✐s ✐♠♣♦rt❛♥t❡s✱ ♣❛r❛ ♠♦str❛r ❡ss❡ r❡s✉❧t❛❞♦✱ é ♦ ❡st✉❞♦ ❞❛s ✜❜r❛çõ❡s
πl ✐♥❞✉③✐❞❛ ♣♦r ✉♠❛ r❡t❛ l ❝♦♥t✐❞❛ ♥❛ s✉♣❡r❢í❝✐❡✱ ❡ ❛ r❡❧❛çã♦ q✉❡ ❡①✐st❡ ❡♥tr❡ ❛
❝❛r❛❝t❡ríst✐❝❛ ❞❡ ❊✉❧❡r ❞❛ ❜❛s❡ ✭❡♠ ♥♦ss♦ ❝❛s♦P1✮✱ ❞❛s ✜❜r❛s s✐♥❣✉❧❛r❡s ❡ ❛ ❞❛ s✉♣❡r❢í❝✐❡
❡♠ q✉❡stã♦✳
■♥ 1943 ❇❡♥✐❛♠✐♥♦ ❙❡❣r❡ ❜❡❧✐❡✈❡❞ t♦ ❤❛✈❡ s❤♦✇♥ t❤❛t t❤❡ ♠❛①✐♠✉♠ ♥✉♠❜❡r ♦❢
❧✐♥❡s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ s♠♦♦t❤ q✉❛rt✐❝ s✉r❢❛❝❡ ✐♥P3 ✐s64✱ ✭❬✶✻❪✮✳ ❇✉t r❡❝❡♥t❧②✱ t❤❡r❡ ✇❛s ❛
♠❛❥♦r ♦✈❡rt✉r♥ ♦♥ t❤❛t t❤❡♠❡ ✇❤❡♥ t❤❡ ♠❛t❤❡♠❛t✐❝✐❛♥s ❘❛♠s ❛♥❞ ❙❝❤✉tt ❢♦✉♥❞ t❤❛t ❙❡❣r❡ ❤❛❞ ♠❛❞❡ ❛ ♠✐st❛❦❡ ✐♥ ❤✐s ✇♦r❦ t♦ ❢♦r❣❡t t❤❡ q✉❛rt✐❝✬s ❢❛♠✐❧② Z ✱ ✭❬✶✹❪✮✱ ✇❤✐❝❤
❡ss❡♥t✐❛❧❧② ❝♦rr❡s♣♦♥❞s t♦ t❤♦s❡ q✉❛rt✐❝s ❝♦♥t❛✐♥✐♥❣ ❛ ❧✐♥❡s t❤❛t ❝❛♥ ❜❡ ✐♥❝✐❞❡♥t t♦ ♠♦r❡ t❤❛♥18❧✐♥❡s ❝♦♥t❛✐♥❡❞ ✐♥ t❤❡ s✉r❢❛❝❡✳ ■♥ t❤✐s ✇♦r❦✱ ❜❛s❡❞ ♦♥ ✭❬✶✹❪✮✱ ✇❡ s❤♦✇ t❤❛t ❡✈❡r②
s♠♦♦t❤ q✉❛rt✐❝ s✉r❢❛❝❡✱ ✇❤✐❝❤ ❞♦❡s ♥♦t ❜❡❧♦♥❣ t♦ ❢❛♠✐❧②Z ❝♦♥t❛✐♥s ❛ ♠❛①✐♠✉♠ ♦❢ 64
❧✐♥❡s✳ ❖♥❡ ♦❢ t❤❡ ♠♦st ✐♠♣♦rt❛♥t t♦♦❧s t♦ s❤♦✇ t❤✐s r❡s✉❧t✱ ✐s t❤❡ st✉❞② ♦❢ ✜❜r❛t✐♦♥sπl
✐♥❞✉❝❡❞ ❜② ❛ ❧✐♥❡ l ❝♦♥t❛✐♥❡❞ ♦♥ t❤❡ s✉r❢❛❝❡✱ ❛♥❞ t❤❡ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ t❤❡ ❊✉❧❡r
❝❤❛r❛❝t❡r✐st✐❝ ♦❢ t❤❡ ❜❛s❡ ✭P1 ✐♥ ♦✉r ❝❛s❡ ✮✱ t❤❡ ✜❜❡rs ❛♥❞ t❤❡ s✉r❢❛❝❡ ❝♦♥❝❡r♥❡❞✳
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ✶
✶ ❘❡t❛s ❡♠ s✉♣❡r❢í❝✐❡s ❡♠ P3 ✸
✶✳✶ ❙✉♣❡r❢í❝✐❡s ❞❡ ❣r❛✉ d≤3❡♠ P3 s❡♠♣r❡ ❝♦♥té♠ r❡t❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸
✶✳✷ ❊①❡♠♣❧♦ ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❞❡ ❣r❛✉ d≥4 ❡♠ P3 q✉❡ ♥ã♦ ❝♦♥té♠ r❡t❛s ✳ ✶✸
✶✳✸ ❙✉♣❡r❢í❝✐❡s ❡♠P3 ❝♦♥t❡♥❞♦ ✉♠❛ r❡t❛ l ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✷ ❆ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ❊✉❧❡r ❞❛s ✜❜r❛s s✐♥❣✉❧❛r❡s ✸✶
✷✳✶ ❆ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ❊✉❧❡r✿ Pr♦♣r✐❡❞❛❞❡s ❜ás✐❝❛s ❡ ❛❧❣✉♥s ❝á❧❝✉❧♦s s✐♠♣❧❡s ✸✶ ✷✳✷ ❈á❧❝✉❧♦ ❞❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ❊✉❧❡r ❞❛s ✜❜r❛s s✐♥❣✉❧❛r❡s ❞♦ ♠♦r✜s♠♦ πl ✳ ✸✺
✸ ❈♦♥t❛❣❡♠ ❞♦ ♥ú♠❡r♦ ♠á①✐♠♦ ❞❡ r❡t❛s ♥❛s q✉árt✐❝❛s ♥ã♦ s✐♥❣✉❧❛r❡s
❡♠ Zc ✸✾
❆ ❍❡ss✐❛♥❛ ❞❡ ✉♠❛ ❝✉r✈❛ ♣❧❛♥❛ ❡ r❡s✉❧t❛❞♦s ❛✜♥s ✺✶
❆✳✶ ❘❡t❛ t❛♥❣❡♥t❡✱ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ ❞❡ ✐♥t❡rs❡çã♦ ❞❡ ✉♠❛ r❡t❛ ❝♦♠ ✉♠❛ ❝✉r✈❛ ❡ ♣♦♥t♦s ❞❡ ✐♥✢❡①ã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ❆✳✷ ❆ ❝✉r✈❛ ❍❡ss✐❛♥❛ ❞❡ ✉♠❛ ❝✉r✈❛ C ❡ ❆❧❣✉♥s r❡s✉❧t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼
❇ ❆ r❡s✉❧t❛♥t❡ ❞❡ ❞♦✐s ♣♦❧✐♥ô♠✐♦s ✻✺
❈ ❇❧♦✇ ✉♣ ✼✵
✶✳✶ P❧❛♥♦H ∈Ω(l) ✐♥t❡rs❡❝t❛♥❞♦ S. ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✶✳✷ P❧❛♥♦s ❞❡ Ω(l)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✶✳✸ ❈✉r✈❛CH ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
✶✳✹ ❚✐♣♦s ❞❡ ✜❜r❛s s✐♥❣✉❧❛r❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵
✷✳✶ ❋✐❜r❛ ❞♦ t✐♣♦ IV✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺
✷✳✷ ❋✐❜r❛ ❞♦ t✐♣♦ I2✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✷✳✸ ❋✐❜r❛ ❞♦ t✐♣♦ II✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✷✳✹ ❋✐❜r❛ ❞♦ t✐♣♦ I1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✷✳✺ ❋✐❜r❛s ❡ ❝❛r❛❝t❡ríst✐❝❛s ❞❡ ❊✉❧❡r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
◆♦t❛çõ❡s
❆ s❡❣✉✐r✱ ❧✐st❛♠♦s ❛❧❣✉♠❛s ♥♦t❛çõ❡s ✉t✐❧✐③❛❞❛s ♥❡st❡ tr❛❜❛❧❤♦✳
• C[x0, . . . , xn]k ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ♣♦❧✐♥ô♠✐♦s ❤♦♠♦❣ê♥❡♦s ❞❡ ❣r❛✉ k ❞♦ ❛♥❡❧
C[x0, . . . , xn].
• A ♦ ❛♥❡❧ C[x0, x1, x2, x3] ❡ Ak ♦ ❝♦♥❥✉♥t♦C[x0, x1, x2, x3]k.
