exercices : Géométrie fondamentale

G H G G
G/H G H g ∈G
xH H g.(xH) = gxH
{g∈G|∀x∈G , g.(xH) = xH }
x∈G Stab(xH)
G
p n ∈N\{0}
pnp2
V n GL(V)
X V
W∈X
X n ∈N∗p
c= (1, . . . ,p)∈ Sp
Z/pZXp
∀k∈Z/pZ∀x= (x1, . . . ,xp)∈Xp, k.x = (xck(1), . . . ,xck(p)).
GΩ = {1, . . . ,n}N(g)
gPg∈GN(g) = |G|
G X 2∀(x,y)∈
X2∀(x0,y0)∈X2∃g∈G(g.x,g.y) = (x0,y0)
G2 Ω
i∈ΩGi=stab(i) Ω1= Ω \ {1}
G1Ω1
x∈G1N0(x) Ω1
xPx∈G1N0(x) = |G1|

Px∈G1N(x) = 2|G1|N(x)
Ωx
Pn
j=1 Px∈GjN(x) = 2n|G1|= 2|G|
x∈G x N(x)
Gj
X
g∈G
N(g)2=
n
X
j=1 X
x∈Gj
N(x) = 2|G|.
4
3 2
Z/pZX p
P(X)

X
X a,b X
a b
−→
ab
a b
ab {Bar{(a,t),(b,1−t)}|t∈[0,1]}
[ab]bar{(a,1),(b,1)}a,b,c
a0,b0c0bc,ac,ab
aa0,bb0,cc0
X{(a1,λ1), . . . ,(an,λn)}
X L :X−→ −→ X:x7→ Pn
i=1 λi−→
xai
Pn
i=1 λi= 0 L
Pn
i=1 λi6= 0 L
(ai,λi)L
Pn
i=1 λi= 0 {1, . . . ,}=I∪JPi∈Iλi6=
O G1=bar{(ai,λi)|i∈I}G2=Bar{(aj,λj)|jinJ}
G16=G2Aff{G1,G2}
(I,J)
X1,X2X
X1∩X26=∅dim(Aff(X1∪X2)) = dim(−→ X1+−→ X2)
X1∩X2=∅dim(Aff(X1∪X2)) = dim(−→ X1+−→ X2)+1
X Y
Y Y
3
a,b,c,d M1, . . . ,M4ab,bc,cd,da
M1, . . . ,M4
ac bd

A,B,C,D 4
3E,F,G
E AB, −−→ BF = 2/3−−→ BC G =Bar{(C,1),(D,3)}.
E, F, G
A, B, C, D
H,M,N
A,B,C,D
M(EG).
N(HF )H
(AD)
H(AD) (EG)
(HF )
X A,B,C
D(u,v,w),D(u0,v0,w0),D00(u00,v00,w00).
d=¯
¯
¯
¯
¯
¯
1 1 1
u v w
u0v0w0
¯
¯
¯
¯
¯
¯
∆ = ¯
¯
¯
¯
¯
¯
u v w
u0v0w0
u00 v00 w00
¯
¯
¯
¯
¯
¯
D∩D0d6= 0
P(vw0−wv0,wu0−uw0,uv0−vu0)
D D0d= 0 D6=D0−→ D=−→ D0
a(vw0−v0w,wu0−uw0,uv0−vu0)
D,D0,D00
∆ = 0
A1,A2,A3
P1=λ12A2+λ13A3∈(A2A3)P2=λ21A1+λ23A3∈(A1A3)P3=λ31A1+
λ32A2∈(A1A2)
(AiPi)
λ13λ32λ21 =λ12λ23λ31.
P1,P2,P3
λ13λ32λ21 =−λ12λ23λ31.

n= 2 {A1,A2,A3}Pi∈(AjAk)Mi
[AiPi]MiPi
f X −→ f=id.
f X
f f2=f
f f2=id.
X2{P,Q,R}X
A∈(QR)A0A(P R)−−−→ (P Q)
A00 A0(P Q)−−−→ (QR)B
A00 (QR)−−−→ (P R)s: (QR)−→
(QR) : A7→ B
β∈R\ {1}X u X
X
∀x∈X ,−→ u(−−−→ xu(x)) = β−−−→ xu(x)
u2= (1 + β)u−βId
u F (u)ker(−→ u−β−→ Id)
X u X X D
X M ∈D I M u(M)
I M D
X n u X
X F (u)u
F(u)X
n= 2 F(u)u
n= 3
dimF (u) = 1 u