La droite projective

P1(C)
P1(C)
C C2π:C2\ {0} →
P1(C)x6= 0 C2\ {0}
C2P1(C)π
S2⊂R33
R3
Cb C:= C∪ {∞} C
C
C∞C
C
X Y =Z=C
U=Y\ {0}V=Z\ {0}ϕ:U→V
ϕ(x) = 1/x X Y qZ
y∼z y ∈Y z ∈Z y =ϕ(z)
YqZ
X Y f :X→Y
A⊂Y f−1(A)X
Y X f
f:X→Y
Z g :Y→Z g ◦f:X→Z

Y
Z g :Y→Z g
g◦f
g◦f g W ⊂Z
V:= g−1(W)Y
f−1(V)f−1(V)=(g◦f)−1(W)g◦f
X
X X X
∞X
Xb X∞ ∞
Xb X∞b X
b
X
b
X
{Ui}b X
Ui0∞Ui0
K X Ui1∩K,...,Uin∩K
UikUik0≤k≤nb X
X, Y
f:X→Y f
x∈X{Ui}i∈I
x{f(Ui)}f(x)
f
x y f
f f
f U X x ∈U
UxU f(Ux)f(U) = ∪x∈Uf(Ux)
Xa →P1(C)b →S2σ →b C
E=R3S2
Pσ:S2\ {N} → P
N σ(M)=(NM)∩P
EC⊕R C (z, t)∈C⊕R
M∈E σ
σ σ−1
S2b C=P∪ {∞} σ σ(N) = ∞

z M M
Pm O M0=σ(M)
O M N P
[Om)λz λ ∈R+
ONM0
t
1=λ−1 λ
λ=1
1−tσ(z, t) = z 1−t
z0M0∈P
(OMN)M=σ−1(M0) (µz0, t)
µ∈R+
µ2|z0|2+t2= 1 M∈S2
µz0
1−t=z0σ(M) = M0
µ= 1 −t t = 1 −µ
µ2|z|2+µ2−2µ= 0 .
µ= 0 z0= 0 µ6= 0
µ=2
|z0|2+ 1 σ−1(z0) = 2z0 |z0|2+ 1,|z0|2−1 |z0|2+ 1.
z0µ= 0
v= (a, b)∈C2\ {0}
π(x)=(a:b)P1(C)a, b
λ∈C∗
a:X→P1(C)a|Yy∈Y(y: 1) a|Zz∈Z
(1 : z)b:P1(C)→S2
b(u:v) = 2uv
|u|2+|v|2,|u|2− |v|2 |u|2+|v|2∈S2⊂C⊕R'R3
a b
0∈C=Z⊂X B(0, ) = {z∈Z, |z|< } ⊂ X
 > 0a, b, σ
Xa →P1(C)b →S2σ →b C
0∈Z7→ (1 : 0) 7→ (z, t) = (0,1) 7→ ∞
∞
a, b, σ

aea:YqZ→
P1(C)a|Ya|ZX
a|Ya|Zλ∈C∗(λa :λb) = (a:b)
y=ϕ(z) = 1/z
a|Y(y) = (y: 1) = (1/z : 1) = (1 : z) = a|Z(z).
b b(u:v)S2
B(0, )⊂X
P1(C){(a:b), |a|>|b|}
S2{(z, t)∈C⊕R, t > 1−2
1+2}
b
C0∈C1/