❆s ❞❡♠❛✐s ♥♦t❛çõ❡s ❡ t❡r♠✐♥♦❧♦❣✐❛s ♣r❡s❡♥t❡s ♥♦ tr❛❜❛❧❤♦ t❡rã♦ s❡✉ s✐❣♥✐✜❝❛❞♦ ❡①♣r❡ss♦ ♥♦ ❞❡❝♦rr❡r ❞♦ ♠❡s♠♦✳
❊♠ ✶✾✹✸ ♦ ♠❛t❡♠át✐❝♦ ✐t❛❧✐❛♥♦ ❇❡♥✐❛♠✐♥♦ ❙❡❣r❡ ✭✶✾✵✸✲✶✾✼✼✮ ♣✉❜❧✐❝♦✉ ✉♠ ❢❛♠♦s♦ ❛rt✐❣♦✱ ❝✉❥♦ tít✉❧♦ é ✏❚❤❡ ♠❛①✐♠✉♠ ♥✉♠❜❡r ♦❢ ❧✐♥❡s ❧②✐♥❣ ♦♥ ❛ q✉❛rt✐❝ s✉r❢❛❝❡✑✱ ❛✜r♠❛♥❞♦ q✉❡ ♦ ♥ú♠❡r♦ ♠á①✐♠♦ ❞❡ r❡t❛s ❝♦♥t✐❞❛s ♥✉♠❛ q✉árt✐❝❛ ♥ã♦ s✐♥❣✉❧❛r ❡♠ P3 é ✻✹✳ Pr♦❜❧❡♠❛s ❞❡ss❡ t✐♣♦✱ ❣❡r❛❧♠❡♥t❡ ❝❤❛♠❛❞♦s ❞❡ ♣r♦❜❧❡♠❛s
❞❡ ●❡♦♠❡tr✐❛ ❊♥✉♠❡r❛t✐✈❛✱ ❝♦♥tr✐❜✉✐r❛♠ ❢♦rt❡♠❡♥t❡ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ●❡♦♠❡tr✐❛ ❆❧❣é❜r✐❝❛✳ ❉❡ ❢❛t♦✱ ❡♠ ✶✽✹✼✱ q✉❛s❡ ❝❡♠ ❛♥♦s ❛♥t❡s ❞♦ ❛rt✐❣♦ ❞❡ ❙❡❣r❡✱ ❈❛②❧❡② ❡ ❙❛❧♠♦♥ ♣r♦✈❛r❛♠ q✉❡ t♦❞❛ s✉♣❡r❢í❝✐❡ ❝ú❜✐❝❛ ♥ã♦ s✐♥❣✉❧❛r ❡♠ P3 ❝♦♥té♠
❡①❛t❛♠❡♥t❡ ✷✼ r❡t❛s✳ ▼✉✐t♦s ❛❝r❡❞✐t❛♠ q✉❡ ❛ ♣❛rt✐r ❞❡ss❡ ♠♦♠❡♥t♦ s❡ ✐♥✐❝✐♦✉ ❛ ●❡♦♠❡tr✐❛ ❆❧❣é❜r✐❝❛ ♠♦❞❡r♥❛✳
❆♦ ❝♦♠♣❛r❛r♠♦s ❛s s✉♣❡r❢í❝✐❡s ❝ú❜✐❝❛s ❡ q✉árt✐❝❛s ♥ã♦ s✐♥❣✉❧❛r❡s ❝♦♥st❛t❛♠♦s q✉❡ t♦❞❛ s✉♣❡r❢í❝✐❡ ❝ú❜✐❝❛ ♥ã♦ s✐♥❣✉❧❛r ❡♠ P3 ❝♦♥té♠ r❡t❛s ✭ ❝❢✳ ❚❡♦r❡♠❛ ✶✳✹✮✱ ♦ q✉❡
♥ã♦ ♦❝♦rr❡ ❝♦♠ ❛s s✉♣❡r❢í❝✐❡s q✉árt✐❝❛s ♥ã♦ s✐♥❣✉❧❛r❡s✳ ❉❡ ❢❛t♦✱ ♥♦ ❊①❡♠♣❧♦ ✶✳✺✱ ♠♦str❛r❡♠♦s ✉♠❛ s✉♣❡r❢í❝✐❡ ❞❡ ❣r❛✉d≥4q✉❡ ♥ã♦ ❝♦♥té♠ r❡t❛s✳ ❊♥tr❡t❛♥t♦ ❋r✐❡❞r✐❝❤
❙❝❤✉r ✭✶✽✺✻✲✶✾✸✷✮ ❡♠ ✶✽✽✷ ❡①✐❜✐✉ ✉♠❛ s✉♣❡r❢í❝✐❡ ♥ã♦ s✐♥❣✉❧❛r ❞❡ ❣r❛✉ 4 q✉❡ ❝♦♥té♠
✻✹ r❡t❛s ✭ ❝❢✳ ❊①❡♠♣❧♦ ✶✳✶✮✳ ■ss♦ ♣r♦✈❛✈❡❧♠❡♥t❡ ❢♦✐ ✉♠ ❞♦s ✐♥❞í❝✐♦s q✉❡ ❧❡✈❛r❛♠ ❙❡❣r❡ ❛ ♣❡♥s❛r ♥❛ ♣❡sq✉✐s❛ ❛♣r❡s❡♥t❛❞❛ ❡♠ ❬✶✻❪✳
❆ ❞❡♠♦♥str❛çã♦ q✉❡ ❙❡❣r❡ t✐♥❤❛ r❡❛❧✐③❛❞♦ ❡♠ s❡✉ ❛rt✐❣♦ ❡st❛✈❛ ❡rr❛❞❛✱ ❡ ❡ss❡ ❡rr♦ ❢♦✐ ❞❡s❝♦❜❡rt♦ ❛♣❡♥❛s r❡❝❡♥t❡♠❡♥t❡ ♣♦r ❙➟❛✇♦♠✐r ❘❛♠s ❡ ▼❛tt❤✐❛s ❙❝❤ütt✳ ❊♠ ✷✵✶✸ ❡❧❡s ♣r♦✈❛r❛♠ q✉❡ ❛ ❛✜r♠❛çã♦ ❞❡ ❙❡❣r❡ ❡r❛ r❡❛❧♠❡♥t❡ ✈❡r❞❛❞❡✐r❛✱ ❡♠❜♦r❛ q✉❡ s❡❣✉✐♥❞♦ ♦s s❡✉s ❛r❣✉♠❡♥t♦s✱ ❙❡❣r❡ só ♣♦❞❡r✐❛ ❝♦♥❝❧✉✐r q✉❡ ♦ ♥ú♠❡r♦ ♠á①✐♠♦ ❞❡ r❡t❛s ♥✉♠❛ s✉♣❡r❢í❝✐❡ q✉árt✐❝❛ ❡r❛ ✼✷✱✭ ❬✶✹❪ ✮✳
◆❡st❡ tr❛❜❛❧❤♦ ❛❜♦r❞❛♠♦s ❛s q✉árt✐❝❛s ❡st✉❞❛❞❛s ♣♦r ❙❡❣r❡ ✉t✐❧✐③❛♥❞♦ ❛s té❝♥✐❝❛s ❞♦ ❛rt✐❣♦ ♣✉❜❧✐❝❛❞♦ ♣♦r ❘❛♠s ❡ ❙❝❤ütt✳ ◆❡ss❡ ❛rt✐❣♦ ❢♦r❛♠ ✉t✐❧✐③❛❞❛s ❞✐✈❡rs❛s ✐❞❡✐❛s ❞♦ ❛rt✐❣♦ ❞❡ ❙❡❣r❡✱ ❝♦♠♦ t❛♠❜é♠ ❛ ❚❡♦r✐❛ ❞❛s ❋✐❜r❛çõ❡s ❊❧í♣t✐❝❛s ❞❡s❡♥✈♦❧✈✐❞❛ ❛♣❡♥❛s ❡♠ ✶✾✺✵✳ ❉❡ ❢❛t♦✱ ♠♦str❛r❡♠♦s q✉❡ s❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ♥ã♦ s✐♥❣✉❧❛rS ❝♦♥té♠ ✉♠❛ r❡t❛
❡♥tã♦ ♣♦❞❡♠♦s ❡st❛❜❡❧❡❝❡r ✉♠ ♠♦r✜s♠♦ ❞❡S ❡♠ P1,❡ t❛✐s ♠♦r✜s♠♦s sã♦ ❡①❡♠♣❧♦s ❞❡
✜❜r❛çõ❡s ❡❧í♣t✐❝❛s✳ ❙❛❧✐❡♥t❛♠♦s q✉❡ ♦ ❛rt✐❣♦ ❞❡ ❘❛♠s ❡ ❙❝❤ütt ❣❡♥❡r❛❧✐③❛ ❛ ❛✜r♠❛çã♦ ♦r✐❣✐♥❛❧ ❞❡ ❙❡❣r❡✱ ♣♦✐s ♠♦str❛ q✉❡ ♦ r❡s✉❧t❛❞♦ ❛✐♥❞❛ é ✈❡r❞❛❞❡✐r♦ ♣❛r❛ ✉♠ ❝♦r♣♦ ❛❧❣❡❜r✐❝❛♠❡♥t❡ ❢❡❝❤❛❞♦ ❞❡ ❝❛r❛❝t❡ríst✐❝❛ ❞✐❢❡r❡♥t❡ ❞❡ 2❡ 3✳
♥ã♦ s✐♥❣✉❧❛r ❝♦♠ ❣r❛✉d≥5 ❛✐♥❞❛ ❡stá ❡♠ ❛❜❡rt♦✳
❊st❛ ❞✐ss❡rt❛çã♦ ❡stá ❞✐✈✐❞✐❞❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ♥♦ ❈❛♣ít✉❧♦ ✶ ♠♦str❛r❡♠♦s q✉❡ t♦❞❛ s✉♣❡r❢í❝✐❡ ♥ã♦ s✐♥❣✉❧❛r ❞❡ ❣r❛✉ ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛ 3 ❝♦♥té♠ r❡t❛s✳ ❆❧é♠
❞✐ss♦✱ ❡①✐❜✐♠♦s ✉♠❛ s✉♣❡r❢í❝✐❡ ♥ã♦ s✐♥❣✉❧❛r ❞❡ ❣r❛✉ d ≥ 4 q✉❡ ♥ã♦ ❝♦♥té♠ r❡t❛s ❡
t❛♠❜é♠ ❛ q✉árt✐❝❛ ❞❡ ❙❝❤✉r✱ q✉❡ ❝♦♥té♠ 64 r❡t❛s✳ ◆❛ ú❧t✐♠❛ s❡çã♦ ❞❡ss❡ ❝❛♣ít✉❧♦
❡st✉❞❛♠♦s ❛ ✜❜r❛çã♦ s♦❜r❡P1 ✐♥❞✉③✐❞❛ ♣♦r ✉♠❛ s✉♣❡r❢í❝✐❡ ♥ã♦ s✐♥❣✉❧❛r ❝♦♥t❡♥❞♦ ✉♠❛
r❡t❛✳ ❊♥❝♦♥tr❛r❡♠♦s ❝♦♠ ❡ss❡ ❡st✉❞♦ r❡s✉❧t❛❞♦s ✐♠♣♦rt❛♥t❡s ♣❛r❛ ♣r♦✈❛r ♦ ♣r✐♥❝✐♣❛❧ ❚❡♦r❡♠❛✱ ✭❚❡♦r❡♠❛ ✸✳✹✮✳ ◆♦ ❈❛♣ít✉❧♦ ✷ ❡st✉❞❛r❡♠♦s ❛s ❞❡✜♥✐çõ❡s ❡ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❛ ❝❡r❝❛ ❞❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ❊✉❧❡r ♣❛r❛ ❝✉r✈❛s ❡ s✉♣❡r❢í❝✐❡s✳ ▼♦str❛r❡♠♦s ✉♠❛ ✐♠♣♦rt❛♥t❡ ❝♦♥❡①ã♦ ❡♥tr❡ ❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ❊✉❧❡r ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡S ❝♦♥t❡♥❞♦ ✉♠❛ r❡t❛l❝♦♠ ❛
❝❛r❛❝t❡ríst✐❝❛ ❞❡ ❊✉❧❡r ❞❛s ✜❜r❛s s✐♥❣✉❧❛r❡s ❞♦ ♠♦r✜s♠♦πl ✐♥❞✉③✐❞❛ ♣♦r ❡ss❛ r❡t❛ ✭❝❢✳
❉❡✜♥✐çã♦ ✶✳✸✮✳
❖ ❈❛♣ít✉❧♦ ✸ t❡♠ ❝♦♠♦ ❢♦❝♦ ❛ ❞❡♠♦♥str❛çã♦ ❞❡ q✉❡ t♦❞❛ s✉♣❡r❢í❝✐❡ ♥ã♦ s✐♥❣✉❧❛r S
q✉❡ ♥ã♦ ♣❡rt❡♥❝❡ à ❢❛♠í❧✐❛Z ✭❝❢✳ ❉❡✜♥✐çã♦ ✸✳✸✮ ❝♦♥té♠ ♥♦ ♠á①✐♠♦64r❡t❛s ✭❚❡♦r❡♠❛
✸✳✹✮✳ ❖s ❛♣ê♥❞✐❝❡s ❆✱ ❇ ❡ ❈ tr❛t❛♠ s♦❜r❡ ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s r❡❧❛t✐✈♦s ❛ ❝✉r✈❛ ❤❡ss✐❛♥❛✱ r❡s✉❧t❛♥t❡ ❞❡ ❞♦✐s ♣♦❧✐♥ô♠✐♦s ❡ ❇❧♦✇ ✉♣✳ ❊❧❡s ❢♦r❛♠ ✐♥s❡r✐❞♦s ♣❛r❛ q✉❡ ♦ ❧❡✐t♦r✱ ♥ã♦ ❢❛♠✐❧✐❛r✐③❛❞♦s ❝♦♠ ❡ss❡s ❝♦♥❝❡✐t♦s✱ ♣♦ss❛ ❡♥❝♦♥tr❛r ✉♠ ♠❛t❡r✐❛❧ ❝♦♥s✐st❡♥t❡ s♦❜r❡ ❡ss❡s t❡♠❛s✳
❘❡t❛s ❡♠ s✉♣❡r❢í❝✐❡s ❡♠
P
3
■♥✐❝✐❛r❡♠♦s ❡st❡ ❝❛♣ít✉❧♦ ♠♦str❛♥❞♦ q✉❡ t♦❞❛ s✉♣❡r❢í❝✐❡ ❡♠P3❞❡ ❣r❛✉d≤3s❡♠♣r❡
❝♦♥té♠ r❡t❛s✳ ◆❛ s❡❣✉♥❞❛ s❡çã♦✱ ♠♦str❛r❡♠♦s ✉♠❛ té❝♥✐❝❛✱ ❝❤❛♠❛❞❛ ❡str❛t✐✜❝❛çã♦✱ q✉❡ ♥♦s ♣❡r♠✐t✐rá ❞❡ ♠❛♥❡✐r❛ ❜❛st❛♥t❡ ❡✜❝✐❡♥t❡ ❞❡❝✐❞✐r q✉❛♥❞♦ ✉♠❛ s✉♣❡r❢í❝✐❡ S ❡♠
P3 ❝♦♥té♠ r❡t❛s✳ ❉❡ ❢❛t♦✱ ✉s❛r❡♠♦s ❞✐t❛ ❡str❛t✐✜❝❛çã♦ ♣❛r❛ ❡①✐❜✐r ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ S ❡♠ P3 ❞❡ ❣r❛✉ d ≥ 4 q✉❡ ♥ã♦ ❝♦♥té♠ r❡t❛s✳ ◆❛ t❡r❝❡✐r❛ s❡çã♦ ❝♦♥s✐❞❡r❛r❡♠♦s ✉♠❛
s✉♣❡r❢í❝✐❡ ♥ã♦ s✐♥❣✉❧❛rS❡♠P3 ❞❡ ❣r❛✉d≥3q✉❡ ❝♦♥té♠ ✉♠❛ ❝❡rt❛ r❡t❛l✱ ❡st✉❞❛r❡♠♦s
❛ ✐♥t❡rs❡çã♦ ❞❛ s✉♣❡r❢í❝✐❡ S ❝♦♠ ♣❧❛♥♦s q✉❡ ❝♦♥té♠ ❛ r❡t❛ l. ❊ss❛ ✐♥t❡rs❡çã♦ ❝♦♥s✐st❡
❞❛ r❡t❛ l ❡ ❞❡ ✉♠❛ ❝✉r✈❛ r❡s✐❞✉❛❧ r❡❞✉③✐❞❛ ❞❡ ❣r❛✉ d −1 q✉❡ ♥ã♦ ❝♦♥té♠ l ❝♦♠♦
❝♦♠♣♦♥❡♥t❡✳ ❱❡r❡♠♦s q✉❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❝✉r✈❛s r❡s✐❞✉❛✐s q✉❡ sã♦ s✐♥❣✉❧❛r❡s é ✜♥✐t❛✳
✶✳✶ ❙✉♣❡r❢í❝✐❡s ❞❡ ❣r❛✉
d
≤
3
❡♠
P
3s❡♠♣r❡ ❝♦♥té♠
r❡t❛s
❉❛❞♦ W ✉♠ s✉❜❡s♣❛ç♦ ❞❡ Cn+1 ❞❡♥♦t❛r❡♠♦s ♣♦r P(W), ❡ ❝❤❛♠❛r❡♠♦s ❞❡
♣r♦❥❡t✐✈✐③❛çã♦ ❞❡W, ♦ ❝♦♥❥✉♥t♦
P(W) ={[w]∈Pn
|w∈W − {0}}.
❉✐r❡♠♦s q✉❡ Λ ⊂ Pn é ✉♠❛ ✈❛r✐❡❞❛❞❡ r✲❧✐♥❡❛r s❡ ❡①✐st❡ ✉♠ s✉❜s❡♣❛ç♦ W ⊂ Cn+1
❞❡ ❞✐♠❡♥sã♦r+ 1 t❛❧ q✉❡ Λ =P(W).
❉❡✜♥✐çã♦ ✶✳✶ ❉✐r❡♠♦s q✉❡ l ⊂ Pn é ✉♠❛ r❡t❛✱ s❡ l ❢♦r ✉♠❛ ✈❛r✐❡❞❛❞❡ 1✲❧✐♥❡❛r✱ ♦✉
s❡❥❛ s❡ ❡①✐st❡ W ⊂Cn+1 ✉♠ s✉❜❡s♣❛ç♦ ❞❡ ❞✐♠❡♥sã♦ 2, t❛❧ q✉❡ l=P(W).
❱❡❥❛♠♦s ❛❣♦r❛ ♦✉tr❛s ❝❛r❛❝t❡r✐③❛çõ❡s ❞❡ ✉♠❛ r❡t❛ ❡♠Pn.❆ ♣r✐♠❡✐r❛ ❞❡❧❛s ♠♦str❛
q✉❡✱ ❛ss✐♠ ❝♦♠♦ ♥❛ ❣❡♦♠❡tr✐❛ ♣❧❛♥❛✱ ❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s ❞❡t❡r♠✐♥❛♠ ✉♠❛ ú♥✐❝❛ r❡t❛ ❡♠ Pn.
Pr♦♣♦s✐çã♦ ✶✳✶ ❙❡❥❛♠ p, q ∈ Pn ♣♦♥t♦s ❞✐st✐♥t♦s✳ ❊♥tã♦ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ r❡t❛ ❡♠
Pn ♣❛ss❛♥❞♦ ♣♦r p ❡ q. ❊ss❛ r❡t❛ s❡rá ❞❡♥♦t❛❞❛ ♣♦r l p,q.
❉❡♠♦♥str❛çã♦✿ ❙❡ p = [u] ❡ q = [v] sã♦ ♣♦♥t♦s ❞✐st✐♥t♦s ❡♠ Pn, s❡❣✉❡ q✉❡ u ❡ v
sã♦ ✈❡t♦r❡s ▲✳■ ❡♠ Cn+1. ❙❡❥❛ W = [u, v] ♦ s✉❜❡s♣❛ç♦ ❣❡r❛❞♦ ♣♦r u ❡ v. ❚❡♠♦s q✉❡ W ♣♦ss✉✐ ❞✐♠❡♥ssã♦ 2. ▲♦❣♦ l =P(W) ❞❡t❡r♠✐♥❛ ✉♠❛ r❡t❛ ❡♠ Pn. ❆❧é♠ ❞✐ss♦✱ ❝♦♠♦
u, v ∈ W t❡♠♦s q✉❡ p, q ∈l. P❛r❛ ♠♦str❛r ❛ ✉♥✐❝✐❞❛❞❡ s✉♣♦♥❤❛♠♦s q✉❡ l′ = P(W′) é
✉♠❛ ♦✉tr❛ r❡t❛ q✉❡ ❝♦♥té♠ ♦s ♣♦♥t♦s p ❡ q. ❆ss✐♠ ♦s ✈❡t♦r❡s u ❡ v ♣❡rt❡♥❝❡♠ ❛ W′.
▲♦❣♦ W ⊂W′. ▼❛s ❞❡s❞❡ q✉❡ dimW′ = 2,❝♦♥❝❧✉í♠♦s q✉❡ W =W′✳ ❆ss✐♠ l=l′.
❉❛❞♦s p = [u] ❡ q = [v] ♣♦♥t♦s ❞✐st✐♥t♦s ❞❡ Pn, ❝♦♠ u = (u
0, . . . , un) ❡
v = (v0, . . . , vn), ❡♥tã♦ ❝♦♠♦ [u, v] ={αu+βv |α, β ∈C}. ❙❡❣✉❡ q✉❡
lp,q ={[αu0+βv0 :. . .:αun+βvn]|[α :β]∈P1}.
■ss♦ ♠♦str❛ q✉❡ t♦❞❛ r❡t❛ ♥♦ ❡s♣❛ç♦ ♣r♦❥❡t✐✈♦Pn é ❛♣❡♥❛s ✉♠❛ ❝ó♣✐❛ ❞❛ r❡t❛ ♣r♦❥❡t✐✈❛
P1.
❆ ♣ró①✐♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ♠♦str❛ q✉❡ ❛s r❡t❛s sã♦ ❝♦♥❥✉♥t♦s ❛❧❣é❜r✐❝♦s ❞❡t❡r♠✐♥❛❞♦s ♣❡❧♦s ③❡r♦s ❞❡ ❞♦✐s ♣♦❧✐♥ô♠✐♦s ❤♦♠♦❣ê♥❡♦s ❞❡ ❣r❛✉1 ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✳
❚❡♦r❡♠❛ ✶✳✶ ❙❡❥❛ Λ ✉♠❛ ✈❛r✐❡❞❛❞❡ r✲❧✐♥❡❛r ❡♠ Pn✱ ❡♥tã♦ ❡①✐st❡♠ L1, . . . , L n−r
♣♦❧✐♥ô♠✐♦s ❤♦♠♦❣ê♥❡♦s ❞❡ ❣r❛✉ 1 ❡♠ C[x0, . . . , xn] ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s t❛✐s q✉❡
Λ =Z(L1, . . . , Ln−r).
❉❡♠♦♥str❛çã♦✿ ❱❡r ♣á❣✳ 6❡♠ ❬✶✶❪✳
❯♠ ❤✐♣❡r♣❧❛♥♦ ❡♠ Pn é ❛ ♣r♦❥❡t✐✈✐③❛çã♦ ❞❡ ✉♠ s✉❜❡s♣❛ç♦ W ⊂Cn+1 ❞❡ ❞✐♠❡♥sã♦ n. ❖✉ s❡❥❛✱ ✉♠ ❤✐♣❡r♣❧❛♥♦ ❡♠ Pn é ✉♠❛ ✈❛r✐❡❞❛❞❡ (n
−1)✲❧✐♥❡❛r✳ ❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛
❞♦ t❡♦r❡♠❛ ❛❝✐♠❛ s❡❣✉❡ q✉❡ ✉♠ ❤✐♣❡r♣❧❛♥♦ ❡♠ Pn é ❞❛❞♦ ❝♦♠♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ③❡r♦s
❞❡ ✉♠ ♣♦❧✐♥ô♠✐♦ L∈C[x0, . . . , xn]1.
Pr♦♣♦s✐çã♦ ✶✳✷ ❙❡❥❛♠L1, . . . , Lk ♣♦❧✐♥ô♠✐♦s ❤♦♠♦❣ê♥❡♦s ❞❡ ❣r❛✉1 ❡♠C[x0, . . . , xn].
❙✉♣♦♥❤❛♠♦s q✉❡ α = {L1, . . . , Lk} é ✉♠ ❝♦♥❥✉♥t♦ ▲✳■ ♥♦ s✉❜❡s♣❛ç♦ [x0, . . . , xn]
❞❡ C[x0, . . . , xn] ❞♦s ♣♦❧✐♥ô♠✐♦s ❤♦♠♦❣ê♥❡♦s ❞❡ ❣r❛✉ 1✳ ❊♥tã♦ ♦ ✐❞❡❛❧ ❣❡r❛❞♦ ♣♦r
L1, . . . , Lk, I =hL1, . . . , Lki é ✉♠ ✐❞❡❛❧ ♣r✐♠♦✳ ❊♠ ♣❛rt✐❝✉❧❛r
√
I =I.
❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ α ={L1, . . . , Lk} é ▲✳■✳ ❡♠ [x0, . . . , xn] s❡❣✉❡ q✉❡ k ≤ n+ 1.
❆❧é♠ ❞✐ss♦✱ ❝♦♠♣❧❡t❛♥❞♦ ❛α ♣❛r❛ ♦❜t❡r ✉♠❛ ❜❛s❡ ❞❡ [x0, . . . , xn]✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r
T : W −→ W✱ ♦♥❞❡ W = [x0, . . . , xn], ✉♠ ✐s♦♠♦r✜s♠♦ t❛❧ q✉❡ T(Li) = xi−1, ❝♦♠
i= 1, . . . , k. ❆ ❛♣❧✐❝❛çã♦ ϕ:C[x0, . . . , xn]−→C[x0, . . . , xn] ❞❡✜♥✐❞❛ ♣♦r
ϕ(Xai0,...,inx
i0
0.· · · .x
in
n) =
X
ai0,...,inT(x0)
i0.· · ·.T(x
n)in,
❞❡✜♥❡ ✉♠ ❛✉t♦♠♦r✜s♠♦ ❞❡ ❛♥é✐s ❡♠ C[x0, . . . , xn]. P♦rt❛♥t♦✱ ❝♦♠♦ ♦ ✐❞❡❛❧
hx0, . . . , xk−1i = T(I) é ♣r✐♠♦ ❡♠ C[x0, . . . , xn], s❡❣✉❡ q✉❡ I = hL1, . . . , Lki é ✐❞❡❛❧
♣r✐♠♦✳
❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞❛ ♣r♦♣♦s✐çã♦ ❛❝✐♠❛ ♦❜t❡♠♦s q✉❡ t♦❞❛ ✈❛r✐❡❞❛❞❡ r✲❧✐♥❡❛r ❞❡
Pn é ✉♠ ❝♦♥❥✉♥t♦ ✐rr❡❞✉tí✈❡❧✳ ❊♠ ♣❛rt✐❝✉❧❛r r❡t❛s sã♦ ✐rr❡❞✉tí✈❡✐s ❡♠Pn.
◆♦ ❝❛s♦ ❞❡ P3 ✉♠❛ r❡t❛ é ❞❛❞❛ ♣❡❧♦ ❝♦♥❥✉♥t♦ ❞♦s ③❡r♦s ❞❡ ❞✉❛s ❢♦r♠❛s ❧✐♥❡❛r❡s
L1, L2 ∈C[x0, x1, x2, x3] ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✳
◆♦ q✉❡ s❡❣✉❡ ❞♦ t❡①t♦ ✉s❛r❡♠♦s s❡♠♣r❡ ❛ ❧❡tr❛ A ♣❛r❛ ❞❡♥♦t❛r ♦ ❛♥❡❧
C[x0, x1, x2, x3]✱ ✭✈❡r ◆♦t❛çõ❡s✮✳
❖❜s❡r✈❛çã♦ ✶✳✶ ❙❡❥❛♠f ∈ A ✉♠ ♣♦❧✐♥ô♠✐♦ ❤♦♠♦❣ê♥❡♦ ❞❡ ❣r❛✉d≥1 ❡ S =Z(f)⊂
P3 ❛ s✉♣❡r❢í❝✐❡ ❞❡✜♥✐❞❛ ♣♦rf. ❆ r❡t❛l=Z(L1, L2)❡stá ❝♦♥t✐❞❛ ❡♠ S s❡✱ ❡ s♦♠❡♥t❡ s❡✱ f =AL1+BL2,♣❛r❛ ❛❧❣✉♠A, B ∈ Ad−1.❉❡ ❢❛t♦✱ s❡f =AL1+BL2,❝♦♠A, B ∈ Ad−1,
❡♥tã♦ s❡❣✉❡ q✉❡ l⊂S. ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡ l⊂S, ❡♥tã♦ ❛♣❧✐❝❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❞♦s ③❡r♦s
❞❡ ❍✐❧❜❡rt ♦❜t❡♠♦s q✉❡ I(S)⊂ I(l). ▼❛s ❞❡s❞❡ q✉❡ I(l) =hL1, L2i ❡ ❝♦♠♦ f ∈ I(S),
t❡♠✲s❡ q✉❡ f ∈ hL1, L2i. ▲♦❣♦ ❡①✐st❡♠ A′, B′ ∈ A t❛✐s q✉❡ f =A′L1 +B′L2. P♦❞❡♠♦s
❡s❝r❡✈❡r A′ =Pk
i=0A′i ❡ B′ =
Pl
i=0Bi′✱ ♦♥❞❡ ❝❛❞❛ A′i ❡ Bj′ sã♦ ❤♦♠♦❣ê♥❡♦s ❞❡ ❣r❛✉ i ❡
j, r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❉❛í
f =
k
X
i=0
A′iL1+ l
X
i=0 Bi′L2.
❈❛❞❛ ♣❛r❝❡❧❛ A′
iL1 ❡ Bi′L2 ♣♦ss✉✐ ❣r❛✉ i+ 1 ❡ j + 1, r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❉❡s❞❡ q✉❡ f é
✉♠ ♣♦❧✐♥ô♠✐♦ ❤♦♠♦❣ê♥❡♦✱ ❡ ✉s❛♥❞♦ q✉❡ ❞♦✐s ♣♦❧✐♥ô♠✐♦s sã♦ ✐❣✉❛✐s q✉❛♥❞♦ s✉❛s ♣❛rt❡s ❤♦♠♦❣ê♥❡❛s sã♦ ✐❣✉❛✐s✱ ♦❜t❡♠♦s q✉❡f =AL1+BL2, ♦♥❞❡ A, B ∈ Ad−1.
P❛r❛ ❝❛❞❛ ♣♦❧✐♥ô♠✐♦f ∈ Ad♥ã♦ ♥✉❧♦ ❝♦♥s✐❞❡r❡♠♦s ♦ ❝♦♥❥✉♥t♦Rf ❞❡ t♦❞❛s ❛s r❡t❛s
❝♦♥t✐❞❛s ❡♠ Z(f)✳ ❆ss✐♠
Rf ={l|l é ✉♠❛ r❡t❛ ❝♦♥t✐❞❛ ❡♠ Z(f)}.
▼♦str❛r❡♠♦s ❛❞✐❛♥t❡ q✉❡ ♥♦ ❝❛s♦ ❡♠ q✉❡ d∈ {1,2,3} ❡♥tã♦ Rf 6=∅.
❙✉♣❡r❢í❝✐❡s ❞❡ ❣r❛✉ ✶
❙✉♣♦♥❤❛♠♦s q✉❡ f ∈ A1 é ✉♠ ♣♦❧✐♥ô♠✐♦ ♥ã♦ ♥✉❧♦✳ ❊♥tã♦
f =a0x0+a1x1+a2x2+a3x3,
❝♦♠ ai 6= 0♣❛r❛ ❛❧❣✉♠ i= 0,1,2 ♦✉3. ❙❡ l =Z(L1, L2) é ✉♠❛ r❡t❛ ❝♦♥t✐❞❛ ❡♠ Z(f),
❡♥tã♦ f =αL1+βL2,❝♦♠ α, β ∈ C ♥ã♦ ❛♠❜♦s ♥✉❧♦s✳ ❙✉♣♦♥❤❛♠♦s q✉❡α 6= 0✱ ❡♥tã♦
L1 = α−1f −α−1βL2. ❖❜s❡r✈❡♠♦s q✉❡ s♦❜ ❡ss❛s ❝♦♥❞✐çõ❡s t❡♠♦s q✉❡ f ❡ L2 sã♦ ▲✳■
❡♠ A1✳ ❊ss❛ r❡❧❛çã♦ ♠♦str❛ q✉❡ hL1, L2i = hf, L2.i✳ P♦rt❛♥t♦ t♦❞❛ r❡t❛ ❝♦♥t✐❞❛ ❡♠ Z(f) é ❞❛ ❢♦r♠❛ l =Z(f, L), ♦♥❞❡L, f ∈ A1 sã♦ ▲✳■✳
❚❡♦r❡♠❛ ✶✳✷ ❙❡❥❛ f ∈ A1 ♥ã♦ ♥✉❧♦✳ ❙✉♣♦♥❤❛♠♦s q✉❡ M1, M2, M3 ∈ A1 sã♦
t❛✐s q✉❡ α = {f, M1, M2, M3} é ❜❛s❡ ❞❡ A1. ❊♥tã♦ ϕ : P2 −→ Rf, ❞❡✜♥✐❞❛ ♣♦r
ϕ([a:b:c]) =Z(f, aM1+bM2+cM3), é ✉♠❛ ❜✐❥❡çã♦✳
❉❡♠♦♥str❛çã♦✿ ❈♦♠♦α ={f, M1, M2, M3}é ▲✳■✳ ❡♠A1s❡❣✉❡ q✉❡aM1+bM2+cM3✱
♦♥❞❡ [a :b : c] ∈ P2, é ▲✳■ ❝♦♠ f. ❉❡ss❛ ❢♦r♠❛ s❡❣✉❡ q✉❡ l =Z(f, aM1+bM2+cM3)
é ✉♠❛ r❡t❛✳ ❆❧é♠ ❞✐ss♦✱ é ❝❧❛r♦ q✉❡l ⊂Z(f). ❖❜s❡r✈❡♠♦s t❛♠❜é♠ q✉❡ s❡[a:b :c] = [a′ :b′ :c′]✱ ❡♥tã♦Z(f, aM1+bM2+cM3) = Z(f, a′M1+b′M2+c′M3)✳ ■ss♦ ♠♦str❛ q✉❡ ❛
❛♣❧✐❝❛çã♦ϕ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ❡ é ✐♥❥❡t✐✈❛✳ P❡❧♦ q✉❡ ✜③❡♠♦s ❛❝✐♠❛ t❡♠♦s q✉❡ s❡l⊂Z(f)
é ✉♠❛ r❡t❛✱ ❡♥tã♦l =Z(f, L),❝♦♠L❡f ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s ❡♠ A1✳ P♦r ♦✉tr♦
❧❛❞♦ ❞❡✈❡♠ ❡①✐st✐rα, a, b, c∈C✱ ♥ã♦ t♦❞♦s ♥✉❧♦s✱ t❛✐s q✉❡L=αf+aM1+bM2+cM3✳
◆❡ss❛s ❝♦♥❞✐çõ❡s ❞❡✈❡♠♦s t❡r[a:b :c]∈P2 ✳ ❉❛í
l =Z(f, αf +aM1+bM2+cM3) = Z(f, aM1+bM2+cM3).
▼♦str❛♥❞♦ ❛ss✐♠ q✉❡ ϕ é s♦❜r❡❥❡t✐✈❛✳
❖ t❡♦r❡♠❛ ❛❝✐♠❛ ♠♦str❛ q✉❡ t♦❞❛ s✉♣❡r❢í❝✐❡ Z(f), ❝♦♠ f ∈ A1 ♥ã♦ ♥✉❧♦✱ ❝♦♥té♠
✉♠❛ q✉❛♥t✐❞❛❞❡ ✐♥✜♥✐t❛ ❞❡ r❡t❛s✳ ❊♠ ♣❛rt✐❝✉❧❛rRf 6=∅.
❙✉♣❡r❢í❝✐❡s ❞❡ ❣r❛✉ ✷
P❛r❛ ❡st✉❞❛r ♦ ❝❛s♦ ❡♠ q✉❡ ❛ s✉♣❡r❢í❝✐❡ S = Z(f) é ❞❛❞❛ ♣♦r ✉♠ ♣♦❧✐♥ô♠✐♦
❤♦♠♦❣ê♥❡♦ ♥ã♦ ♥✉❧♦ ❞❡ ❣r❛✉ 2 ❡♠ A. ❱❛♠♦s ♣r❡❝✐s❛r ❞❡ ✉♠ t❡♦r❡♠❛ ❞❡ ❝❧❛ss✐✜❝❛çã♦
❜❛st❛♥t❡ ✐♠♣♦rt❛♥t❡✳
❚❡♦r❡♠❛ ✶✳✸ ✭❚❡♦r❡♠❛ ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❞❛s ❤✐♣❡rs✉♣❡r❢í❝✐❡s q✉á❞r✐❝❛s ❡♠ Pn✮ ❙❡❥❛
f ∈ C[x0, . . . , xn] ✉♠ ♣♦❧✐♥ô♠✐♦ ♥ã♦ ♥✉❧♦ ❤♦♠♦❣ê♥❡♦ ❞❡ ❣r❛✉ 2. ❊♥tã♦ ❡①✐st❡ ✉♠❛
♠✉❞❛♥ç❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ♣r♦❥❡t✐✈❛s T : Pn −→ Pn t❛❧ q✉❡ T
•f é ✉♠ ❞♦s s❡❣✉✐♥t❡s
♣♦❧✐♥ô♠✐♦s
f0 =x20,
f1 =x2
0+x21,
✳✳✳
fn =x20+. . .+x2n.
❉❡♠♦♥str❛çã♦✿ ❱❡r ♣á❣✳ 411 ❞❡ ❬✸❪✳
❈♦r♦❧ár✐♦ ✶✳✶ ❙❡❥❛ f ∈ A2 ♥ã♦ ♥✉❧♦✳ ❊♥tã♦ ❡①✐st❡ ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s
♣r♦❥❡t✐✈❛s T :P3 −→P3 t❛❧ q✉❡ T
•f é ✉♠ ❞♦s s❡❣✉✐♥t❡s ♣♦❧✐♥ô♠✐♦s
f0 =x2
0,
f1 =x2
0+x21, f2 =x20+x21 +x22
f3 =x2
0+x21 +x22+x23.
❊♥tã♦✱ ❞❡s❞❡ q✉❡ ♠✉❞❛♥ç❛ ❞❡ ❝♦♦r❞❡♥❛❞❛ ♣r♦❥❡t✐✈❛ ♣r❡s❡r✈❛ r❡t❛s✱ ♣❛r❛ ♠♦str❛r q✉❡ ❝❛❞❛ s✉♣❡r❢í❝✐❡ q✉á❞r✐❝❛ S =Z(f) ❡♠ P3 ❝♦♥té♠ r❡t❛s ❜❛st❛ ♠♦str❛r q✉❡ S
i =Z(fi),
❝♦♠ i= 0,1,2,3, ❝♦♥té♠ r❡t❛s✳
◆♦t❡♠♦s q✉❡S0 =Z(x2
0) = Z(x0).❉❛íl =Z(x0, ax1+bx2+cx3),❝♦♠[a:b:c]∈P2,
é ✉♠❛ r❡t❛ q✉❡ ❡stá ❝♦♥t✐❞❛ ❡♠S0 =Z(f0).◆❛ r❡❛❧✐❞❛❞❡ t❡♠♦s q✉❡
Rf0 ={Z(x0, ax1+bx2+cx3)|[a:b:c]∈P
2
}.
P❛r❛ ♠♦str❛r q✉❡ S1 = Z(f1) ❝♦♥té♠ r❡t❛s✱ ♦❜s❡r✈❡♠♦s q✉❡ f1 = x2
0 + x21 =
(x0 +ix1)(x0 − ix1). ■ss♦ ♠♦str❛ q✉❡ ❛ r❡t❛ l = Z(x0 +ix1, x0 − ix1) = Z(x0, x1)
❡stá ❝♦♥t✐❞❛ ❡♠ S1. P♦rt❛♥t♦ Rf1 6= ∅. ❆❧é♠ ❞✐ss♦✱ ❝♦♠♦ S1 é ❛ ✉♥✐ã♦ ❞♦s ♣❧❛♥♦s
Z(x0+ix1)❡Z(x0−ix1),❡ ❡ss❡s ❝♦♥tê♠ ✐♥✜♥✐t❛s r❡t❛s✳ ❙❡❣✉❡ q✉❡S1 ❝♦♥té♠ ✐♥✜♥✐t❛s
r❡t❛s✳
❱❛♠♦s ❡st✉❞❛r ❛❣♦r❛ ❛ s✉♣❡r❢í❝✐❡S2 =Z(f2). ❖❜s❡r✈❡♠♦s q✉❡ ♦ ♣♦♥t♦ v = [0 : 0 : 0 : 1]é ♦ ú♥✐❝♦ ♣♦♥t♦ s✐♥❣✉❧❛r ❞❡S2. ❈♦♥s✐❞❡r❡♠♦s ❛ ❝✉r✈❛C′ =Z(x2
0+x21+x22)⊂P2.
P❛r❛ ❝❛❞❛ ♣♦♥t♦p′ = [p0 :p1 :p2]∈C′,❝♦♥s✐❞❡r❡♠♦s ❛ r❡t❛ l
p′,v ⊂P3 ❞❛❞❛ ♣♦r
lp′,v ={[αp0 :αp1 :αp2 :β]|[α:β]∈P1}.
❖❜s❡r✈❡♠♦s q✉❡ lp′,v ❡stá ❝♦♥t✐❞♦ ❡♠ S2, ♣♦✐s ❞❛❞♦ [αp0 : αp1 : αp2 : β]∈ lp′,v t❡♠✲s❡
q✉❡ f2(αp0, αp1, αp2, β) =p2
0 +p21+p22 = 0. ❉❡ss❛ ❢♦r♠❛ ❝♦♥str✉í♠♦s ✉♠❛ ❢❛♠í❧✐❛ ❞❡
r❡t❛s{lp′,v}p′∈C′ ❝♦♥t✐❞❛s ❡♠S2 ♣❛ss❛♥❞♦ ♣❡❧♦ ♣♦♥t♦v.◆❛ r❡❛❧✐❞❛❞❡ é ♣♦ssí✈❡❧ ♠♦str❛r
q✉❡ s❡ l é ✉♠❛ r❡t❛ q✉❛❧q✉❡r ❝♦♥t✐❞❛ ❡♠ S2 ❡♥tã♦ v ∈ l ❡ l ∈ {lp′,v}p′∈C′✳ ▲♦❣♦ S2
❝♦♥té♠ ✐♥✜♥✐t❛ r❡t❛s✳ P♦rt❛♥t♦Rf2 6=∅.
❱❛♠♦s ❛❣♦r❛ ❡st✉❞❛r S3 = Z(f3). ◆♦t❡♠♦s q✉❡ g0 = x0+ix1, g1 = x0−ix1, g2 =
x2 + ix3 ❡ g3 = −x2 + ix3 sã♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s ❡♠ A1. ❆❧é♠ ❞✐ss♦✱ f3 = g0g1 −g2g3. ■ss♦ ♥♦s ♠♦str❛✱ ♣♦r ❡①❡♠♣❧♦✱ q✉❡ l = Z(g0, g2) ❡stá ❝♦♥t✐❞❛ ❡♠
S3. ❆ss✐♠ Rf3 6=∅.
❙❡❥❛T :P3 −→P3 ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ♣r♦❥❡t✐✈❛s t❛❧ q✉❡ T
•gi =xi,❝♦♠
i= 0,1,2,3.❉❡ss❛ ❢♦r♠❛T•f3 =x0x3−x1x2.❖✉ s❡❥❛✱ S3 é ♣r♦❥❡t✐✈❛♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡
❛ S =Z(f), ♦♥❞❡ f = x0x3−x1x2. P❛r❛ ❝❛❞❛ ♣♦♥t♦ p = [a0 :a1] ∈P1 ❝♦♥s✐❞❡r❡♠♦s
❛s r❡t❛s (
Lp ={[a0u:a0v :a1u:a1v]|[u:v]∈P1},
Mp ={[a0u:a1u:a0v :a1v]|[u:v]∈P1}.
❖❜s❡r✈❡♠♦s q✉❡ ❛ r❡t❛ Lp, ♣♦r ❡①❡♠♣❧♦✱ é ❞❡t❡r♠✐♥❛❞❛ ♣❡❧♦s ♣♦♥t♦s [a0 : 0 :a1 : 0] ❡
[0 :a0 : 0 :a1]. ◆ã♦ é ❞✐❢í❝✐❧ ✈❡r✐✜❝❛r q✉❡ Lp ❡ Mp ❡stã♦ ❝♦♥t✐❞❛s ❡♠ S = Z(f), ♣❛r❛
t♦❞♦ p ∈ P1. ▲♦❣♦ S3 ❝♦♥té♠ ✉♠❛ q✉❛♥t✐❞❛❞❡ ✐♥✜♥✐t❛ ❞❡ r❡t❛s✳ ◆❛ ✈❡r❞❛❞❡ ♣♦❞❡✲s❡
♠♦str❛r q✉❡ ❛s ❢❛♠í❧✐❛s ❞❡ r❡t❛s L={Lp}p∈P1 ❡ M={Mp}p∈P1 ♣♦ss✉❡♠ ❛s s❡❣✉✐♥t❡s
♣r♦♣r✐❡❞❛❞❡s✿
✶✳ Lp ∩Lq =∅ ❡Mp∩Mq =∅, ♣❛r❛ t♦❞♦p, q ∈P1, ❝♦♠ p6=q.
✷✳ Lp ∩Mq ❝♦♥s✐st❡ ❡♠ ✉♠ ú♥✐❝♦ ♣♦♥t♦✱ ♣❛r❛ t♦❞♦ p, q ∈P1.
✸✳ ❉❛❞♦ x∈S, ❡①✐st❡ ✉♠❛ ú♥✐❝❛ r❡t❛ L∈ L ❡ M ∈ M t❛✐s q✉❡ L∩M ={x}.
✹✳ ❙❡ l⊂S é ✉♠❛ r❡t❛✱ ❡♥tã♦ l ∈ L ♦✉l ∈ M.
P♦r t✉❞♦ ♦ q✉❡ ✜③❡♠♦s ❛té ❛❣♦r❛ ♠♦str❛♠♦s q✉❡ Rf é ✉♠ ❝♦♥❥✉♥t♦ ✐♥✜♥✐t♦ s❡♥❞♦
f ✉♠ ♣♦❧✐♥ô♠✐♦ ❤♦♠♦❣ê♥❡♦ ♥ã♦ ♥✉❧♦ ❞❡ ❣r❛✉1 ♦✉2.
❙✉♣❡r❢í❝✐❡s ❞❡ ❣r❛✉ ✸
■r❡♠♦s ❛❣♦r❛ ❡st✉❞❛r ♦ ❝❛s♦ ❡♠ q✉❡ S =Z(f), ♦♥❞❡ f ∈ A3− {0}. ❊ss❡ ♣r♦❜❧❡♠❛
é ❜❡♠ ♠❛✐s ❞✐❢í❝✐❧ q✉❡ ♦s ♣r♦❜❧❡♠❛s ❛♥t❡r✐♦r❡s✳ P❛r❛ r❡s♦❧✈ê✲❧♦ ♣r❡❝✐s❛r❡♠♦s ✐♥tr♦❞✉③✐r ♥♦✈♦s ❝♦♥❝❡✐t♦s ❡ ❞❡s❡♥✈♦❧✈❡r ✉♠ ♣♦✉❝♦ ♠❛✐s ❛ t❡♦r✐❛✳
❉❡✜♥✐çã♦ ✶✳✷ ❙❡❥❛ V ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ s♦❜r❡ C ❞❡ ❞✐♠❡♥sã♦ n. P❛r❛ ❝❛❞❛ 0≤d≤
n, ❝♦♥s✐❞❡r❡♠♦s ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s s✉❜❡s♣❛ç♦s ❞❡ ❞✐♠❡♥sã♦ d ❞❡ V✱
Gd(V) = {W |W ≤V ❡ dimW =d}.
❚❛❧ ❝♦♥❥✉♥t♦ é ❝❤❛♠❛❞♦ ❞❡ ❣r❛ss♠❛♥♥✐❛♥❛ ❞❡ s✉❜❡s♣❛ç♦s ❞❡ V ❞❡ ❞✐♠❡♥sã♦ d.
❊①✐st❡ ✉♠❛ ✐❞❡♥t✐✜❝❛çã♦ ♥❛t✉r❛❧ ❞❡G1(Cn+1)❝♦♠Pn✳ ❈♦♠ ❡❢❡✐t♦✱ ❛ ❛♣❧✐❝❛çã♦ ❞❛❞❛
♣♦r
ϕ: Pn −→ G1(Cn+1)
¯
v −→ [v].
é ✉♠❛ ❜✐❥❡çã♦✳ ❆ ♣❛rt✐r ❞❡ss❛ ❜✐❥❡çã♦ ♣♦❞❡♠♦s ✐♥❞✉③✐r ❡♠ G1(Cn+1) ✉♠❛ t♦♣♦❧♦❣✐❛✱
❡①♣❧✐❝✐t❛♠❡♥t❡ t❡♠♦s
U ⊂G1(Cn+1)é ❢❡❝❤❛❞♦
⇐⇒ϕ−1(U)
⊂Pn é ❢❡❝❤❛❞♦✳
❈♦♥s✐❞❡r❡♠♦s P = {l : l é ✉♠❛ r❡t❛ ❡♠ P3}. ❙❛❜❡♠♦s q✉❡ ❝❛❞❛ r❡t❛ ❞❡ P3 é ❛
♣r♦❥❡t✐✈✐③❛çã♦ ❞❡ ✉♠ s✉❜❡s♣❛ç♦ ❞❡ ❞✐♠❡♥sã♦ 2 ❞❡ C4. ❊ss❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ é ✉♠❛
❜✐❥❡çã♦✳ ❊①♣❧✐❝✐t❛♠❡♥t❡ t❡♠♦s q✉❡ ψ :G2(C4)−→P, W 7→P(W)é ✉♠❛ ❜✐❥❡çã♦✳ ❉❡
♠♦❞♦ ❛♥á❧♦❣♦ ❛♦ q✉❡ ❢♦✐ ❢❡✐t♦ ❛♥t❡r✐♦r♠❡♥t❡✱ ♣♦❞❡♠♦s ✐♥❞✉③✐r ✉♠❛ t♦♣♦❧♦❣✐❛ ❡♠G2(C4)
s❡♥❞♦ ❡st❛ ❞❛❞❛ ♣♦r ✉♠❛ ❛♣❧✐❝❛çã♦ ❝❤❛♠❛❞❛ ♠❡r❣✉❧❤♦ ❞❡ P❧ü❝❦❡r q✉❡ ❞❡✜♥✐r❡♠♦s ❛ s❡❣✉✐r✳
❙❡❥❛♠ W ∈ G2(C4) ❡ ∆ = {u, v} ✉♠❛ ❜❛s❡ ❞❡ W, ♦♥❞❡ u = (u0, u1, u2, u3)
❡ v = (v0, v1, v2, v3). ❈♦♥s✐r❡♠♦s M∆ ❛ ♠❛tr✐③ ❝✉❥❛s ❧✐♥❤❛s sã♦ ❞❡t❡r♠✐♥❛❞❛s ♣❡❧❛s
❝♦♦r❞❡♥❛❞❛s ❞♦ ✈❡t♦ru ❡v r❡s♣❡❝t✐✈❛♠❡♥t❡
M∆=
"
u0 u1 u2 u3
v0 v1 v2 v3
#
.
❙❡❥❛♠ pij ♦s ❞❡t❡r♠✐♥❛♥t❡s ❞♦s ♠❡♥♦r❡s 2×2 ❞❡M∆
pij =
ui uj
vi vj
=uivj−ujvi, ❝♦♠ 0≤i < j ≤3.
❆❣♦r❛ s❡∆′ ={u′, v′}é ♦✉tr❛ ❜❛s❡ ❞❡W,♦♥❞❡u′ = (u′
0, u′1, u′2, u′3)❡v′ = (v0′, v1′, v2′, v′3).
❊♥tã♦ ❡①✐st❡♠α, β, γ ❡ δ∈C✱ ❝♦♠ λ=αδ−γβ 6= 0,t❛✐s q✉❡
(
u=αu′+βv′
v =αu′+βv′.
▲♦❣♦ ui =αu′i+βv′i ❡vj =γu′j +δvj′✳ ◆♦t❡♠♦s q✉❡
uivj−ujvi = (αu′i+βvi′)(γu′j+δvj′)−(αu′j+βv′j)(γu′i+δvi′) = (αδ−γβ)(u′iv′j−u′jv′i).
❉✐ss♦ r❡s✉❧t❛ q✉❡pij =λp′ij. ❉❡ss❛ ❢♦r♠❛ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ❛ ❢✉♥çã♦ω:G2(C4)−→
P5, ♦♥❞❡ ω(W) = [p01 :p02: p03 :p12 :p13 :p23], ❝♦♠ p
ij =uivj−ujvi ,0≤i < j ≤3✱
♦♥❞❡ {(u0, u1, u2, u3),(v0, v1, v2, v3)} é ✉♠❛ ❜❛s❡ ❞❡ W. ❆ ❢✉♥çã♦ ω é ❝❤❛♠❛❞❛ ❞❡
▼❡r❣✉❧❤♦ ❞❡ P❧ü❝❦❡r✳
Pr♦♣♦s✐çã♦ ✶✳✸ ❆ ❛♣❧✐❝❛çã♦ ω é ✐♥❥❡t✐✈❛ ❡ Im(ω) = Q, ♦♥❞❡ Q = Z(f), ❝♦♠
f =x0x5−x1x4+x2x3 ∈C[x0, x1, x2, x3, x4, x5].
❉❡♠♦♥str❛çã♦✿ ❱❡r ♣á❣✳ 13❞❡ ❬✶✶❪ ✳
❆ ❤✐♣❡rs✉♣❡r❢í❝✐❡ q✉á❞r✐❝❛ Q = Z(f) ⊂ P5 é ❝❤❛♠❛❞❛ ❞❡ q✉á❞r✐❝❛ ❞❡ P❧ü❝❦❡r✳
❚❡♠♦s ❛s ❝♦rr❡s♣♦♥❞ê♥❝✐❛s ❜✐❥❡t✐✈❛s ψ−1 : P −→ G2(C4) ❡ ω : G2(C4) −→ Q✳
❙❛❧✐❡♥t❛♠♦s q✉❡ t❛♠❜é♠ ✉s❛r❡♠♦s ω ♣❛r❛ ❞❡♥♦t❛r ❛ ❝♦♠♣♦s✐çã♦ ω ◦ ψ−1✳ ▲♦❣♦ ♦
♠❡r❣✉❧❤♦ ❞❡ P❧ü❝❦❡r ♥♦s ♣❡r♠✐t❡ ✐❞❡♥t✐✜❝❛r ❝❛❞❛ r❡t❛ ❡♠P3 ❝♦♠ ✉♠ ♣♦♥t♦ ❡♠ Q.❖✉
s❡❥❛✱ ♣♦❞❡♠♦s ✈❡r P ❝♦♠♦ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❛❧❣é❜r✐❝❛ ❡♠ P5✳ ❖❜s❡r✈❡♠♦s ❛✐♥❞❛ q✉❡
❝♦♠♦dimQ= 4, ❡♥tã♦ G2(C4)t❡♠ ❞✐♠❡♥sã♦ 4.
▲❡♠❜r❡♠♦s q✉❡ A = C[x0, x1, x2, x3] ❡ Ad = C[x0, x1, x2, x3]d, ♣❛r❛ ❝❛❞❛ d ≥ 0
✐♥t❡✐r♦✱ ❛❧é♠ ❞✐ss♦ Ad é s✉❜❡s♣❛ç♦ ✈❡t♦r✐❛❧ A. ❆❧é♠ ❞✐ss♦✱ dimAd= d+33
.
❉❛❞❛l ✉♠❛ r❡t❛ ❡♠P3.❙❡❥❛I(l)
d❛ ❝♦❧❡çã♦ ❞❡ t♦❞♦s ♦s ♣♦❧✐♥ô♠✐♦s ❤♦♠♦❣ê♥❡♦s ❞❡
❣r❛✉ d ❞❡I(l), ♦✉ s❡❥❛✱ I(l)d =I(l)∩ Ad.
Pr♦♣♦s✐çã♦ ✶✳✹ ❙❡❥❛ l⊂P3 ✉♠❛ r❡t❛✳ ❊♥tã♦ ❛ ❞✐♠❡♥sã♦ ❞❡ I(l)
d é ❞❛❞❛ ♣♦r
dimI(l)d=
d(d+ 1)(d+ 5)
6 .
❉❡♠♦♥str❛çã♦✿ ❙❡♥❞♦ l ⊂ P3 ✉♠❛ r❡t❛✱ ❡①✐st❡♠ L1, L2 ∈ A
1 ❧✐♥❡❛r♠❡♥t❡
✐♥❞❡♣❡♥❞❡♥t❡s t❛✐s q✉❡ l = Z(L1, L2). ▲♦❣♦ I(l) = hL1, L2i✳ P❛r❛ d = 0, t❡♠♦s
q✉❡I(l)0 =C∩ I(l) ={0}✳ ▲♦❣♦ dimI(l)0 = 0. P❛r❛d= 1 t❡♠♦s q✉❡
I(l)1 =A1 ∩ I(l) = [x0, x1, x2, x3]∩ hL1, L2i= [L1, L2].
❆ss✐♠✱ dimI(l)1 = 2. P❛r❛d≥2 ❞❡✜♥❛♠♦s
ϕ: Ad−1× Ad−1 −→ I(l)d
(P, Q) 7−→P L1+QL2.
❖❜s❡r✈❡♠♦s q✉❡ ϕ é ✉♠❛ ❛♣❧✐❝❛çã♦ ❧✐♥❡❛r s♦❜r❡❥❡t✐✈❛✳ P❡❧♦ t❡♦r❡♠❛ ❞♦ ♥ú❝❧❡♦ ❡ ❞❛
✐♠❛❣❡♠
dim(Ad−1× Ad−1) = dim ker(ϕ) + dimI(l)d.
❆❣♦r❛ ker(ϕ) = {(P, Q) ∈ Ad−1 × Ad−1 | P L1 +QL2 = 0}. ▲♦❣♦ (P, Q) ∈ ker(ϕ)
❡♥tã♦P L1 =−QL2.❉❡s❞❡ q✉❡Aé ✉♠ ❞♦♠í♥✐♦ ❞❡ ❢❛t♦r❛çã♦ ú♥✐❝❛✱ s❡❣✉❡ q✉❡P =M L2
❡Q=−M L1,❝♦♠ M ∈ Ad−2. ❈♦♥❝❧✉í♠♦s ❛ss✐♠ q✉❡ker(ϕ) = {(M L2,−M L1) :M ∈
Ad−2}✳ P♦rt❛♥t♦ dim ker(ϕ) = dimAd−2. ❉❡ss❛ ❢♦r♠❛
dimI(l)d = dim(Ad−1× Ad−1)−dim ker(ϕ)
= 2 dim(Ad−1)−dimAd−2
= 2 d+23 − d+1 3
= 2d(d+1)(6 d+2) − (d−1)d6(d+1)
= d(d6+1)(2(d+ 2)−(d−1)) = d(d+1)(6d+5).
P♦rt❛♥t♦dimI(l)d= d(d+1)(6d+5) ♣❛r❛ t♦❞♦d≥0 ✐♥t❡✐r♦✳
❙❡❥❛ P(Ad) ❛ ♣r♦❥❡t✐✈✐③❛çã♦ ❞♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ Ad, ♦✉ s❡❥❛✱ P(Ad) = {[g] : g ∈
Ad − {0}}, ♦♥❞❡ [g] é ❛ ❝❧❛ss❡ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ g. ❈♦♠♦ dimAd = d+33
✱ ❡♥tã♦
dimP(Ad) = d+33
−1✳ ❚❡♠♦s q✉❡P(Ad) é ✉♠❛ ✈❛r✐❡❞❛❞❡ ♣r♦❥❡t✐✈❛✳ ❉❡✜♥❛♠♦s
∆ ={([g], W)∈P(Ad)×G2(C4)|l=P(W)⊂Z(g)}.
❈♦♥s✐❞❡r❡♠♦s ❛s ♣r♦❥❡çõ❡s π1 : ∆−→P(Ad) ❡ π2 : ∆−→G2(C4).❖❜s❡r✈❡♠♦s q✉❡ π1
é s♦❜r❡❥❡t✐✈❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♣❛r❛ t♦❞♦f ∈ Ad, ❝♦♠f = 06 ,❡①✐st❡ ✉♠❛ r❡t❛ l=P(W)
❝♦♥t✐❞❛ ❡♠ Z(f). ❖✉ s❡❥❛✱ s❡ π1 é s♦❜r❡❥❡t✐✈❛ ❡♥tã♦ t♦❞❛ s✉♣❡r❢í❝✐❡ ❞❡ ❣r❛✉ d ❡♠ P3
❝♦♥té♠ ❛♦ ♠❡♥♦s ✉♠❛ r❡t❛✳ ◆♦t❡♠♦s t❛♠❜é♠ q✉❡ ❛ ♣ré✲✐♠❛❣❡♠ ❞❡[g] ♣♦r π1 é ❞❛❞❛
♣♦r
π−1
1 ([g]) = {[g]} × {W ∈G2(C4)|l=P(W)⊂Z(g)}.
❖✉ s❡❥❛✱π−1
1 ([g])❡stá ❡♠ ❜✐❥❡çã♦ ❝♦♠ ♦ ❝♦♥❥✉♥t♦ ❞❛s r❡t❛s q✉❡ ❡stã♦ ❝♦♥t✐❞❛s ❡♠Z(g).
❆ ❛♣❧✐❝❛çã♦ π2 é s❡♠♣r❡ s♦❜r❡❥❡t✐✈❛✳ ❉❡ ❢❛t♦✱ ❞❛❞♦ l =P(W)✱ ❝♦♠ W ∈ G2(C4),
❡♥tã♦ ❡①✐st❡♠ L1, L2 ∈ A1 ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s t❛✐s q✉❡ l = Z(L1, L2). ❆ss✐♠
g =Ld
1 ∈Ad ❡ ❛ s✉♣❡r❢í❝✐❡Z(g) ❝♦♥té♠ l. ❆❣♦r❛ ❞❛❞♦ W ∈G2(C4) t❡♠♦s q✉❡
π2−1(W) = {([g], W)∈P(Ad)×G2(C4) :l =P(W)⊂Z(g)}
= {[g]∈P(Ad) :l =P(W)⊂Z(g)} × {W}
= {[g]∈P(Ad) :g ∈ I(l)d} × {W}
= P(I(l)d)× {W}.
❱❛♠♦s ❛♣❧✐❝❛r ♦ ❚❡♦r❡♠❛ ❞❛ ❞✐♠❡♥sã♦ ❞❛ ✜❜r❛✱ ✈❡r ♣á❣✳ 75❞❡ ❬✶✼❪✱ ♣❛r❛ ❝❛❧❝✉❧❛r
❛ ❞✐♠❡♥sã♦ ❞❡ ∆. ❚❡♠♦s q✉❡ π2 : ∆ −→ G2(C4) é ✉♠ ♠♦r✜s♠♦ s♦❜r❡❥❡t✐✈♦✳ ▲♦❣♦
♦ ❚❡♦r❡♠❛ ❞❛ ❞✐♠❡♥sã♦ ❞❛s ✜❜r❛s ♥♦s ❣❛r❛♥t❡ q✉❡ ❡①✐st❡ ✉♠ ❛❜❡rt♦ ♥ã♦ ✈❛③✐♦ U ❞❡ G2(C4) t❛❧ q✉❡
dimπ−1
2 (W) = dim ∆−dimG2(C4), ∀ W ∈U.
▼❛s dimπ−1
2 (W) = dimP(I(l)d). ❆❣♦r❛ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✹ t❡♠♦s q✉❡ dimI(l)d = d(d+1)(d+5)
6 , ❧♦❣♦ dimP(I(l)d) =
d(d+1)(d+5)
6 −1. ❙❛❜❡♠♦s t❛♠❜é♠ q✉❡ dimG2(C
4) = 4.
P♦rt❛♥t♦
dim ∆ = dimπ−1
2 (W) + dimG2(C4)
= dimP(I(l)d) + dimG2(C4)
= d(d+1)(6 d+5) −1 + 4 = d(d+1)(6 d+5) + 3.
❏á ✈✐♠♦s ❛❝✐♠❛ q✉❡ ❛ s♦❜r❡❥❡t✐✈✐❞❛❞❡ ❞❡ π1 ✐♠♣❧✐❝❛ q✉❡ t♦❞❛ s✉♣❡r❢í❝✐❡ ❞❡ ❣r❛✉ d ❝♦♥té♠ r❡t❛s✳ ❆❧é♠ ❞✐ss♦✱ ♦❜t❡♠♦s ❛❝✐♠❛ ✉♠❛ ❢ór♠✉❧❛✱ ♣❛r❛ ❝❛❞❛ d, q✉❡ ♥♦s ❞á
❛ ❞✐♠❡♥sã♦ ❞❡ ∆✳ ❱❛♠♦s ❛❣♦r❛ ♣r♦❝✉r❛r ♣❛r❛ q✉❛✐s ✈❛❧♦r❡s ❞❡ d π1 ♣♦❞❡r✐❛ s❡r
s♦❜r❡❥❡t✐✈❛✳ ❙✉♣♦♥❤❛♠♦s q✉❡π1 : ∆ −→P(Ad)é s♦❜r❡❥❡t✐✈❛✳ ◆♦✈❛♠❡♥t❡ ♣❡❧♦ ❚❡♦r❡♠❛
❞❛ ❞✐♠❡♥sã♦ ❞❛s ✜❜r❛s s❡❣✉❡ q✉❡
dimπ−11([g])≥dim ∆−dimP(Ad), ∀ [g]∈P(Ad).
P❛r❛ ❝♦♥t✐♥✉❛r ♦ ♥♦ss♦ ❡st✉❞♦ ❝♦♥s✐❞❡r❡♠♦s ♦s s❡❣✉✐♥t❡s ❡①❡♠♣❧♦s✿
❊①❡♠♣❧♦s ✶✳✶ ✶✳ ❙❡❥❛S =Z(g)✱ ❝♦♠g =x3
0+x31+x32+x33 ∈ A,❝♦♥té♠ ❡①❛t❛♠❡♥t❡
27 r❡t❛s✱ ✈❡r ♣á❣✳ 1 ❞❡ ❬✶❪✳
✷✳ ❙❡❥❛ Z(f)⊂P3, ❝♦♠ f =xd
3+x0x
d−1
1 +x1x
d−1
2 +x2x
d−1
0 ∈ A,♦♥❞❡ d≥4✳ ❊♥tã♦ Z(f) é ✉♠❛ s✉♣❡r❢í❝✐❡ ♥ã♦ s✐♥❣✉❧❛r q✉❡ ♥ã♦ ❝♦♥té♠ r❡t❛s✳
❆ ✈❡r✐✜❝❛çã♦ ❞♦ ❊①❡♠♣❧♦ 2 s❡rá ❢❡✐t❛ ♠❛✐s ❛ ❢r❡♥t❡ ✭✈❡r Pr♦♣♦s✐çã♦ ✶✳✺✮✳ ◆♦s
❡①❡♠♣❧♦s ❛❝✐♠❛ t❡♠♦s q✉❡dimπ−1
1 ([f]) = 0,♣❛r❛ t♦❞♦d≥3✱ ♣♦✐s ♥♦ ♣r✐♠❡✐r♦ ❡①❡♠♣❧♦ π−1
1 ([f]) é ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦ ❡ ♥♦ s❡❣✉♥❞♦ ❡①❡♠♣❧♦ π1−1([f]) é ✈❛③✐♦✳ ▲♦❣♦✱ ❝❛s♦ π1
s❡❥❛ s♦❜r❡❥❡t✐✈❛✱ ❞❡✈❡♠♦s t❡r
dim ∆ = dimP(Ad).
■ss♦ ✐♠♣❧✐❝❛ q✉❡ d(d+1)(d+5)
6 + 3 =
(d+1)(d+2)(d+3)
6 −1. ❆ss✐♠
4 = (d+1)(d+2)(6 d+3) − d(d+1)(6d+5) = (d+1)[(d+2)(d6+3)−d(d+5)] = (d+1)[d2+56d+6−d2−5d] = d+ 1.
❈♦♥❝❧✉í♠♦s ❛ss✐♠ q✉❡ d = 3. ❖✉ s❡❥❛✱ ♣❛r❛ q✉❡ π1 s❡❥❛ s♦❜r❡❥❡t✐✈❛ é ♥❡❝❡ssár✐♦ q✉❡ d= 3✳ P♦r t✉❞♦ q✉❡ ✜③❡♠♦s ❛❝✐♠❛ ❝❤❡❣❛♠♦s ❛♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿
❈♦r♦❧ár✐♦ ✶✳✷ ❊①✐st❡♠ s✉♣❡r❢í❝✐❡s ❞❡ ❣r❛✉ d≥4 q✉❡ ♥ã♦ ❝♦♥té♠ r❡t❛s✳
❚❡♦r❡♠❛ ✶✳✹ ❙❡ d = 3, ❡♥tã♦ π1 : ∆ −→ A3 é s♦❜r❡❥❡t✐✈❛✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ t♦❞❛
s✉♣❡r❢í❝✐❡ ❞❡ ❣r❛✉ 3 ❝♦♥té♠ r❡t❛s✳
❉❡♠♦♥str❛çã♦✿ ❈♦♠♦dimA3 = 3+33
= 63= 20s❡❣✉❡ q✉❡dimP(A3) = 20−1 = 19.
❆❧é♠ ❞✐ss♦✱ ❛ ❞✐♠❡♥sã♦ ❞❡∆♣❛r❛ d= 3 é ❞❛❞❛ ♣♦r 3(3+1)(3+5)6 + 3 = 19. ❉❡s❞❡ q✉❡ π1
é ♠♦r✜s♠♦ ❡♥tr❡ ✈❛r✐❡❞❛❞❡s ♣r♦❥❡t✐✈❛s s❡❣✉❡ q✉❡π1(∆)é ✉♠ ❢❡❝❤❛❞♦ ❞❡P(A3).❚❡♠♦s
♣❡❧♦ ❊①❡♠♣❧♦ 2 ❛❝✐♠❛ q✉❡ π−1
1 (f)✱ ♦♥❞❡ f = x30+x31+x32+x33 ∈ A, é ✉♠ ❝♦♥❥✉♥t♦
✜♥✐t♦✳ ▲♦❣♦✱ ♣❡❧♦ t❡♦r❡♠❛ ❞❛ ❞✐♠❡♥sã♦ ❞❛s ✜❜r❛s✱ t❡♠♦s q✉❡dim ∆ = dimπ1(∆) = 19.
P♦rt❛♥t♦π1(∆)⊂P(A3)é ✉♠ ❢❡❝❤❛❞♦ ❞❡ ♠❡s♠❛ ❞✐♠❡♥sã♦ ❞❡P(A3)✳ ❉❡s❞❡ q✉❡P(A3)
é ✐rr❡❞✉tí✈❡❧ s❡❣✉❡ q✉❡ P(∆) =P(A3).
▲♦❣♦ Rf 6=∅✱ ♣❛r❛ t♦❞♦ f ♣♦❧✐♥ô♠✐♦ ❤♦♠♦❣ê♥❡♦ ❞❡ ❣r❛✉1,2 ♦✉3.
✶✳✷ ❊①❡♠♣❧♦ ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❞❡ ❣r❛✉
d
≥
4
❡♠
P
3q✉❡ ♥ã♦ ❝♦♥té♠ r❡t❛s
❏á ✈✐♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s r❡t❛s P ❞❡ P3 ❡stá ❡♠ ❜✐❥❡çã♦ ❝♦♠ G 2(C4)✳
❱✐♠♦s t❛♠❜é♠ q✉❡ ♦ ♠❡r❣✉❧❤♦ ❞❡ P❧ü❝❦❡r ω ✭ ✈❡r Pr♦♣♦s✐çã♦ ✶✳✸✮✱ ❡st❛❜❡❧❡❝❡ ✉♠❛
❜✐❥❡çã♦ ❡♥tr❡ G2(C4) ❡ ❛ q✉á❞r✐❝❛ ❞❡ P❧ü❝❦❡r Q = Z(x0x5 −x1x4 +x2x3)✳ ❯s❛r❡♠♦s
♦ ♠❡r❣✉❧❤♦ ❞❡ P❧ü❝❦❡r ♣❛r❛ ❡st❛❜❡❧❡❝❡r ✉♠❛ ♣❛rt✐çã♦ ❝♦♥✈❡♥✐❡♥t❡ ❡♠P✳ ❚❛❧ ♣❛rt✐çã♦ ♥♦s ❞❛rá ✉♠❛ ❢❡rr❛♠❡♥t❛ ♣❛r❛ ❞❡❝✐❞✐r q✉❛♥❞♦ ✉♠❛ s✉♣❡r❢í❝✐❡S ❝♦♥té♠ ♦✉ ♥ã♦ r❡t❛s✳
❉❡✜♥❛♠♦s ♦s s❡❣✉✐♥t❡s s✉❜❝♦♥❥✉♥t♦s ❞❡G2(C4)
E6 ={W ∈G2(W) :w01=w02 =w03=w12=w13 = 0 ❡ w23 6= 0}
E5 ={W ∈G2(W) :w01=w02 =w03=w12= 0 ❡ w136= 0}
E4 ={W ∈G2(W) :w01=w02 =w03= 0 ❡ w12 6= 0}
E3 ={W ∈G2(W) :w01=w02 = 0 ❡ w036= 0}
E2 ={W ∈G2(W) :w01= 0 ❡w026= 0}
E1 ={W ∈G2(W) :w016= 0},
♦♥❞❡ ω(W) = [w01 : w02 : w03 : w12 : w13 : w23]. ❉❡ss❛ ❢♦r♠❛ t❡♠♦s q✉❡ G2(C4) =E1 ∪E2∪E3∪E4∪E5∪E6✱ ❛❧é♠ ❞✐ss♦✱ ❡ss❛ ✉♥✐ã♦ é ❞✐s❥✉♥t❛✳ ❈❤❛♠❛♠♦s
❡ss❛ ♣❛rt✐çã♦ ❞❡ G2(C4) ❞❡ ❡str❛t✐✜❝❛çã♦ ❞❡ G2(C4)✳
❙❡ W ∈ E6 ❡ ∆ = {u, v}✱ ❝♦♠ u= (u0, u1, u2, u3), v = (v0, v1, v2, v3), ❢♦r ✉♠❛ ❜❛s❡
❞❡W, ❡♥tã♦
M∆=
"
u0 u1 u2 u3
v0 v1 v2 v3
#
t❛❧ q✉❡ w01 =w02 =w03 =w12 =w13= 0 ❡ w23 =u2v3−u3v2 6= 0. ❆ ú❧t✐♠❛ ❝♦♥❞✐çã♦
♥♦s ❞✐③ q✉❡ ♣♦❞❡♠♦s ❛♣❧✐❝❛r ♦♣❡r❛çõ❡s ❡❧❡♠❡♥t❛r❡s ❞❡ ♠♦❞♦ q✉❡
" u2 u3 v2 v3 # −→ " 1 0 0 1 # .
❆♣❧✐❝❛♥❞♦ ❡ss❛s ♠✉❞❛♥ç❛s à ♠❛tr✐③M∆,♦❜t❡♠♦s ✉♠❛ ♥♦✈❛ ♠❛tr✐③✱ ❝✉❥❛s ❧✐♥❤❛s ❛✐♥❞❛
❞❡t❡r♠✐♥❛♠ ♦✉tr❛ ❜❛s❡ ❞❡W, ❞❛❞❛ ♣♦r
M∆′ =
"
u′
0 u′1 1 0
v′
0 v1′ 0 1
#
.
❯s❛♥❞♦ ❛❣♦r❛ ❛s ♦✉tr❛s r❡❧❛çõ❡s✱ s❡❣✉❡ q✉❡u′
0 =u′1 =v0′ =v1′ = 0.P♦rt❛♥t♦✱
W = [(0,0,1,0),(0,0,0,1)].
❋❛③❡♥❞♦ ❛❧❣✉♥s ❝á❧❝✉❧♦s s❡♠❡❧❤❛♥t❡s ❛♦s ❛♣❧✐❝❛❞♦s ❛❝✐♠❛ ❞❡s❝♦❜r✐♠♦s q✉❡ ♦s s✉❜❡s♣❛ç♦s ❞❡ss❛ ♣❛rt✐çã♦ ♣♦ss✉❡♠ ❛ s❡❣✉✐♥t❡ ❜❛s❡✿
E6 ={[(0,0,1,0),(0,0,0,1)]},
E5 ={[(0,1, α,0),(0,0,0,1)] :α ∈C}, E4 ={[(0,1,0, α),(0,0,1, β)] :α, β ∈C}, E3 ={[(1, α, β,0),(0,0,0,1)] :α, β ∈C}, E2 ={[(1, α,0, β),(0,0,1, γ)] :α, β, γ ∈C}, E1 ={[(1,0, α, β),(0,1, γ, δ)] :α, β, γ, δ ∈C}.
❙❡❥❛Z(f)⊂P3✉♠❛ s✉♣❡r❢í❝✐❡ ❝♦♥t❡♥❞♦ ❛ r❡t❛l ⊂P3.❊♥tã♦l=P(W),♣❛r❛ ❛❧❣✉♠ W ∈G2(C4).▼❛s ❞❡✈❡♠♦s t❡r W ∈ Ei, ♣❛r❛ ✉♠ ú♥✐❝♦i∈ {1,2,3,4,5,6}. ❈♦♠ ❡ss❛s
✐❞❡✐❛s é ♣♦ssí✈❡❧ ✉s❛r ❡ss❛ ❡str❛t✐✜❝❛çã♦ ❞❡ G2(C4) ♣❛r❛ ❡♥❝♦♥tr❛r ❛s r❡t❛s ❝♦♥t✐❞❛s
♥✉♠❛ s✉♣❡r❢í❝✐❡ ❡♠P3, ❝❛s♦ ❡❧❛ ❝♦♥t❡♥❤❛ ❛❧❣✉♠❛ r❡t❛✳
Pr♦♣♦s✐çã♦ ✶✳✺ ❙❡❥❛ Z(f)⊂P3, ❝♦♠ f =xd
3+x0xd1−1+x1x2d−1+x2xd0−1 ∈ A, ♦♥❞❡ d≥4✳ ❊♥tã♦ Z(f) é ✉♠❛ s✉♣❡r❢í❝✐❡ ♥ã♦ s✐♥❣✉❧❛r q✉❡ ♥ã♦ ❝♦♥té♠ r❡t❛s✳
❉❡♠♦♥str❛çã♦✿ ❈❛❧❝✉❧❛♥❞♦ ❛s ❞❡r✐✈❛❞❛s ♣❛r❝✐❛✐s ❞❡f ❡ ✐❣✉❛❧❛♥❞♦ ❛ ③❡r♦ ♦❜t❡♠♦s ♦
s❡❣✉✐♥t❡ s✐st❡♠❛✿
(1) ∂0f =xd−1
1 + (d−1)x2x
d−2
0 = 0
(2) ∂1f = (d−1)x0xd−2
1 +xd2−1 = 0
(3) ∂2f = (d−1)x1xd−2
2 +xd0−1 = 0
(4) ∂3f =dxd3−1 = 0.
❆ ❡q✉❛çã♦ (4) ♥♦s ❞✐③ q✉❡ x3 = 0. ❙❡ t✐✈❡r♠♦s x0 = 0, ❡♥tã♦✱ ♣♦r (1)✱ x1 = 0. ❊
♣♦r (2), s❡❣✉❡ q✉❡ x2 = 0✳ ❖❜t❡♠♦s ❛s ♠❡s♠❛s ❝♦♥❝❧✉sõ❡s s❡ x1 = 0 ♦✉ x2 = 0.
❙✉♣♦♥❤❛♠♦s ♣♦r ❛❜s✉r❞♦ q✉❡ ♦ s✐st❡♠❛ ❛❝✐♠❛ t❡♥❤❛ s♦❧✉çã♦ ❝♦♠ xi 6= 0, ♣❛r❛
i = 0,1,2. ▼✉❧t✐♣❧✐❝❛♥❞♦ ❛ ❡q✉❛çã♦ (1) ♣♦r x0 ♦❜t❡♠♦s✱ x0xd−1
1 + (d−1)x2x
d−1
0 = 0.
❆❣♦r❛ s✉❜st✐t✉✐♥❞♦ (3) ♥❡ss❛ ❡q✉❛çã♦ s❡❣✉❡ q✉❡ x0xd−1
1 + (d−1)x2(1−d)x1xd2−2 = 0.
❆ss✐♠✱x1(x0xd−2
1 + (d−1)(1−d)x2d−1) = 0. ❉❛í✱ ❞✐✈✐❞✐♥❞♦ ❡ss❛ ❡q✉❛çã♦ ♣♦rx1,t❡♠✲s❡ x0xd1−2+ (d−1)(1−d)xd2−1 = 0. ❙✉❜st✐t✉✐♥❞♦(2) ♥❛ ❡q✉❛çã♦ ❛♥t❡r✐♦r ❝♦♥❝❧✉í♠♦s q✉❡
x0xd−2
1 + (d−1)(1−d)2x0x
d−2
1 =x0x
d−2
1 (1 + (d−1)3) = 0.
❖ q✉❡ ✐♠♣❧✐❝❛ q✉❡ 1 + (d−1)3 = 0. ▼❛s ✐ss♦ é ✉♠ ❛❜s✉r❞♦ ♣♦✐s d ≥4, ❡ x
0xd1−2 6= 0✳
P♦rt❛♥t♦ ❛ s✉♣❡r❢í❝✐❡ Z(f) é ♥ã♦ s✐♥❣✉❧❛r✳
❱❛♠♦s ♠♦str❛r ❛❣♦r❛ q✉❡Z(f) ♥ã♦ ❝♦♥té♠ r❡t❛s✳ ❙✉♣♦♥❤❛♠♦s ♣♦r ❛❜s✉r❞♦ q✉❡ l
s❡❥❛ ✉♠❛ r❡t❛ ❡♠ P3 ❝♦♥t✐❞❛ ❡♠ Z(f).❊♥tã♦ l=P(W), ♦♥❞❡W ∈E
i,♣❛r❛ ✉♠ ú♥✐❝♦
i∈ {1,2,3,4,5,6}.
❈❛s♦ ✶✿ W ∈E6.
❯s❛♥❞♦ q✉❡E6 ={[(0,0,1,0),(0,0,0,1)]}❡ q✉❡ ❛ ❡q✉❛çã♦ ♣❛r❛♠étr✐❝❛ ❞❡l=P(W)
é ❞❛❞❛ ♣♦r l = {[0 : 0 : u : v] : [u : v] ∈ P1}. ❉❛í ❝♦♠♦ l ⊂ Z(f) ❞❡✈❡♠♦s t❡r
f(0,0, u, v) = vd= 0, ∀ [u:v]∈P1. ❖ q✉❡ ❝❧❛r❛♠❡♥t❡ ♥ã♦ ♦❝♦rr❡✳ P♦rt❛♥❞♦W /∈E1.
❈❛s♦ ✷✿ W ∈E5.
❊♥tã♦ W = [(0,1, a,0),(0,0,0,1)], ♣❛r❛ ❛❧❣✉♠ a ∈ C. ❆ss✐♠ l = {[0 : u : au : v] : [u :v]∈P1}✳ ❉❛í ❞❡✈❡♠♦s t❡r f(0, u, au, v) =vd+ad−1ud = 0, ♣❛r❛ t♦❞♦ [u:v]∈ P1.
▼❛s t♦♠❛♥❞♦[0 : 1] ❡♠ P1,❝❤❡❣❛♠♦s ❛♦ ❛❜s✉r❞♦ 0 = 1. ❉❡ss❛ ❢♦r♠❛ W /∈E5.
❈❛s♦ ✸✿ W ∈E4.
❊♥tã♦W = [(0,1,0, a),(0,0,1, b)],♣❛r❛ ❛❧❣✉♠a, b∈C.▲♦❣♦ ♦s ♣♦♥t♦s ❞❡l=P(W)
sã♦ ❞❛ ❢♦r♠❛ [0 : u : v : au+bv], ❝♦♠ [u : v] ∈ P1. ◆♦✈❛♠❡♥t❡✱ ❝♦♠♦ l ⊂ Z(f),
❡♥tã♦ f(0, u, v, au +bv) = (ua + bv)d +uvd−1 = 0, ♣❛r❛ t♦❞♦ [u : v] ∈ P1. ❙❡❥❛ q(x, y) = (xa+by)d+xyd−1, ❡♠ C[x, y]. ❆ ❝♦♥❞✐çã♦ ❛♥t❡r✐♦r s♦❜r❡ f ♥♦s ♠♦str❛ q✉❡ q(x, y) é ♦ ♣♦❧✐♥ô♠✐♦ ♥✉❧♦✳ ❆♣❧✐❝❛♥❞♦ ♦ ❜✐♥ô♠✐♦ ❞❡ ◆❡✇t♦♥ ♣❛r❛q(x, y) ♦❜t❡♠♦s q✉❡
q(x, y) =adxd+
· · ·+bdyd+xyd−1.
■ss♦ r❡✈❡❧❛ q✉❡ ♦s ❝♦❡✜❝✐❡♥t❡s a ❡ b sã♦ ♥✉❧♦s✳ ❉❡ss❛ ❢♦r♠❛ q(x, y) = xyd−1. ▼❛s ❡ss❡
♣♦❧✐♥ô♠✐♦ ♥ã♦ é ♥✉❧♦✳ ❖ q✉❡ ♥♦s ❝♦♥❞✉③ ❛ ✉♠ ❛❜s✉r❞♦✳ P♦rt❛♥t♦ W /∈E4.
❈❛s♦ ✹✿ W ∈E3.
◆❡st❡ ❝❛s♦ t❡♠♦s q✉❡ W = [(1, a, b,0),(0,0,0,1)], ❝♦♠ a, b∈C. ❉❡ss❛ ❢♦r♠❛
l={[u:au:bu:v] : [u:v]∈P1}.
❈♦♠♦ l ⊂ Z(f), t❡♠♦s q✉❡ f(u, au, bu, v) = vd+u(au)d−1 +au(bu)d−1 +buud−1 =
0, ∀ [u : v] ∈ C. P♦r❡♠ ♣❛r❛ [u : v] = [0 : 1] ∈ P1, t❡♠✲s❡ 0 = 1. ❈♦♥❝❧✉í♠♦s✱ ❞❡ss❡
❛❜s✉r❞♦✱ q✉❡ W /∈E3.
❈❛s♦ ✺✿ W ∈E2.
▲♦❣♦ l ={[u :au :v : bu+cv] : [u :v]∈ P1}, ❝♦♠ a, b, c∈ C. ❆❣♦r❛✱ ❞❡ l ⊂Z(f),
❞❡✈❡♠♦s t❡r
f(u, au, v, bu+vc) = (bu+cv)d+u(au)d−1+auvd−1+vud−1 = 0, ∀[u:v]∈P1.
❈♦♥s✐❞❡r❡♠♦s ♦ ♣♦❧✐♥ô♠✐♦q(x, y) = (bx+cy)d+x(ax)d−1+axyd−1+yxd−1 ∈C[x, y].
